Wang, Xiwen

2021 English Thesis

A Neighborhood of the Missing Knot

Advisor Cassandra Cleghorn

Additional Advisor

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Contains Copyrighted Material? No

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A NEIGHBORHOOD OF THE MISSING KNOT

by

XIWEN WANG

Cassandra Cleghorn, Advisor

A thesis submitted in partial fulfillment of the requirements for the Degree of Bachelor of Arts with Honors in English

WILLIAMS COLLEGE

Williamstown, Massachusetts

February 22, 2021

Contents

#1 Eversion Proposition ...... 1

#2 When Knots Becomes Personal ...... 13

#3 On Sleeping on It ...... 27

#4 Aloft the Confirmed Knot Tier ...... 39

#5 A Neighborhood of the Missing Knot ...... 55

Acknowledgements ...... 69

Bibliography ...... 70

knot n. complex 46.2 v. join 47.5 tangle, tangled skein, mess, snarl; couple, pair, accouple, copulate, knot, Gordian knot; maze, meander, conjugate, marry, link, yoke, knot, Chinese puzzle, labyrinth; Rube splice, tie, chain, bracket; put Goldberg contraption, wheels within together, fix together, lay together, wheels; rat’s nest, can of worms, piece together, clap together, tack snake pit. together, stick together, lump connection 47.3 together, roll into one. attachment; binding, bonding, gluing, sticking, tieing, lashing, trussing, girding, hooking, clasping, zipping, buckling, buttoning. dilemma 731.6 paradox, oxymoron; asses’ bridge, pons asinorum. puzzle 549.8 knotty point, crux, point to be solved; puzzler, poser, brain twister or teaser, sticker; mind-boggler, floorer or stumper; nut to crack, hard or tough nut to crack; tough proposition, “a perfect nonplus and baffle to all human understanding” [Southey].

—Roget’s International Thesaurus, fourth edition

#1

Eversion Proposition

If you want to study Zen, you should forget all your previous ideas

and just practice zazen and see what kind of experience you have

in your practice. That is naturalness. Sometimes we say nyu nan

shin, “soft or flexible mind.” As long as you have some fixed idea

or are caught by some habitual way of doing things, you cannot

appreciate things in their true sense.

—Shunryu Suzuki, Zen Mind, Beginner’s Mind

On Tuesday nights, I attend a knot theory research group. My Knot Friend and I bike to the math building after dinner, plonk our bags on the floor of the common room, and join the group huddled before the board. It’s only the third time we’ve met, but already I’m not sure what

I’m doing there. I’m staring blankly at the flurry of chalk when Professor Adams asks me, “Do you buy this?” I open my mouth, seconds of silence elapse. He smiles and says, “You haven’t decided yet.”

We’re supposed to be theorizing virtual triple crossing knots, a new type of knot which grew out of an office hour conversation between Professor Adams and his thesis student Jonah.

Questions about its definition and properties were deemed sufficiently interesting to warrant exploration in a paper. The next week in Intro to Knot Theory, Professor Adams passed around a sign-up sheet. I was curious what a mathematician’s work looked like, how math is made. I would lurk, for journalistic purposes, I convinced myself. At our first meeting, Professor Adams

1 said that there were no stupid questions, that he knew about virtual triple crossing knots as much as we did. Our help was enlisted. Passive observation, it seemed, was not an option.

But so far I’ve made no contribution. I’m still tripping over the very idea of virtual knots, the imaginary numbers of knot theory. How to wrap one’s head around it?

Bear with me: take a rope, put a knot in it, and glue the two ends together. There, we have one of the simplest knots, the trefoil. That’s knot theory 101.

At each point where the rope intersects in the diagram, one strand must go over, and another strand must go under. These intersections are called classical crossings. A virtual crossing, however, looks like this:

We can never realize a virtual knot out of a rope in the real world, but mathematicians have knitted an entire field out of these virtual knots. The strands in a virtual crossing are neither under nor over. But what on earth does this mean?

Knots continue to flash in the back of my eyes as I hurry across campus to the Zendo for my meditation session. Students and townies on zafus and wooden seiza benches already line the small, dim room. Our teacher, Bernie kneels in the center, his eyes closed. I tuck myself into a corner, and notice the resemblance he bears in his thin, shaved-headed equanimity to the buddha in a drawing on the wall, loosely robed, meditating with eyes closed, the back of one hand resting in the palm of the other, thumbs lightly pointed together before a protruding belly.

2 As usual, the session begins with a guided meditation:

To settle into your posture, take a few deep breaths. Let your shoulders rise and fall. As you inhale, imagine a string running along your spine, extending out from the top of your head, pulling you upward; as you exhale, imagine your spine as a coat rack, and let your whole body just hang.

Here we go. Do not fall asleep again today. Let the sludge settle . . .

In the next minute or so, I’d like you to open yourself up to all the sounds in the environment. Just hear, let all the sounds pass through your body.

Well, I hear laughter from the interfaith common room. A guy is speaking and girls are laughing. Whirl of the ventilation system. Breathing of the lady to my right. Breathing. A rustle of clothes. A swallow. What else . . . a cough. More laughter. Bernie’s deep voice—I really need to start my essay later tonight. Why am I doing this at this hour?

When you notice that you are carried away by a thought, just label it with the phrase

“having a thought” and repeat, as verbatim as you can, the thought you just had. “Having a thought: Meditation is hard.” “Having a thought: I can’t do this.” Then, gently return to the open listening, to the sounds of the environment.

Having a thought: planning, planning. A deep breath. Having a thought: what was going on there with that isotopy problem? Having a thought: this is not the time to think about knots.

Another: at least wait until you have pen and paper. And: mechanical revving in the room . . .

So now I’d like you to bring your awareness to your breath. Feel the passage of the air over the soft tissue inside your nose. See how granular and moment-by-moment you can make your sensation. Sense the difference of the beginning, the middle, and the end of each inhalation and exhalation.

3 Okay . . . Breathing in, a cool, minty sensation. Breathing out, warmth. Breathing in, a prick, discomfort, stretchiness. Breathing out, not much to note. How much more granular can I go?

Here’s a question: why are these two nubbins isotopic?

When Bernie rings the bell at the end of the meditation, the string running up my spine has long tied itself into a knot. I thought I had an inkling of a solution to the isotopy problem— thinking of the two nubbins as extremely malleable rubber, a way to deform one into the other without cutting and gluing. Maybe I can just pinch in here, bring that part over . . . But then my mind’s eye isn’t convinced this is legal . . .

In the infinite catch and throw of meditation, that night I failed.

!

The recent Tuesday night series marks my third attempt at getting into meditation, and I must admit, what I’ve most enjoyed this time round is the logic behind it, though I hesitate to use this word. Meditation is about seeing what I otherwise cannot see. Let me explain: it took me a while to make peace with the fact that my mind wanders during sitting. But that’s what the mind does, and in fact, it is never the point of meditation to suppress thinking. Instead, meditation is largely about entering a state where we can best observe not just the thoughts themselves, but their patterns. When I attune my attention to sounds in the environment and sensations in my body, I invite the agitated glass of river water that is my mind to settle. When thoughts float up and swim into my consciousness, I try to label each with curiosity. I say, having a thought. This

4 tag is a gentle reminder that whatever the thought is, however gripping and compelling it may seem, in the end it is just a thought. “Having a thought: I suck at math.” This micro-exercise returns our thought-beliefs to their proper status. It is not about disbelieving; it is about awareness of the possibility that “I suck at math” might not be true.

Thought labeling aims to dissociate thoughts from beliefs. The hope is that, if we can just notice such self-beliefs for what they are, instead of believing them as truth, we can start to imagine a roomier relationship between our judgements and ourselves. Eventually, I might see that I am not the whirlpool in the glass; I am the water where grit and dirt and sand act out their little drama. I am the empty sky where clouds drift through. And Annie Dillard’s formulation: “I am the skin of water the wind plays over.”

It seems that I resort to the word “see” a lot when I describe meditation, when in fact it is hardly visual as a practice. We close our eyes and privilege the other senses: the chill of a breeze, the sting near a knee cap, the tingle in the nostrils, the buzz of a bug. When I saw the pair of isotopic rubber nubbins, unlike the other physical sensations that ground me in the present, the brain image only served to transport me away, give me up to the self-feeding torrent of the mind.

This rambling sort of “lost in thought,” of course, is not very meditative. Shockingly, the attachment to seeing may not even prove constructive to the study of mathematical topology.

Consider Bernard Morin. This French mathematician first found a way to turn a sphere inside out without cutting or tearing, and he was blind since age six. I should be careful and add that we are allowing self-intersection of the surface. Don’t worry, this doesn’t make it any easier.

In fact, the sphere eversion problem is a veridical paradox: something that runs against every gut feeling, and yet, is true. It took mathematicians seventy years after proving the existence of such an eversion to figure out how to do it.

5 There is a twenty-minute video on the Internet animating the procedure; I couldn’t get step 1. Please allow me to give you a taste with a simpler problem:

Claim. These two loops living on the surface of a donut are actually the same.

Proof. Buy a sock. Make a hollow donut by cutting off the toe part and

stitching it end to end. Mark the loop on the sock donut with a sharpie. Cut a hole in

the sock, avoiding the loop. Now pull the inside of the sock out through the hole!

Q.E.D.

I suppose it shouldn’t be surprising that Morin’s work was guided by tactility. When asked how he had derived a sign which took his colleague pages of computation, he responded,

“By feeling the weight of the thing, by pondering it.” It turns out that most blind mathematicians work in topology, the most visual branch of mathematics. Their blindness seems only to enhance their mathematical intuition. Morin believed that, unlike the sighted, whose spatial imagination is mostly based on the brain’s analysis of two-dimensional projections onto the retina of a three- dimensional world, “our spatial imagination is framed by manipulating objects.” As mathematician Alexei Sossinski notes, it is the sighted who tend to have misconceptions about three-dimensional space, and who, for example, see only the outside of geometric objects, and not the outside and the inside together. The usual way of seeing consists only ever of disjoint pictures.

Dillard tried to see like the newly sighted, after reading their accounts in Marius von

Senden’s Space and Light. She wanted to see the world in color-patches, without depth. “I couldn’t sustain the illusion of flatness. I’ve been around for too long. . . . I couldn’t unpeach the

6 peaches,” she reports. My challenge with knots goes the other way: once I see a flat knot diagram, I can’t sustain the illusion of three-dimensionality. The fact is that, as long as I don’t snip the rope and re-tie it, a knot is the same no matter how messy I manipulate it to seem:

These are called different confirmations of the same knot. If we were together, I would slip a hair tie off my wrist and show you how the twist of the second picture springs back into the unknot, or the trivial knot, of the first picture. It turns out that, by moving and pulling strands around, in a sequence of 18 basic operations known as Reidemeister moves, the tangle of the last picture can be similarly resolved. It takes mental effort to come up with the intervening pictures:

I can make it uglier, but even when I’m done toying with the hair tie, there’s still the question of the angle from which I view it. A confirmation viewed from the top sure gives a different picture than from the side. If we think of a knot as living inside a sphere, then placing our eyes at every point on the sphere would give a different picture, or projection. What happens then, when a single knot can have infinitely many confirmations, each with infinitely many projections—what is the right way of seeing?

7

The good news, thanks to Hass and Lagarias (2001), is that given any projection of the unknot, we can arrive at its trivial, hair tie projection in a finite number of Reidemeister moves.

This means that no matter how I view a knot, there is a limit to the number of moves I need to try before declaring that the knot is not trivial, that it is in fact knotted. The bad news is that, given a projection with � crossings, the number is 2100,000,000�.

When I look at a crossing, my eyes compel an assignment of over-and-under statuses to the strands. It seems like there can only be two options. A neither-under-nor-over crossing bends my mind, because my eyes don’t buy it. The virtual crossing comes from a different viewing system, complete with its own axioms. With my lay eyes I could never have come up with the proof technique of projecting a knot made of sticks from its vertex onto a hemisphere; my eyes project sticks onto a plane.

That night, when meditation was haunted by knots, this is what came to me: both meditation and knot theory are asking me to slip out of my actually very limited way of seeing for a moment, to suspend its authority as the only way, the truthful way. Just as Morin turned the sphere inside out, meditation gives light to my inner ego structure, potentially overturning what I used to believe as my central truth.

!

8 I have to be careful here: math has a reputation as the ambassador of a brand of machismo. Mind conquering the world! I don’t wish to convey anything like that about meditation. Elizabeth Gilbert was my first meditation teacher. My mother bought her book for me at an airport bookstore thinking it was a children’s book. Eat Pray Love is unfortunately translated into Chinese as Be A Girl for Your Entire Life. I was grabbed by its mention of a sketch, drawn by Gilbert’s Indonesian medicine man friend/mentor Ketut Liyer. I studied the glimpses of it shown in the movie:

It was an androgynous human figure, standing up, hands clasped in prayer. But

this figure had four legs, and no head. Where the head should have been, there was only a

wild foliage of ferns and flowers. There was a small, smiling face drawn over the heart.

“To find the balance you want,” Ketut spoke through his translator, “this is what

you must become. You must keep your feet grounded so firmly on the earth that it’s like

you have four legs, instead of two. That way, you can stay in the world. But you must

stop looking at the world through your head. You must look through your heart, instead.

That way, you will know God.”

Meditation is bent on the present, a more or less ungraspable dose of time made up of no more than myriad physical and environmental indices. This is why Bernie insists on dual awareness practice—awareness of breath and of sounds in the environment. They are the four legs that ground us in the present instead of the usual muddy flow of mind.

Besides the return to sensations, meditation loosens the mind’s grip in yet another sense: you can’t really will your way into it. You don’t want to be telling yourself to be better at meditation! focus more! try harder!, Bernie says, because that would be the exact opposite of the right path, which is letting be. Meditation is not about coercing the world like a piece of

9 malleable rubber. Newly ejected from the regimentation of high school into Bernie’s English class, I began to suspect that “happiness” is more a function of our capacity for happiness and less of outward circumstances. A concurrent strain of my teenage philosophy was a relentless squashing of expectations. The first meditation I ever did was a five-minute YouTube video sent me by my friend Wendy the night before a college interview. Positive affirmations narrated in a deep British voice was set against spa music. Much cathartic tears were sobbed on the floor of my room. Why would the “Honest” Guys say they “believed in” me when they obviously don’t know me? Why do they intentionally want to make me feel bad? . . . In short, my two tenets combined rendered as myths the virtue of striving for self-improvement and the general upward trajectory of life. I wanted to be a person who’s capable of happiness. “I wanted to find a sustainable way to live” (Norman Fischer).

Such handwavy ideas explain part of my attraction to Buddhism. The other reason is simply that Bernie has a most calming, sincere yet hilarious energy, and I aspire via Buddhism to copy his outlook on life. It was a moment of horror when I realized that my worldview had been modeled on how-to articles. The obsession began with such titles as The Great Big Glorious

Book for Girls and The Girls’ Book: How to Be the Best at Everything. When I had thrice read about tracking animal footprints after a fresh snow from my room in a city with neither snow nor wild animals, I turned to wikiHow. I printed out entire webpages and went at them with a highlighter. When DIY tutorials bottomed out, I arrived at the self-help genre with “How to Be a

Kind Person” and “How to Truly Be There for Your Friends.” I was the kid who enjoyed following Lego instruction booklets. I consume life advice.

It took multiple attempts across months before sitting wasn’t completely boring and full of pain. I kept reaching for it, though, and in January of 2018, convinced I was too goal-oriented,

10 I started showing up at Bernie’s evening meditation. Things had become truly grave, I thought, as I was taking breaks for a utilitarian reason: so that I can work better later. I recall a particular point of confusion when Bernie told us during a guided meditation that we were perfect right at this moment, just as we were. No need for change. But there’s obviously so much room for improvement, I thought. All the tasks that need to get done aside, I could and should try to become a much better person in so many ways . . . I wanted to cry, for I felt like a fake. That was my beginner’s mind.

Of course, my turn to meditation is oxymoronic: I wished that a practice which purports

“just letting be” to change and fix me, so that I can “just let be.” For a while I persistently antagonized and purged goal-oriented thoughts that came up during meditation. Having a thought: I should be studying rather than sitting right now. Having a thought: that right there is exactly the kind of thinking that makes me miserable! That is what gets me into trouble in the first place. I desperately wanted it to leave my thought pattern, but of course this was only another symptom of needing to get somewhere, as if that’s the only way to squeeze meaning out of living.

But then again, the proclamation that “Goal-drivenness is the source of my problems!” is, as always, only a thought. If thoughts are knotty, the eyes need to be lifted to the level of the sphere surrounding the knot. Nothing macho here, only softness.

!

If only I could will myself to believe that one of these two surfaces can be rubber deformed into the other:

11

The isotopy was not obvious to me. Knot Friend and I debated this problem in the department common room. We spoke of worms drilling their way through an apple. Stabbing the board with chalk nubbins, we might have scared a few students working there. My brain felt like a pretzel, I wanted to drop the class. These days I can’t remember what we scribbled on the board; I see full well how to slide, in the second picture, the openings of the leftmost tube towards and eventually onto the middle tube, at which point I can undo the knot to obtain the first picture, no scratchwork needed.

I took Italian junior year, thanks to Gilbert’s memoir. After her, I learned to converse with myself on notebook pages. I was a little sad when I grew up to hear Gilbert mention in an interview that she didn’t meditate daily anymore, that she hadn’t meditated in a long while. The world was also a bit shook when she shared that she had separated from “Felipe” of Eat Pray

Love upon realizing the depth of her love for her best friend, who was ill. She later wrote a novel titled City of Girls, which I inhaled last summer.

12 #2

When Knots Becomes Personal

The wise have nothing to do,

While the unwise tie themselves in knots.

—Seng-ts’an, “Relying on Mind”

A black, wooden knot sits in our living room, at the foot of a giant copper-colored fluted vase. About the size of a dinner plate, it has three protruding ears, which connect into gnarlier tangles toward the center. The lump of wood has no uniform diameter; its surface is not of constant curvature. One can imagine where chunks were shaved away, yet the whole thing has no sharp points or edges. The black lacquer gives it the sheen of playdough, as if one could put a thumbprint on it. My mother picked it up at a homeware store. It is perhaps the most baffling decor item we own, neither memorabilium nor art, zero function.

I’d considered the sculpture quite an eyesore before I was a student of knots. But when my line of vision landed on it again two summers ago, I couldn’t help but see that it had a triple crossing, where three strands crossed at one point, in addition to five usual, or double crossings.

A triple crossing dissolves into three double crossings, if we give it a wiggle. What I had there, in my living room, was a projection of a knot with eight double crossings, I started to wonder if this was indeed a knot with crossing number 8 (i.e., a knot with a minimum of 8 crossings in all its projections), and if so, which 8-crossing knot it was. Who is this knot? Where is it sitting in the Rolfsen knot table?

13

That was the day when, tracing it with a piece of shoestring, copying its shape into my notebook, the mystery knot became somehow personal.

!

I met my first Buddhist knots in The Roaring Stream anthology. Beneath each chapter title was a pleasingly symmetric, squarish emblem, an alternating link of 12 crossings known as

L12a1908. A link, by the way, is just any number of knotted loops tangled together. L12a1908,

2 for example, is said to have three components. Copies of its diminutive cousin, the 41 link, were jauntily strewn over the pages as sectional end marks.

A trefoil knot may also serve as a halmos, the symbol declaring the end of a proof, named after mathematician Paul Halmos who first popularized its use. Some like to write Q.E.D., short for quod erat demonstrandum, “that which was to be demonstrated.” Others prefer to draw a hollow or filled rectangle or square, called a “tombstone,” likely for its shape, but perhaps also to signify finality. They are the exclamation marks of mathematical writing. Triumphant feelings arise in me when I see little knots embedded in writing.

2 I had to know more about what L12a1908 and 41 were doing here in the Zen reader, why they were chosen. I found out that within the same family as these two is the glorious śrīvatsa:

14

Also known as the endless, eternal, or mystic knot, the śrīvatsa is one among the eight auspicious symbols in Buddhist traditions. In the Tibetan branch, the endless knot adorns monasteries and private homes. Its symbolic meanings are profuse. Some interpretations appeal to its lack of a beginning or an end, and understand it to signify the endless cycle of saṃsāra, or alternatively, the infinite wisdom of the Buddha. Some others draw on its conjoining property, and explain the symbol as the intertwining of emptiness and dependent arising, of wisdom and compassion. Still another sums it up as representing “the ultimate unity of everything.”

Mathematically, its name is the 74. Professor Adams has expressed his fondness for this knot, citing its many symmetries, vertical, horizontal, and central. I think it is the prime contender should he ever get a knot tattoo. Drawing the 74 was an extra credit problem on his midterm for Knot Theory.

I’d wager that another knot with a special place in Professor Adams’s heart is the 52. The following story I’ve heard on no fewer than half a dozen occasions. He was a starving graduate student when his professor offered five dollars as reward for the first person to find the hyperbolic volume of the 52 knot. At that time, only a single knot, the figure-eight (41), had a known volume, and 52 was next in the table. Professor Adams worked on it for a month, before a postdoc solved it. He was crushed, but when he heard the solution presented, he realized what he had been doing wrong, and after the fix his method could be used to find the volume of every single knot ever. “So he got the five dollars, but I got a PhD, so I thought that was a pretty good deal,” he ends the story every time, his eyes shining.

!"

15 Despite the visual felicity knots lend to Buddhism, they are seen as problematic in the

Buddhist theory of interpersonal relationships. As Zen master Thich Nhat Hanh explicates in

Peace Is Every Step:

There is a term in Buddhist psychology that can be translated as “internal formations,”

“fetters,” or “knots.” When we have a sensory input, depending on how we receive it, a

knot may be tied in us. When someone speaks unkindly to us, if we understand the reason

and do not take his or her words to heart, we will not feel irritated at all, and no knot will

be tied. But if we do not understand why we were spoken to that way and we become

irritated, a knot will be tied in us. The absence of clear understanding is the basis for

every knot.

As prescribed by Buddhism’s Four Noble Truths, attachment is the cause of suffering. Our internal knots, Thich tells us, shackle us to saṃsāra, tie us up, and constrict our freedom. The word in Sanskrit is saṃyojana, which literally means “to crystallize.” To internally knot, then, is to solidify, to separate from the solution, to form crystals—the opposite of non-form.

This perhaps helps to explain why the physical discomfort we experience during meditation is often described as “muscle knots” or “tightening.” Perhaps this is why the Buddha himself compares a being trapped in saṃsāra to “a tangled skein, a knotted ball of thread, a weave of grass and rushes.” Perhaps this is why sometimes, at the beginning of meditations, we are instructed to imagine that a string, rising through the spine out of the top of the head, is given a gentle tug.

Unsuspectingly, the mathematical counterparts of Thich Nhat Hanh’s knots live in the universe of a pit-less peach: the three-dimensional space �2 × �1, also known as a circle (�1)’s worth of spheres (�2). To make this universe, start with a solid peach, but throw out the pit.

16 Now—this gets a bit funny, but stay with me—glue the inner sphere to the outer sphere, point-to- point. What does the result look like? I’m not sure, since it doesn’t exist in our ordinary three- space. I can only ask that you do the gluing abstractly in your imagination.

Consider now this family of knots in the universe of the glued pit-less peach: a string rises out of the north pole of the inner sphere, knots around itself a couple times, and finishes at the north pole of the outer sphere. Note that, by the rules of this universe that we defined earlier, the north poles of the two spheres are in fact a single point. The effect, then, is not unlike in

Doodle Jump, where as you leap out of the right edge, you land back from the left edge. And so the string knots itself, and loops back to where it starts.

Looks hairy enough, right? Can we untangle this knot, without cutting the string? A minute ago, we didn’t even know this peach universe existed, and now there’s a knot in it? Well, if we sit with it for a while, we might start toying with the string and eventually come to this solution:

17 The Light Bulb Theorem. This family of knots, no matter how messy the

tangle in the string looks, can always be untied.

Proof. All we need is a shift in perspective: instead of wielding the knotted

string itself, imagine the inner sphere as a light bulb, hanging down from the ceiling of

the outer sphere with a bunch of knots in the cord. Then, we can always push the inner

sphere back along the cord! The bulb just needs to follow the path before it. Q.E.D.

David Gabai proved the Light Bulb Theorem in four-dimensional space in 2017; it’s

mind-boggling stuff. Q.E.D.

W ithout the right way of seeing, we are quite entangled. What’s worse, Thich tells us, “if we do not untie our knots when they form, they will grow tighter and stronger.” This is not a wholly unfamiliar idea. Think of your devious earphones. In your exasperation you decide for the last time to untangle them, coil them up nice and even, and tuck them into their nook in the drawer. A week later you’re cursing again. In 2007, Raymer and Smith studied the factors governing the spontaneous knotting of an agitated string. They found that the knotting of earphones, left to its own device in the corner of the drawer, is attributable to the unintended knocking about from day-to-day use of the drawer. Science has it that there is no escape from our knotted earphones, try as we may “to repress them, to push them into remote areas of our consciousness in order to forget them” (Thich). We can neatly box up the mess, ship it away

18 from our field of vision with every intention not to stir it up again, yet the disorder stirs in the back corner of our head. For, long as we live, how can we be free of life’s vibrations?

!

Back home, three hours weren’t enough to ID the black, wooden mystery knot. Running with my suspicion that it was an 8-crossing prime knot (think prime numbers), I pulled out the knot table and cross-examined the twenty-one suspects. My bare eyes failed me; I couldn’t see a match. The inequalities weren’t looking promising, either. My last resort was to use brute force, calculating the Jones polynomial of the knot. The process involves, in broad strokes, repeatedly invoking two equations known as skein relations until the mystery knot is dissolved into the unknot, and expanding layers of brackets.

This is time-consuming and ridiculously rife with opportunities for arithmetic mistakes for an amateur like me. But should I correctly calculate it, then it’d really only be a matter of matching the formula to the information on the table. I sent along a picture of my project to my Knot

Friend, then spent fifteen minutes a day plugging in skein relations and unfolding brackets. I wanted to know the name of this knot, how to address it, who it was. When eventually I was done—was it eight days or sixteen?—the result didn’t match any existing knot in the table. I had made a mistake somewhere. At my wit’s end, I pushed the pile of scratch paper into the bottom of a drawer to tangle themselves.

For a year I was in the dark. The next summer, I heard about a computer program called

SnapPy, with which one could draw a knot with the cursor as input, and out would pop its

19 hyperbolic volume. I flipped through my journal to find the mystery knot. In seconds I came to know it as the 820, and the mystery was solved. But there was no one to share the news with, as my Knot Friend was not my friend no more.

!

It was Knot Friend, avid knot correspondent, who first told me of R. D. Laing’s dabble in poetry. In the front matter of Knots (1970), Laing tells us of his subject: “The patterns delineated here have not yet been classified by a Linnaeus of human bondage. Words that come to mind to name them are: knots, tangles, fankles, impasses, disjunctions, whirligigs, binds.” The pages that follow present us with what might be called distillations: of dialogues between parents and children, between lovers, as well as of monologues from patients to analysts. Knots was popular enough to engender a play and then a movie adaptation by Edward Petherbridge, which scholars tend to neglect, along with Life Before Death (1978), Laing’s twelve-track LP of spoken word poetry. In his Wikipedia page photo, Laing was reading The Ashley Book of Knots, bible of practical knots. “This picture was taken in Stockholm on board the Norwegian author Axel

Jensen’s ship Shanti Devi in 1983,” the caption reads. “The Ashley Book of Knots was given wittily to Laing by Jensen as a gift.”

Highly formal in style, his poems invoke if-clauses, discuss by cases, and prove by contradiction. Recursions, negations, and parentheses abound:

JILL I’m upset you are upset

JACK I’m not upset

JILL I’m upset that you’re not upset that I’m upset you’re upset

JACK I’m upset that you’re upset that I’m not upset that you’re upset that I’m

upset, when I’m not.

20 Laing explains his method: “I could have distilled them further towards an abstract logico- mathematical calculus. I hope they are not so schematized that one may not refer back to the very specific experiences from which they derive; yet that they are sufficiently independent of

‘content’, for one to divine the formal elegance in these webs of maya.” I looked up the word.

Māyā: Sanskrit for “illusion,” also the name of Gautama Buddha’s mother.

Laing was right on: mathematics really may bind one in a crazy deadlock. As a logical system, the mathematical mode asserts, necessitates, and demands. For example, I collapsed at the last problem on our first knot theory homework. The problem asked for a four-component

Brunnian link, namely, four unknots non-trivially linked together, such that snipping any one of the four unknots leaves us with three unlinked unknots. The Borromean rings is the most famous three-component Brunnian link, so named because it dresses the coat of arms of the Borromeo family in northern Italy. Imagine erasing any of the three rings; the remaining two would no longer be linked together. Today, knot pilgrims can find the emblem on Isola Bella, dotting the palazzo Vitaliano Borromeo had built in the seventeenth century. Beware, though, as the builders must have messed up quite a few, and flower pots would feature counterfeit rings, which wouldn’t free upon the removal of any one ring:

The task at hand: to generalize the Borromean rings, to preserve its Brunnian property while slipping in one more ring. Cue one more actor in the drama of knots in 70s psychoanalysis.

Jacques Lacan had long had topology on his mind. He’d been thinking about the torus in the 50s.

A decade later, it was the Möbius strip. Take a strip of paper, put a half twist in it, and glue the

21 ends together. We now have what is called a non-orientable surface, the paper strip’s front side running into its backside. It is indeed disorienting to live in a Möbius strip universe. Print a stick figure on a Möbius strip, with his heart on the left side of his body, so that the ink bleeds through to the other side. Now slide the stick figure once around the Möbius strip. The figure now finds his heart on his right side. Chillingly, there is no exact moment at which the flip of the heart to the other side of the body happens. In the folded space, the subject runs into its own reverse,

Lacan says, and it is no longer clear which of them came first.

By the 70s, topology wasn’t simply a metaphor for Lacan’s theory, but its flesh and bone.

He turned to knots. If he had a personal knot it’d be the Borromean rings, which he saw as the triad that makes up the subject—the symbolic, the real, and the imaginary. In his 1975–76 seminar, he pegged psychosis as the unravelling of the Borromean rings. Moreover, Lacan had it figured out: he slipped in a fourth ring, the sinthome, or the symptom. “Inscribed in a writing process,” the sinthome marries topology with James Joyce, Lacan says.

Knot Friend and I had less luck. It was the first and last private problem Professor Adams assigned, which meant we couldn’t collaborate on it. I buried my face in my notepad of worthless pencil scribbles, but I knew the sky had darkened outside the library study room window. I also didn’t need to look to know Knot Friend’s frustration at my frustration, yet I was

22 convinced I must not relent. Emotion follows its own calculus. I was upset, and upset that he was upset that I was upset. It was his move.

Consequently, we gave Laing a prompt: Halfway into the 24-hour window, Jack took out a pair of scissors from his drawer and cut up his knot exam. “And since Jill is afraid / that Jack will think that / Jill is afraid / Jill pretends that / Jill is not afraid of Jack.” Can Jack and Jill peel back the layers and become, eventually, less terrified of terror?

!

On my screen was the knot volume calculator which realizes the result of Professor

Adams’ dissertation—on the left, the 820 drawn in hasty red line segments, and on the right, the non-terminating, non-repeating decimal 4.12490325 . . . that is its volume—and I’m at a loss what to make of it all. Would life be worse off if the mystery knot always remained in the box I sent it to, in the deep of my mind? If all I did was plug and chug, can I really be said to have known 820? Was the brute force unfolding of brackets paying the problem the right kind of attention, or was it a waste of time?

In the last chapter of Knots, Laing turns to deal with explicitly Zen notions such as no- self, nirvana, and dharmas. His final koan riffs off of classic metaphors for the concept of

Perfection of Wisdom, where Buddhist thought takes a meta turn. The idea is that the teaching is only the finger pointing to the moon, and we shouldn’t mistake it for the moon itself. Thus, even the Buddhist worldview is not to be attached to, turned into yet another form of bondage.

By those who know the discourse on dharmas

as like unto a raft

dharmas should be forsaken, still more so

no-dharmas

23

Put the expression

a finger points to the moon, in brackets

(a finger points to the moon)

Put all possible expressions in brackets

Put all possible forms in brackets

and put the brackets in brackets

Not,

as finger to moon

so form to formless

but,

as finger is to moon

so

all possible expressions, forms, propositions, including this one, made or yet to be made, together with the brackets

to

But the bracket was starting to feel overstuffed, with layers and layers of littler brackets of littler knots. I went and culled from a box under my bed the letter from Knot Friend. There are many reasons I don’t love to look at it: I don’t love how, coming home from snow at four in the morning to these pages slipped under my door, I wanted to read them. I don’t love that you woke from my shuffling and came next door to speak to me. I don’t love the fact that we were still

24 neighbors then. I don’t love how your very first paragraph ended by saying, “we were soon knotted together, you and I.” I felt offended when you started drafting a list of special knots. I managed to still feel indignant about how you twisted events, having known you who always lied so much you needed a journal to keep track of them. I was upset with how some of your nonsense rings of a truth that is too true, and how I felt disgusted but had to devour it standing there and then.

As though dressed in a Brunnian fabric of little rings, when one ring snagged and broke, the rest of it all came unraveling off and I was naked. As though dressed in webs of māyā, illusory clothing.

Your therapist thought it might be a good idea to write me a letter, you said. You sent it to New York Times’ Modern Love. You submitted this as part of your application for summer knot research with Professor Adams. But above all I mustn’t be furious, I determined. You wanted me to lash out at you, which was exactly why I wouldn’t.

I tried to get you on the Zen wagon, and you said you didn’t think you were that kind of person.

Tell me: what am I supposed to do with this, the note where you copied “Nothing Gold

Can Stay” and appended, “Call me. I’m on a hike.” The note you stuck on top of a box, in which you placed my phone, where you set an alarm for 2:35 a.m.—a number of meaning for you, apparently, as you used to get out of practice at 2:35. The box was near your bed, where I’d fallen into a nap after the dinner you made me. I’d gotten back kind of late, for you ended things that day and I’d been out drinking. Next to the box was a fruit knife. The plan was that, you’d go on a hike to the Hoosic river, and at 2:35 I’d wake, read the note, cut the tape, call you, and you were going to tell me that you were walking into the river. But you never got the call; I only

25 woke when you threw open the door demanding to know how the alarm didn’t go off—what am

I supposed to with that note? Should I let it go?

You forgot to mention: the 52 knot holds commemorative value for you and me, too. It took mad combined effort to determine which of the two 5-crossing knots a specimen was. After that you bought us personal chalkboards, and we started swiping Hagoromo chalk from classrooms and carrying them in gum boxes, so that we could knot on the go.

!

Louise Bourgeois made a diptych titled The Endless Loop. The left fabric sheet features three figures in red, an impression from her drawing “The Family I.” The right sheet has mounted on a collage of fabric: a length of white, knotted loop, with two unknots, one black and one red, hooked on. “How often, in my private mind, have I choreographed ribbons of black and red in water, two serious ropes of heart and mind. The ink and the blood in the turquoise water: these are the colors inside the fucking” (Maggie Nelson).

26 #3

On Sleeping on It

Satori can thus be had only through our once personally

experiencing it. Its semblance or analogy in a more or less feeble

and fragmentary way is gained when a difficult mathematical

problem is solved, or when a great discovery is made, or when a

sudden means of escape is realized in the midst of most desperate

complications; in short, when one exclaims “Eureka! Eureka!”

—D. T. Suzuki, “Satori, or Enlightenment”

In school, I always wished that I could tuck a book under my pillow at night, and set my mind to memorize its contents while I slept. Seems too good to be true, but now I know it happens: I have gone to sleep stuck on a math problem, and woken up with its solution. In fact it has happened three times. The first two instances corresponded to two 24-hour take-home exams. The way I like to do those is to schedule sleep for after 8 straight hours of attempts; when

I wake, it’s time for the final sprint. The reward is fantastic if I do get a question after another hour of work in the morning. I commend my rested brain for the expedient collaboration—it really is a different beast. I take an extra moment to appreciate our work before handing it in.

Sometimes it’s a solid proof, elaborate twists and turns complete with majestic inevitability.

Other times the sheer strangeness of our morning mind’s creation is the source of wonder:

27 “Consider a prism of the curly Greek cross. Let a prism of a scalene triangle jut out the front and scoop a corresponding dent in the back. Have three cylinders coming out the three right sides and three corresponding indentations on the left . . .” Thus begins the construction of a dimorphic prototile, with copies of which we can tile space in two and only two ways.

Recently I experienced a more providential version of waking up with math insight.

Something not-so-obvious presented itself as I was trying to prove the “obvious primeness” of a class of links, called generalized augmented fully alternating links—don’t worry about them. A novice research student, I resorted to bed after an evening in vain. When I woke, floating there amongst the earliest thoughts was a phrase: just make it bigger. The incantation took care of all my problems for the day.

Now, mine was a minimal discovery. If I’d asked for help, fellow researchers would probably have seen the way out on the spot. And as Professor Adams explains in the endnotes to his book, Riot at the Calc Exam and Other Mathematically Bent Stories, “That’s what doing research in mathematics is all about—reaching a level where you get stuck. Then you begin puzzling over it and struggling with it, sometimes for months or years at a time until you eventually figure it out or give up. Often, students think that if they can’t get a problem in an hour, they are stupid. An hour is nothing.” Alas, for me the early morning insight remains a tiny victory, less because I can prove that G.A.F.A. links are hyperbolic, and more because now I know what Professor Adams means when he regularly reports, “So I’ve been up since 2 last night because I had this idea . . .”

Isaac Newton might have been slumbering under the apple tree. Albert Einstein treasured his ten hours’ sleep every night, plus naps. Thomas Edison liked to catnap with ball bearings in his hands and a pie tin at his feet, so that just as he drifted into the unconscious, he would drop

28 the balls and wake from the clatter. And there’s John Steinbeck, who said, “It is a common experience that a problem difficult at night is resolved in the morning after the committee of sleep has worked on it.” In a 2019 study, Sanders et al. invited 57 participants to evening puzzle solving sessions. Participants cracked away on a set of puzzles, while researchers played fifteen second sound clips on loop, an arbitrary and distinct tune for each puzzle. That night, as a participant entered the deep sleep stage, researchers replayed the music to half of her unsolved puzzles. The next morning, participants solved 55% more of the cued puzzles than the uncued ones. I long for sleep to win me a daily increase in my intellectual capacity.

Conventional wisdom says that our creative juices flow more freely as we perform menial tasks, like cleaning and showering. Science backs this up, as Elsbach and Hargadon show in their study to improve workday design. Hence waterproof notepads for in the shower. A selection of Amazon testimonials for AquaNotes:

“I write poetry/lyrics and kept coming up with ideas in the shower and having to

grab my phone to write them down. I was tired of drying my hands off and reaching out

of the shower to type.”

“Ask yourself, which is more important, saving your thoughts before they escape

you or the price?”

“Top tip: After you have used up one side, tear off that piece, turn it around, and

stick it to the wet shower wall: you can write on the back now. It’ll leave some slight

pencil marks on the wall, but that washes right off.”

Upon research, the product line also includes shower-compatible words search pads, maze pads, and Aqua Love Notes, which features a heart on the head strip and a red pencil, instead of the original blue one. I wonder if they came up with all these ideas in the shower.

29 Excuse my smugness, but what’s so charming about waking up to clarity, about the brain swooping in to help us out? When we pause our straining, the mind relaxes into a state ready for inspiration’s visitation, that much makes sense. But I’m guessing there’s a reason why “sleeping on it” worked exceedingly well during my exams. One needs to be able to hand the case to the committee of sleep, somehow. Have it stewing on a back burner. In this way thinking becomes a bodily experience, and given how little we understand about the brain, we feel it as a coincidence, an expediency, a stroke of luck, good karma for showing up for work that day.

!

A related mode of thinking is what David Foster Wallace calls horizontal early-morning abstract thinking. “Abstract thinking tends most often to strike during moments of quiet repose,” he writes. “As in for example the early morning, especially if you wake up slightly before your alarm goes off, when it can suddenly and for no reason occur to you that you’ve been getting out of bed every morning without the slightest doubt that the floor would support you.” The unnerving thought is soothed when you recall the Principle of Induction—you have woken up a thousand times before and the floor has always been there—before you begin to question your faith in the P. of I. What is the ground it gets up to every morning? You suspect that it justifies itself, and the only way out is for you to consider the existence of valid circular justifications, which, in turn, leads you to appraise your need for justification in the first place . . . In Wallecian fashion, you have stepped up the ladder of abstraction to maybe level five, all the while still stuck in bed, or very well because of it.

There’s pleasure in the mismatch between your embryonic physical state and your mental trapeze. Once you get up, what is required of the day-to-day seems almost incompatible with this sort of lucid yet formal, elastic thinking. It takes a meditator to excuse herself from the day’s

30 bustling, to sit and shut her eyes. Now and then in her head she is running up the ladder, though when she catches herself racing, she returns to ground level with one breath: “Having a thought.”

!

“My dad had no idea what my mom was doing and discovering. He would never have spent hours finding patterns when he thought there were other things that needed our attention,” recalls the daughter of Marjorie Rice, a San Diego homemaker. What Rice discovered at her kitchen table between 1975–77 was four of the fifteen families of pentagons that tile the plane.

Rice’s basic mathematical training ended in high school, but she had a lifelong interest in the colors, patterns, and designs of nature, and worked as a commercial artist. She also enjoyed puzzles, crosswords, and jigsaws of all kinds, including Martin Gardner’s column,

“Mathematical Games,” in her son’s subscription of The Scientific American. In July 1975,

Gardner’s column ran an exposition on Robert Kershner’s discovery of pentagonal tiling Type 6,

7, and 8, which he claimed completed the classification problem for all convex pentagons that tile the plane. “The proof that the list is complete is extremely laborious and will be given elsewhere,” Kershner notes. The problem is simpler for triangles: all of them tile, since a triangle glued to a copy of itself, rotated 180°, always gives a parallelogram, and we feel pretty good that any parallelogram tiles. What if we go one step up to quadrilaterals? Turns out the same trick of rotating 180° works again.

Pentagons are sticky. Consider the regular pentagon. Take a tile, put down a second tile next to it, a third after that.

31 We are in trouble: there’s a gap that can never be filled! Well, you might ask, can we squish the pentagons a little bit to make this corner good? This works, as long as we’re careful about how we squish.

In December of the same year, Gardner’s column reported that a reader, Richard James

III, software engineer, had found a ninth result. This set Rice going. Today her discovery of a pentagonal tiling, composed of what she called “the versatile,” can be found when you step into the foyer of the Mathematical Association of America’s headquarter in Washington, D.C. Her website, Intriguing Tessellations, showcases her designs of bees, clovers, and hibiscus, superimposed on pentagonal tilings, in the style of M. C. Escher. It wasn’t until about a month before Rice’s passing in 2017 that Michaël Rao proved with a computer-aided exhaustive search that these fifteen are, in fact, all the families of pentagons that tile.

Marjorie Rice is forever an inspiration for a math commoner like me. Lacking formal mathematical tools, she developed her own notational system of pentagons, in the shape of “little houses children draw,” she lovingly called them. Within the little houses are her own symbols,

“tents” and “forks.” She credited her home-grown tools: “I could often find solutions to problems by unorthodox means, since I did not know the correct procedures.” She nails the place of formal thinking in everyday life. For those of us who sulk, if I could solve this, then mathematicians would have solved this already, ergo I can’t solve this, Marjorie Rice gets us out of bed in the morning.

!

I have certainly spent many a meditation session in a state akin to Edison’s induced hypnagogia. My second ever sitting with Bernie was on a winter’s night, in big leather couches, next to a fireplace . . . The exercise that night was to count up ten breaths, then return to one and

32 start again. I was amused to find myself in the forties. When the bell sounded after twenty minutes, there was no number in my brain. I felt refreshed. Either I was really good at meditation on my second try, or I fell asleep. Later I learned that the official meditation postures are not supposed to be comfortable. Having your left foot on your right thigh and your right foot on your left quickly gets disagreeable. The same applies to the practice itself. Myth: meditation is supposed to feel calm. It’s not supposed to feel any way, in fact. Zen master Shunryu Suzuki opens Zen Mind, Beginner’s Mind with a section on posture, detailing instructions for many parts of the body, from the legs in full lotus to the hands in cosmic mudra:

If you put your left hand on top of your right, middle joints of your middle fingers

together, and touch your thumb lightly together (as if you held a piece of paper between

them), your hands will make a beautiful oval. You should keep this universal mudra with

great care, as if you were holding something very precious in your hand. Your hands

should be held against your body, with your thumbs at about the height of your navel.

Hold your arms freely and easily, and slightly away from your body, as if you held an

egg under each arm without breaking it.

The posture is developed by generations of meditators to be maximally conducive—to mental derivation? to enlightenment? to something. Is meditation going to sleep with a problem?

In the brief history of my meditation practice, after the stage of blissful sleep came the stage of high-strung pain and boredom. Eventually, things got more bearable. At home, I sat every day to Bernie’s recordings, on the balcony futon, on my bed. My back finally got strong enough to support itself. Six months later, a forty-five–minute session felt particularly “good”: I figured out for the first time what was meant by the instructions of “softening around” the

33 physical pain and “attending” to it with a gentle focus. Tears streamed out my shut eyes and down my up-tilted face.

Since then, body scans have been less formidable. Instead of labeling pain as pain, I ask, how does it feel, precisely? I decided that aversion sometimes felt like a buzz, like I have excess energies to spend in my extremities, like I need to get another coffee. I decided that obsession could feel like swimming breast-stroke in a circle, if I had to describe it. The following month, I worked an internship in Beijing, doing work I didn’t know how to care about. On the hour-long subway home, I looked forward to blasting the air conditioning in my dim living room, sitting cross-legged on the orange couch, and tuning in to Bernie’s guided meditations on the Tuesday evening group’s Blogger site. Connection was shoddy, and I clung to the buffering recording even more.

!

Over years we learn to show up for what we don’t fully buy, to practice a means though its end is questionable. In my mind there is Zen, and then there is the Buddhism my family observes. The latter is perhaps more suitably termed Chinese folk religion, and I feel about it the way I feel about the gilded four-headed elephant that sits in an obscured corner of my room, next to the trash can and the spider webs, collecting dust. My mother got the statuette at Jing’an

Temple in the heart of the city, the richest-looking temple I’ve been to. She paid extra for the clerics to take a pass at it, buddhābhiseka, as it were. The Chinese term for this consecration ritual is kaiguang, or the opening of light. She wanted me to have it in my room in America, and she insisted on her point, as if anticipating me to be contrary about it. So we packed the hunk of metal in the suitcases and left to drop me off at school. It was my parents’ first time in America, so we were doing the classic coast to coast. One evening in San Francisco, my father almost got

34 hit by a car. He was out to retrieve a suitcase for me. The crossing in front of the hotel was on the top of a hill, and the driver was going fast. The suitcase took the brunt, flew down the street, was scuffed up. Mom came up with the theory that it was the four-headed elephant in the suitcase taking the hit for him. They launched into a development of this theory. For the rest of the night I felt sick.

On move-in day Mom was a bit vexed to find my dorm next to a graveyard. You don’t drive by hills of gravestones in Shanghai. Near funeral homes you find some of the city’s least expensive real estate. “You’ll place the statuette in the direction of the graveyard,” she decided.

We grabbed a last coffee, and they drove off.

The thing is really terribly gold. There are four copies of the front half of an elephant, eight front legs, four trunks tooting in four directions, but no hind legs. On its back the four- headed elephant carries a lotus. On top of the lotus rests a curvy ruyi scepter. Red jewels decorate the scepter and the elephant’s four foreheads. Around the black granite base the gold writing reads, “Wealth will come in four directions, / whole family will be safe and happy. /

Luck will come in eight directions, / good fortune as you wish.” She could have gotten me a lotus incense bowl. Or one of those ceramic bodhisattva figurines. I considered leaving it in the suitcase, but felt guilty, and also knew she might ask for photographic evidence. Each room I stayed in I’ve managed to find a corner for it.

My family takes biannual incense-burning trips to Buddhist temples in the area. On the second day of the New Year, we leave for North Summit in Hangzhou. It’s a great way to get out of calls on relatives. “But no, we really can’t stay the night, as we must leave for Hangzhou tomorrow morning. We do this every year.” I haven’t actually been to the Summit, as there is some unspoken rule against children and incense-burning, I’m not sure, and I haven’t been home

35 around New Year’s lately. My parents’ religious practice entailed for me hotel pools, and duck soup with dried bamboo shoots. My senior year of high school, my parents flew out to Mount

Wutai, the seat of bodhisattva Wenshu of Wisdom. Often holding a book, this is the bodhisattva one goes to see for school-related prayers, in particular for success in college entrance exams.

The next year we made a trip back to “return the wish.”

Recently, we’ve started driving up to Putuo Island, the seat of the bodhisattva Guanyin of

Compassion. The drive can take up to seven hours one way, depending on traffic, and getting to the temples and the hundred-feet bronze statue of Guanyin involves, in addition, a ferry, a couple long waits for shuttles, and hiking. Speed boats make it nicer. Often this trip is also excuse for

Mom and Dad to spend time with their college friends, who also bring along kids. Over dinner the adults share incense-burning anecdotes, especially tales of wish come true after paying respects to the gods. The trip is timed to avoid the fishing ban from May to July on the East

China Sea. My grandmother, though, refuses to eat the day of and the night before. She was never a part of our stops along the hike for corn-on-the-cob or lamb skewers.

The affair on the mountain is nothing short of a spectacle. The scent of incense assaults the nasal passage upon entering Puji Temple, the front gate to which has remained shut with the rarest exceptions, ever since a little monk refused to open it for the Qianlong Emperor, I overhear a tour guide. Near Huiji Temple is the world’s single Carpinus putoensis. I linger at lotus ponds stocked full of fat carps. I pass on the beaches stocked full of people. On the way to the bronze

Guanyin statue, raucous children trip over devotees, who kowtow once every three steps. Mom buys packets of incense and shows my sister how to light a bunch of three sticks in the bucket of fire, how to hold them in the palms, before the head, and bow in four directions. I worry for everyone’s down jackets.

36 When Mom is done, we toss the incense into a giant furnace, and go prostrate before statues of buddhas and bodhisattvas. She instructs us to kneel on the prayer stool, to press the palms together before the forehead, then unfolding them, to dip the forehead and the back of the palms to the stool. This is done three times, during which I’m supposed to inform the buddha or bodhisattva of my name and exact address, city, street, building, apartment number and all. I don’t know what to think. My sister looks awfully serious. Fifth grade is much tougher than fourth. Later she lets me know that she prayed for a xiaodui zhang position, lowest of three bureaucratic ranks available in an elementary school classroom . . . I don’t know what happened to the rule about kids and temples. Seeing that I am there, on the island and kneeling with a line behind me, I type up a quick message to the gods in my head. Dear Buddha, I pray to the health and happiness of my family. Here is my address, get this: . . .

Sometimes, though, things would come up and we would miss a trip, or it’d get rainy on the mountain before we could make the round. “Your wish will come true as long as your heart is in the right place,” Dad would break out this saying. “The Bodhisattva is compassionate and doesn’t hold a grudge as long as your intention is true.”

!

If I’d had an enlightenment experience, that earth-shattering breaking open of satori, this would be the natural place for a segue. That is okay, though, Shunryu Suzuki-roshi assures us:

“When you have [the] posture, you have the right state of mind, so there is no need to try to attain some special state. It is a kind of mystery that for people who have no experience of enlightenment, enlightenment is something wonderful. But if they attain it, it is nothing. But yet it is not nothing. Do you understand?”

37 Professor Adams has a story, “Journey to the Center of Mathematics,” after the Jules

Verne novel. At the story’s crux, when Professor Lederhosen, his nephew Axle, and their guide

Hansel arrive after mounting grueling theorems, they discover that there’s absolutely nothing at the center of mathematics. After some brow-knitting, the Professor suddenly looks up and smiles. “There is nothing here, because that is how you create the numbers, from nothing. First there is nothing. That is the set ∅, the empty set.” But now there is the non-empty set that has this set as an element, i.e., {∅}. Next comes {∅, {∅}}, a set of two elements . . . In this way, we obtain the natural numbers, and from there, all of mathematics. “Now we have created something,” the Professor exclaims. “Something from nothing. Does it not make you feel like a god?”

38 #4

Aloft the Confirmed Knot Tier

Since all the tools for my untying

In four-dimensioned space are lying,

Where playful fancy intersperses

Whole avenues of universes;

Where Klein and Clifford fill the void

With one unbounded, finite homaloid,

Whereby the Infinite is hopelessly destroyed.

—James Clerk Maxwell, “A Paradoxical Ode”

Quipu is the Quechua word for knot, the way indigenous Andeans in Central and South

American kept record in pre-Columbian time. A quipu has a main cord, from which hangs many knotted pendant cords. The color and the texture of the cord tells us about the type of information stored, while the knots encode a numeric value.

Here’s how the math works. Knots in a cord appear in clusters, corresponding to powers of 10. In the tens place and up, the number of overhand knots signifies the digit from 0 to 9. In the ones place, to signify the end of a number, a long knot is tied. The number of twists in the long knot equals the value, which ranges from 2 to 9. Since the long knot cannot be tied with only one twist, a figure-eight knot is used to represent 1. For an example, I give you 314 in quipu notation.

39

The quipu counting scheme is somewhere in between the Roman and the Hindu systems.

On one end of the spectrum, we have Roman numerals—I, V, X, L, C, D, M for 1, 5, 10, 50,

100, 500, 1000—which are to be read cumulatively, in an iconic mode. By which I mean, before the introduction of IV for 4 and IX for 9, etc., the Roman system interprets concatenation as addition, and so the same 314 would be written as CCCXIIII, a pile of numerals that can essentially withstand a shift in order. On the other end, Hindu numerals separate counting from iteration. Where the Romans carved the four matchsticks of IIII and the Incas tied four knots, the

Indians had the squiggle of “4,” a symbol whose shape has nothing obvious to do with its meaning.

!

In 2016, Jinmao Tower opened its skywalk to the public. By skywalk, I mean a transparent ledge, a meter or so wide, jutting out and rimming the observation deck of

Shanghai’s third tallest building. It has no handrails. Ropes and harnesses bind skywalkers to a sliding rail around the perimeter of the building’s eighty-eighth floor. Knot Friend and I were there on a muggy summer evening. With some time to kill, we went down to the fifty-fourth

40 floor to see the sunset from the Grand Hyatt Hotel. Out the window wall and in between

Lujiazui’s verticality we caught, for some minutes, a sliver of the Huangpu, its golden glimmer, and a miniature black boat gliding past.

Up above, the observation deck was boisterous with visitors. In the bathroom line I saw a boy strutting out, singing a rapturous song about taking a leak from the eighty-eighth floor.

About a dozen of us were skywalking together that hour. We changed out of street shoes, took off jewelry, and attached lanyards to our glasses as the sky darkened and the city’s lights gradually came on. From a vestibule a member of the staff briskly spun me around, strapped me up, buckled me in, and ejected me out of the building.

It wasn’t as windy or cold as I’d expected. The quality of sound was different, though, dampened, like after a snowfall. I took big, deliberate breaths of air, filled my lungs with pollution, and thought of the little fox from my first grade textbook, who fills bottle gourds with forest air and sells them in the city. Sitting on the ledge with dangling legs, I saw head and tail lights from between my feet. A golden canvas of lights interrupted by the river and the gargantuan Pearl Tower, fading between the colors of a rainbow. I understood why they insisted we change shoes—you really don’t want your shoe free-falling from up here.

An instructor, who didn’t seem to appreciate how cool his job was, showed us yoga poses on the skywalk. We placed the strap on our shoulders, stepped both feet to the edge and leaned out, face-first. We reached out sideways, left foot on the edge, left hand grabbing onto the strap that ties us to the building, right side of the body overhanging in air. My leg muscles clenched.

My palms were sweaty. As the right half of me was off the ledge my left hand grabbed my strap in a fist, a pointless action—I was at the mercy of the strap, and whatever mechanism attaching it to the structure of the building.

41 I was positively euphoric for the rest of the evening. I wondered if they were hiring, and regretted not buying an annual pass. I was there again four months later. It was adrenalin, of course, but lately I’ve been thinking that there, at the artificial precipice, I had a taste of the

Kantian “negative pleasure,” which occurs “through the inadequacy of even the greatest effort of our imagination to estimate the magnitude of an object.” The pleasure is negative, in the sense that it hinges on the very “perception of the inadequacy.” We delight in the realization that unmeasurable magnitude can nevertheless be contained in our consciousness. We watch enthralled as logic and sensory experience try to one up each other. We package our “incapacity for grasping it” into another form of knowledge. “It is consequently a pleasure, to find every standard to sensibility falling short of the ideas of reason.” On the eighty-eighth floor, I was an addict of unfeelable immensity, and I didn’t even know it. Had I been asked to give an account of what I was not feeling, I couldn’t have done so; but no one asked me. There I was, dangling half off a ledge 1,115 feet in the air, sweaty fingers grasping my strap, feeling at once the height of self-abnegation and aggrandizement. It was fun when I was doing it, and it was fun to think about what it was that I had done, just as it is fun right now.

!

Over forty years of work compiles into The Ashley Book of Knots, Clifford Ashley’s definitive book on applied knots. “Beautiful illustrations of innumerable knots,” Professor

Adams says. “A treasure, if you can find it.” The quantity Professor Adams calls “innumerable,” turns out, is 3,854. 3,854 knots were handpicked by Ashley according to the following standard:

“Unless a knot serves a prescribed purpose, which may be either practical or decorative, it does not belong here. Knots that cannot hold their form when tied in tangible material are not shown, no matter how decorative they may be.” Lauren Scanlon of Durham University asked how many

42 of these 3,854 knots used in human history are mathematically distinct. With the help of computer, she found that The Ashley Book captures sixty-nine of the total 1,701,936 prime knots with sixteen or fewer crossings, as tabulated by Hoste, Thistlethwaite, and Weeks in 1998.

Ashley pins his fascination with knots to the tradition of Marco Polo and Christopher

Columbus. “To me the simple act of tying a knot is an adventure in unlimited space,” he writes.

An evocative statement for an aspiring topologist, who studies the spatial properties preserved under stretching, smushing, crimping, crumping—but not cutting or pasting. Ashley continues:

A bit of string affords a dimensional latitude that is unique among the entities. For

an uncomplicated strand is a palpable object that, for all practical purposes, possesses one

dimension only. If we move a single strand in a plane, interlacing it at will, actual objects

of beauty and of utility and result in what is practically two dimension; and if we choose

to direct our strand out of this one plane, another dimension is added which provides

opportunity for an excursion that is limited only by the scope of our own imagery and the

length of the ropemaker’s coil.

What can be more wonderful than that?

It’s time. Knots seem to have an existence that spans dimensions. We model a rope with one dimension. We draw diagrams of knots in two dimensions. As an object, and arguably one of the rarer mathematical objects that one can actually point to and say, Here! This is what I study, a knot is three-dimensional with body, thickness, and heft. Ashley’s knotting is thwarted, however, at level four:

But there always seems to be another car ahead in every likely parking space. Here is a

Mr. Klein who claims to have proved (Mathematische Annalen) that knots cannot exist in

space of four dimensions. This in itself is bad enough, but if someone else should come

43 forward to prove that heaven does not exist in three dimensions, what future is there left

for the confirmed knot tier?

I wonder how Ashley would feel if he knew that what he needed was the preservation of codimension. Taking a hint from its prefix, we might guess that codimension is when dimensionality becomes relative. A knotted circle in 3-dimensional space has codimension 2, since its dimensionality is 2 less than that of the space it lies in. By analogy, we would like to know what is to 4-dimensional space as a circle is to 3-dimensional space. In other words, what is the analog of the circle, one dimension up? Hint: a unit circle is the set of all points a distance

1 away from the origin in the 2D plane. What, then, is the set of all points a distance 1 away from the origin in the 3D space?

A sphere. What would Ashley make of the fact that, while there exist no knotted circles in 4-dimensional space, there exist knotted spheres? More: there exist knotted 3-spheres in 5- space, knotted 4-spheres in 6-space, ad infinitum. Is there enough beauty here to make up for the lack of utility?

Would this chain of facts quiet Ashley’s analogous worry for the prospect of a heaven for us meager beings of the “confirmed knot tier”?

What if, when Ashley calls knotting “an excursion that is limited only by the scope of our own imagery,” we take him literally? Is wonder, for Ashley, the knot’s infinite expansion, or is it rather wonderful to come upon the car that blocks our way?

!"

44 I love mathematician Robert Meyerhoff’s statement of the problem of dimensionality.

We feel we have a pretty good grasp on two-dimensionality, he writes, essentially because we live in one dimension more. Consequently, “we can leave it and look down upon it.” We may draw surfaces to our heart’s content. No such luxury is afforded us even one dimension up. In order to better understand three-dimensional space, he argues, we should first get used to the great limitation, and practice seeing two-dimensional surfaces from an intrinsic point of view.

Asked to draw the surface of a donut, I would most likely give you the bird’s-eye version of it. Meyerhoff proposes, on the other hand, that we start thinking about the surface, equivalently, as a flat square, with markings that indicate the gluing strategy. Here, we glue one opposite set of edges to obtain a cannoli shell, and then glue the remaining edges to obtain the donut. Note that in the flat representation, we have eliminated the need to poise ourselves outside the plane, as it were.

This task quickly grows formidable, ergo the sad face on the stick figure trapped in two- dimensionality. Meyerhoff calls this “the recognition problem”: given any surface, what is a good intrinsic recognition scheme? The exact conundrum befalls A. Square, of Edwin Abbott’s

Flatland: A Romance of Many Dimensions. The 1884 social satire and novella has become a

45 mathematical classic. As the name implies, Flatland is a two-dimensional universe populated by geometric shapes from line segments to various polygons. According to Mr. Square, Flatlanders experience “extreme difficulty in recognizing one another’s configuration.” As no extrinsic view of their universe is viable for them, every citizen appears to be a line segment. Indeed, if our eye is exactly level with the penny on the table, how can we tell it is circular and not square?

The plot thickens when Mr. Square encounters the spectral Sphere, inhabitant of the three-dimensional Spaceland. Crucially, the Sphere’s explanation of his country by way of words fails to enlighten Mr. Square. “You call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other.” The wordy representation is precise yet cumbersome, in the same way that “a square with opposite edges identified” is much less intuitive than “donut.” The

Sphere finally has to grab Mr. Square by the stomach and lift him into air. “The higher I mount, and the further I go from your Plane, the more I can see,” explains the Sphere. It is thus that, hovering over Flatland, Mr. Square is awestruck by the comprehensive vision now available to him and exclaims, “Behold, I am become as a God.”

You might be interested to know that, in 1671, philosopher Henry More thought the fourth dimension to be the abode of ghosts. Two hundred years later, psychic Henry Slade became famous for channeling ghosts from the fourth dimension to untie knots. In the same year as Flatland, Charles Hinton published a pamphlet titled “What Is the Fourth Dimension? Ghosts

Explained.”

In W. G. Sebald’s account, Sir Thomas Browne shared Mr. Square’s mission. Browne also “saw our world as no more than a shadow image of another one far beyond,” Sebald writes.

“In his thinking and writing he therefore sought to look upon earthly existence, from the things

46 that were closest to him to the spheres of the universe, with the eye of an outsider, one might even say of the creator.” To achieve such “sublime heights,” Browne resorted to “a parlous loftiness in his language.” In this century, for forty-five dollars I can replicate the experience by putting myself eighty-eight floors above ground. Browne and Mr. Square and I, we’re after a similar sort of sublimity. “And the plunging waters of Niagara—what did their eternal thunder mean if there were not also someone standing at the edge of the cataract conscious of his forlornness in this world” (Sebald, The Rings of Saturn).

!

I suspect Sebald fully intends to pun on the distinct meanings of “cataract,” since pages before, he tells us about minister Thomas Abrams’s marvelously precise reconstruction of the

Temple of Jerusalem:

I have to make every one of the tiny coffers on the ceilings, every one of the hundreds of

columns, and every single one of the many thousands of diminutive stone blocks by hand,

and paint them as well. Now, as the edges of my field of vision are beginning to harden, I

sometimes wonder if I will ever finish the Temple and whether all I have done so far has

not been a wretched waste of time.

Here we flip the page to see a photograph of the model from the inside. The edges are blurry as our vision is compelled down the infinite progression of columns. Another flip of the page:

But on other days, when the evening light streams in through this window and I allow

myself to be taken in by the overall view, then I see for a moment the Temple . . . as if

everything were already completed and as if I were gazing into eternity.

I can’t understand Abrams’s case apart from Kant’s mathematically sublime:

Apprehension is easy: we can always count the next number, build the next column. Soon,

47 though, flakes of snow roll into an icy snowball, and “with the advance of apprehension comprehension becomes more difficult at every step and soon attains its maximum.” The double vision Abrams describes, I wager, must be that awakening of the “supersensible.”

I remember the day I was formally introduced to the different sizes of infinity. It was

Real Analysis with Professor Morgan, who was retired but had returned for the semester. He had the sprightliest walk, his combed, silver hair bobbing along. What would you say—are there more integers or more rationals, numbers that can be expressed as a fraction? Well, there are infinitely many of both . . . Although it is also true that while every integer can be written as a fraction, there are definitely fractions that aren’t integers, like 3, so it seems like there’s an 2 infinity of rationals between every two integers. Comparing infinite quantities is choppy waters.

Georg Cantor’s work from 1874–84 convinced the mathematical society that there are, in fact, equally many integers as rationals, and called this quantity aleph-null, or ℵ0. Further, if we consider every possible combination of the integers—an operation called taking the power set of

ℵ0 the integers—we obtain 2 = ℵ1 different combinations. Aleph-one is another infinite number, one that is strictly greater than ℵ0, and also the size of the reals, i.e., rationals plus irrational numbers like �. Professor Morgan told us that we could, again, take the power set of the power

ℵ1 ℵ2 set, and get 2 = ℵ2. Now take the power set of that for 2 = ℵ3 . . . In fact, just keep taking power sets and eventually we’ll have infinitely many infinite numbers.

Cool as this fact is, what impressed me was just how far Professor Morgan was willing to go up the progression ladder. I believe he was truly amused and enraptured to take the fifth power set. Once, we spent five minutes of class on a video zooming in on an instance of the

Mandelbrot set. Thanks to its self-similar nature, a dive into the set is nothing short of mesmerizing. I stole glances at Professor Morgan, who stole no glances at us. He stared into the

48 projector screen for the entirety of the video, as the class murmured and fell silent, intermittently.

After class, I had the strong urge to make this video my computer screen saver.

!

“The ‘∞’ symbol itself is technically called the lemniscate (apparently from the Greek for

‘ribbon’). Other names for the lemniscate include ‘the love knot’” (David Foster Wallace,

Everything and More: A Compact History of ∞).

!

In Buddhist cosmology, the universe is not one world but made up of many “world- spheres.” How many? Some texts speak of “the thousandfold,” “the twice-thousandfold,” and

“the thrice-thousandfold” world-sphere. Others cite the numbers 1,000,000,000 and

1,000,000,000,000. In the temporal dimension, we might ask, how long is saṃsāra? Well let’s just say that a Mahābrahmā, who is in the realm of the Supreme Gods, is supposed to exit saṃsāra and reach nirvāṇa in 16,000 aeons. On the magnitude of an aeon, the Buddha said:

Suppose there was a great mountain of rock, seven miles across and seven miles high, a

solid mass without any cracks. At the end of every hundred years a man might brush it

just once with a fine Benares cloth. That great mountain of rock would decay and come

to an end sooner than ever the aeon. So long is an aeon.

What would such dimensions spell for Human Beings, of the world of the five senses? Shunryu

Suzuki-roshi explains:

In Buddhist scriptures we sometimes use vast analogies in an attempt to describe empty

mind. Sometimes we use an astronomically great number, so great it is beyond counting.

This means to give up calculating. If it is so great that you cannot count it, then you will

lose your interest and eventually give up. This kind of description may also give rise to a

49 kind of interest in the innumerable number, which will help you to stop thinking of your

small mind.

I reread these lines without clarity: is the number that is beyond counting supposed to curb or fuel our interest? Is the mind supposed to be small or astronomically great?

Reader, I must apologize for the vanishing of Zen thoughts. Goodbye, proverbial raft.

I’ve lost a thread . . . My reading of Zen literature is not extensive. Also I’m little concerned with a holistic theory of how Zen coincides with mathematical knots, I must admit. The project of math as a metaphor for Zen was always of course “feeble and fragmentary” at best. How can I intimate infinite wisdom but by improper metaphors? And so I still must grasp the rope; Thomas

Abrams must build a model of the Temple. Mr. Square must confront the ghostly visitor and

Thomas Browne must write. There is what we want, which is to gain height, to perch, to own the outside perspective. Then there is practice, which is the unfolding of a donut into a sheet of paper, and the hundreds of stones and columns. This discourse on how four-dimensionality awakens supersensibility is really close again to becoming an analogy for satori. I’m out of lessons.

!

“Image over time.” These were the only words I took down on the front matters of Trace when Lauret Savoy came to give a reading, for which I missed Bernie’s Zen and American

Literature class. In one essay, “The View from Point Sublime,” Savoy writes of revisiting the spot at the Grand Canyon. Geologist Clarence Dutton, who named the place, reports from the rim: “As we contemplate these objects we find it quite impossible to realize their magnitude. Not only are we deceived, but we are conscious that we are deceived, and yet we cannot conquer the deception. Dimensions means nothing to the senses, and all that we are conscious of in this

50 respect is a troubled sense of immensity.” On a return trip, Savoy holds up a photograph taken from the exact same vantage point, on a childhood trip with her family. The superimposition stuck with me. She has a different reading of Kant than I do.

These days I think everyone who’s been there has a Grand Canyon story. I was there on a school trip the summer before junior year of high school. The climate was strange to me when I stepped out the bus, like standing under a giant hand dryer. I was mildly confused by the lack of glass panes or handrails around the canyon. I assumed this meant it was safe to go sit on the edge. For a while I watched the crows. I asked my friend Wendy for her phone to take pictures with. She handed it to me, and retreated to safety. In the glare I didn’t notice that there was sunblock smeared on the lens. The blurry pictures and my barmy smile therein grow scarier in retrospect. When I came off the edge, Wendy told me she had been praying that I wouldn’t tumble and strategizing what to do if I did. Later I learned that people fall often at the Grand

Canyon, that it hardly makes the news outside the local area.

!

I want to give you the straight dose: here is what Mr. Klein discovered about knotting in four dimensions. But first, how are we going to picture 4-dimensional space, in our universe which feels pretty 3-dimensional? Recall the Sphere, who describes himself as a single point, followed by a sequence of circles growing and then shrinking in diameter, and finally another point. In effect, he is taking 2-dimensional slices of himself as he moves through Flatland. In this way the Sphere, who lives in 3 dimensions, is able to explain himself in the language of Flatland.

51 We now apply the idea to seeing knotted spheres in 4-space. We can keep the three usual spatial dimensions, �, �, and �, and add a fourth temporal dimension �. Then, taking slices of a knotted sphere in the dimension of time, we obtain a sequence over time of a knotted circle in the

�, �, and � dimensions, the familiar shenanigans of our confirmed knot tier. The experience of a knotted sphere in 4-space is like watching a 3D knot movie, as it were.

Claim. There are no knotted circles in 4-space.

Proof. Let the fourth dimension be color � this time. As we move through the

fourth dimension, instead of � = 1, � = 2, � = 3, . . . , let’s have � = red, � =

orange, � = yellow, . . . Living in the color model of 4-space is like living in 3-space,

only we wear glasses with a knob that shifts between different color lenses, Professor

Adams says. For example, when we have red-colored lenses on, only red light filters

through.

Say we have a knot in color dimension red. In 4-space, we may change any

crossing of the knot from over to under and vice versa, by pushing the top strand

through the bottom strand. Here’s how to do it: at the crossing, push one of the strands

in the fourth dimension to the neighboring color orange. But the red world and the

orange world are different 3-dimensional slices of 4-space. With our glasses set to red,

we can’t see the orange strand anymore, and so we can move the red part of the knot

52 right through the orange part. Performing this operation at enough crossings always

gives the trivial knot. Q.E.D.

“Life is a train of moods like a string of beads, and as we pass through them they prove to be many-colored lenses which paint the world their own hue, and each shows only what lies in its focus” (Emerson).

!

Nuar Alsadir’s theory of the fourth person singular models lyric address with four- dimensionality. She quotes from Kafka’s letter: “There are things we’ve never seen, heard or even felt, and we can’t prove they exist, though no one has yet tried, but we run after them, without knowing which direction to run in, and we catch up with them without reaching them, and, still complete with clothes, family souvenirs and social relationships, we fall into them as into a grave that was only a shadow on the road.” Once again we’re on the run, and we arrive without reaching, and fall into what was past us. Lyric, then, is a flick of the four-dimensional self: different slices of myself in time run into each other, pass through each other. It “removes the time-condition . . . and renders co-existence intuitable,” Kant would say.

!

I tried to have Andreas Gursky’s 99 Cent (1999) as my desktop picture, before it got too loud for the eyes. We are perched mid-air in a 99 Cents Only store in Los Angeles, a ghostly and improbable vantage point, unless you’ve also been to eight-story 24-hour discount stores in

Japan and seen floor-to-ceiling bento boxes from the escalator. Here though, our line of vision runs satisfyingly down colorful shelf after shelf, from the Kit Kats and the Reese’s to rounds of cookies then juices, soaps, and shampoos, skipping over the irregular human head here and there, all the way to a rainbow of plastic chairs and stools mounted atop the furthest shelf. Once again,

53 on the ceiling: reflected squares of color between row after row of fluorescent light. The number

99 dots the place—every price tag big or small, the neon sign on the glass window, the writing on the wall: 99¢ only / 99 Thanks . . . ! / 99¢ only / Nothing over 99¢ ever! / Open 9 to 9 9 days a week . . . The actual photograph is about 100 times larger than my computer screen.

I was trying to find a quieter picture when I clicked on Gursky’s Shanghai (2000). It is the view from inside a hotel, lit golden. Floor upon floor of curved hallway stack together, punctuated by repeating doors and frames. A guest hugs the handrail and looks down the abyss. I learned that the picture was made by manipulating together three pictures taken on different floors, all of them in the thirty-three–story atrium of the Grand Park Hyatt.

54 #5

A Neighborhood of the Missing Knot

The unraveling of a torment: you have to

begin somewhere: the color blue, the

damage is repairable. I cannot throw

it away because I do not want to

put it on the back burner. It is worth

saving. I have gone through this one

hundred times + any piece of tapestry is

worth (not saving, cherishing) result: mountains

of unusable clothes buried under

torn clothes. (I cannot renounce

the past. I cannot and do not want to forget it.)

imitation + homage to the mother. lucky to “enjoy” it duress

resist new clothes. Le Printemps

—Louise Bourgeois, diary entry, 6 June 1994

The summer of COVID I was cooped up on Zoom with Professor Adams and six peers, dicing knots and reflecting the pieces to bound their volumes. Wait. Knots have volume? Allow me to bring you up to speed.

55 Contrary to what you’d expect, the volume of a knot refers, in fact, to the volume of space minus that knot. When I say space, for all intents and purposes you may take me to mean simply ℝ3, three-dimensional space. But because mathematicians like to work with compact, or

“nice,” objects, they cheat and compactify ℝ3 by adding to it a point at infinity. The result of the operation ℝ3 ∪ {∞} turns out to be the three-sphere �3, or the three-dimensional analogy of our day-to-day, common sphere.

I fear I have lost you already. Let us just say now, by way of definition, that the volume of a knot � is the volume of �3 − �, or the complement of � in �3. To use Professor Adams’s metaphor: imagine space is filled with green Jell-O. Suppose we drill a wormhole where � is.

What is the volume of the remaining green Jell-O?

Now, it wasn’t until 1974 that Robert Riley first thought about knot volumes. Four years later, topology giant William Thurston realized that all knots fall into exactly one of three buckets: torus, satellite, or hyperbolic. The vast majority of prime knots (think prime numbers,

56 but for knots) are hyperbolic knots. Torus knots and satellite knots are well understood, and it is among hyperbolic knots that the theory is richest, as is usually the case with hyperbolicity. More: whereas all torus knots and satellite knots have volume 0, every hyperbolic knot has a volume that is a positive real number. For instance, the first hyperbolic knot in the table, the figure-eight, has a volume of 2.02988 . . . , the smallest of all hyperbolic knots.

Time for a sanity check. We might think that space minus a knot should have infinite volume, but, in fact, we will be measuring distance with what is called a hyperbolic metric. This is related to the whole business of a triangle having angles that add up to less than 180°, and the letting go of Euclid’s parallel postulate, which states that given a line and a point outside it there exists a single parallel line. Again the question looms—how do we picture such a universe?

Here is a figure for the upper-half plane model of the hyperbolic plane. Everything above the �-axis is part of our universe. A geodesic, or the curve that gives the shortest path between any two points in that curve, is either a line perpendicular to the �-axis, or one of these half circles. Note we have already departed from the Euclidean plane, where the shortest path is always the straight line. Living in the upper-half plane model is like swimming in a sea of chocolate pudding, Professor Adams tells us. This is because as chocolate pudding settles, it gets denser near the bottom and airier near the top. And so to get from point � to point �, it would

57 require more effort to swim the straight line, and thus makes sense to float a bit and then sink back down.

Note also the failure of the parallel postulate: in my figure, given a line � and a point � outside of �, all these other geodesics are parallel to �. Considered heretical at its inception, hyperbolic geometry has its own fascinating history featuring figures like Gauss and

Lobachevsky and Bolyai. Bolyai’s father, also a mathematician, admonished, “Detest it as lewd intercourse.” To move on with our discussion, rest assured that with a hyperbolic metric, knot volumes really are finite.

A wonderful property of the knot volume comes out of the Mostow-Prasad Rigidity

Theorem (1971): volume is an invariant, meaning that, no matter how we arrange the knot, its complement in space always has the same volume. Meaning also that, if two knots have different volumes, we can say with confidence that they are distinct knots. Herein lies the power of knot volumes. Let us just say that not a lot of distinct knots have the same volume (though it happens). Then, the discovery of volume as an invariant answers the basic question of knot theory: how can we tell if two bunches of tangled strings are same or different? This is one reason why we might care about knot volumes.

!

Another dessert analogy comes courtesy of a friend I sat next to in high school. She wrote me a long letter in her beautiful hand for my 16th birthday. This letter was my first inkling that I was dishing out trivial personal tragedies a bit too willy-nilly. I do remember collapsing on our desk time and again to solicit her reaction. “It’s nice to be able to bounce back with encouragement, but true feeling of security in my opinion comes from within—an endless source of power. To fully replenish one’s health points with no reassurance from others, without

58 sending for outside help is in fact really very difficult.” I was struck by her wisdom at 16, and

I’ve been carrying this letter around since. These days it’s in the Box under the bed. Elsewhere in the letter she writes, “Before we spoke much the fridge at my house was always empty, and so I always starved; one day you showed up at my house and said to me, ‘Hey—the walls of your place are decked with cookies!’ I turned to look and indeed, the walls are neatly tiled by iced cookies. And so we sat on the floor and started eating them. Do you know; long after you left

I’m still working on these cookies.”

I’m not sure how to translate her 安全感—a combination of calmness and wholeness, of safety and emotion—which I’ve rendered as “true feeling of security.” A related term could be

“inner peace,” which I picked up from an English teacher in middle school. He was quoting

Shifu from Kung Fu Panda. Inner peace is one among three invaluable lessons I collected from this particular teacher, the other two being: go to bed today (i.e., before midnight), and, drink a glass of water first thing when you wake up.

Is this the compendium of self-help advice I carry around? Well, as they say, the proof is in the pudding.

!

The idea of drilling a wormhole in the shape of a knot out of �3 reverberates beyond knot identification. So far we’ve seen how volume, as the most natural invariant of a hyperbolic knot, reveals knot theory as the study not merely of one-dimensional knotted strings, but equivalently of three-dimensional spaces-minus-a-knot. The sequel of the saga concerns filling that wormhole back up. How do we go about doing that? Well, we could replace exactly what we dug out, which would give us back �3, thankfully. But what if we change up our gluing scheme? Say we

59 dug out a tubular neighborhood of the figure-eight knot in �3. Now say someone hands us a solid torus, and asks us to fill the hole we made with it.

We have quite a few options. (Warning: such gluing is abstract and can be a pain to visualize.)

′ We can glue a meridian �1 of the donut to a meridian �1 of the figure-eight, glue the meridian

′ right next to �1 to the corresponding meridian right next to �1, and so forth. Alternatively, we can glue meridians of the donut to longitudes of the figure-eight. In fact, meridians of the donut can be glued to copies of any curve on the figure-eight that does not cross itself, which is otherwise known as a (�, �)-curve. Such an operation is called a Dehn surgery on the figure- eight. The punchline is this not-at-all obvious theorem due independently to Lickorish and

Wallace: every compact connected three-manifold can be obtained by Dehn surgery on some knot or link.

I probably should have said much earlier that three-manifold is just the correct word for space that locally looks three-dimensional. The Lickorish-Wallace Theorem, then, fundamentally relates knots to space; it claims that if we understand knots well enough, we will understand all

60 compact three-manifolds. If we believe, as many cosmologists do, that our universe is not infinite, then it is so-called compact and also connected. Meaning that we live in a Dehn- surgered knot. Furthermore, we could live in a knot complement. Do we live in space with a giant pretzel-shaped hole in the sky? At this point we simply don’t know.

Thus the study of a rubber band eerily has to do with the shape of the universe. As Louise

Bourgeois asks in her diary, “What is the shape of this problem?” Consider her Repairs in the

Sky (1999). On a lead plate the titled is engraved in uppercase, surrounded by five round punctures. The punctures are mended with blue thread and fabric. On November 7, 1952, she wrote in her diary, “matter makes itself indispensable / because by definition one cannot / create a shape of a volume without matter but it / is not at the heart of the problem.” What is, then?

What is the missing heart of the problem?

Bourgeois left math behind. Regarding her two years studying math at the Sorbonne, she said, “I had been in a state of anxiety and needed reassurance. Solid geometry . . . that was paradise. It lasted several years, my happiest time.” Yet, even stoic math had its rug pulled from under, when geometry, the system of organization and orientation, became plural. “There is

Euclidean geometry, but there are also a number of other geometries so you can have a way out from the rigidity of the Euclidean towards freedom. The Euclidean is comforting because nothing can go wrong, but it is not the geometry of pleasure.” She explains her change of course to enroll in art school, “You’re told that two parallels never meet, and then you learn that in non-

Euclidean geometry they can easily come together. I was deeply disappointed, and turned toward the certainties of feeling.”

!

61 I must admit to the lure of the proof. The mathematical proof as a genre is anti-literature, so much seems clear. The stuff of academic journals is appalling and little relieved by the figures in between. Even David Foster Wallace loses his charm in the depth of his mathematical exposition in Everything and More, a 300-page “booklet,” as he calls it, on the history of ∞. How is it that, despite profuse apologies to readers, his seven-page solution to Zeno’s paradox, crux of the narrative, is not merely unsexy, but nonsensical and incorrect to boot? Why does the book read like a botched Wikipedia page? Is there something fundamentally at odds between math and writing about math, that cannot be squared away? How do I not lose heart and turn toward the certainties of feeling?

But then why are mathematical terms so suggestive, metaphoric? Crack open a text book: rank, span, nullity. Limit points, closure, smooth. I walk out of lectures, thinking about the aptness of mathematical terms for what they define. What do mathematicians mean when they call a proof elegant?

Topology is “making holes in what is written,” Lacan says. “There is no topology without writing.” Writing is the knitting of knots, then. Writing is writhing.

Meaning is a sweater we knit for ourselves, Knot Friend said to me once. The metaphor stuck; in my journal I drew a turtleneck and wrote the word “meaning” across. Sweater of meaning, where are you? I asked. Self-imposed prerequisites to feeling happy are crippling me, if they don’t already cripple others. The sweater is tattered, in the sense that I can’t listen to playlist 01 again. I still haven’t digitalized the red journal. I don’t talk about fishing knots. There is a hole the size of two years in the sweater, where wind fills in. Yesterday I flinched at your text, asking me to brunch with you.

62 “To unravel a torment you must begin somewhere,” Bourgeois wrote in her diary. What if the torment is a garment? What if the unraveling is another torment—where do you begin?

It has to be said: perhaps writing about math is about mending my sweater. I pull my sweater inside out. With my needle and my skein of yarn, I pick up the loose loops, again and again. I attempt to recover entropy.

Aristotle interjects: “Many poets tie the knot well, but unravel it ill.” Zen interjects: the hole is the way. Radical unraveling—do I dare? “To untangle a snarl, loosen all jams or knots and open a hole through the mass at the point where the longest end leaves the snarl. Then proceed to roll or wind the end out through the center exactly as a stocking is rolled. Keep the snarl open and loose at all times and do not pull on the end; permit it to unfold itself” (Ashley).

To be completely honest, the sexiness of Zen is in the promise to make me feel fully, even hurt. I find the idea of metaphorically feeling the icy gush from the holey sweater irresistible. I imagine that’s why Jon Kabat-Zinn’s manual on meditation is titled Full

Catastrophe Living. “Zen does not give us any intellectual assistance, nor does it waste time in arguing the point with us; but it merely suggests or indicates, not because it wants to be indefinite, but because that is really the only thing it can do for us. If it could, it would do anything to help us come to an understanding” (D. T. Suzuki).

!

Worldmaking desire gave birth to the mathematical field of knot theory. Back in the

1880s, when people still believed ether existed, a cohort of three Victorian scientific minds were early knot advocates. It started when lifelong friends Williams Thomson (later knighted Lord

Kelvin) and Peter Guthrie Tait got interested in German scientist Hermann von Helmholtz’s study of the motion of vortices in ideal fluid. In 1867, Tait demonstrated for Thomson his smoke

63 ring machine. He whacked a wooden box with ammonia and sulfuric acid inside and a round hole on the outside. Wobbly vortex rings floated out of the hole. We now know Thomson for his legacy in thermodynamics, but Tait’s smoke ring experiment gave him the idea for the following atomic model: what if chemical elements are just knotted vortices, floating around in the ether?

Charming idea, no? Thomson thought that the trefoil corresponded to carbon, and that the

Hopf link corresponded to sulfur. He published his “vortex theory of atoms” in the same year. In

1878, Tait started tabulating knots. In his mind, he was making a table of the elements. It is by no means an easy task, and Tait had no tools, or invariants, with which to show that his knots were distinct. In 1883, he came out with “The First Seven Orders of Knottiness,” a table of knots with up to seven crossings. A year later and under joint effort from American mathematician C.

N. Little and others, the table was extended up to ten-crossing knots in “Tenfold Knottiness.”

This was the origin of mathematical knot theory. To the disappointment of knot enthusiasts, in

1887 the Michelson-Morley experiment offered strong evidence that ether was not a real thing, and sealed the fate of the knotted table of elements.

Friend to Thomson and Tait, James Clerk Maxwell is better known for electromagnetism and lesser known for his poetry. He wrote his last poem, “Paradoxical Ode,” in 1878, during the final months of his life.

My soul’s an amphicheiral knot

Upon a liquid vortex wrought

By Intellect in the Unseen residing,

While thou dost like a convict sit

64 With marlinspike untwisting it

Only to find my knottiness abiding;

Filled with references to latest ideas in math and science, the poem offers a glimpse of the spiritual dimension of the knotty business. In particular, “Intellect in the Unseen” hearkens to

The Unseen Universe, or Physical Speculations on a Future State (1875), a book by Tait and fellow Scottish physicist Balfour Stewart. The hastily written and immensely popular book was an attempt to square the rise of science with religion, a scientific proof of God. “Thus, by the help of natural reason, I may know there is a God, the first cause and original of all things; but his essence, attributes, and will, are hid within the veil of inaccessible light, and cannot be discerned by us but through faith in his divine revelation,” the authors declare. Their argument involves again, smoke rings. They provide a diagram of concentric circles. The innermost circle is supposed to represent a smoke ring. The immediately adjacent one, our physical universe. All the rest are deepening layers of invisible universes, going out forever. The idea is that, just as the smoke ring is impermanent in its adjacent layer, our lived universe, the latter is itself part of a more perfect universe, so on and so forth. As such, there exist “things which are not seen [that] are eternal.” Q.E.D. A sequel, Paradoxical Philosophy, was published three years later.

Maxwell’s “Paradoxical Ode” is addressed to the protagonist of Paradoxical Philosophy. The poem crescendos at the end: “Let viewless fancies guide my darkling flight / Through Æon- haunted worlds, in order infinite.” I don’t think Maxwell fully bought this endeavor of his friend

Tait’s; he strikes me as more privately spiritual. Nevertheless, the attraction to knots is shared and undeniable. Indeed, a trefoil appears on the spine and title page of Unseen Universe.

!

65 I’ve been thinking about cusp lately, which has quite a few mathematical meanings, including the point at which a curve takes an abrupt turn. In knot theory, specifically, a cusp is a neighborhood of a missing knot—the stuff that surrounds a drilled-out knot in green Jell-O space. In other words, a cusp is a solid torus missing its core curve. Often it is drawn as a volcano. In the upper-half space model, a cusp looks like a collection of what are called

Horoballs, tangent to the ��-plane:

It’s high time to stop being precious and start gouging holes. I’ve graduated from the holey sock donut. The apple with three channels eaten through by worms doesn’t faze me anymore. I thought I was studying knotted loops; all this while, I might as well have been studying cusped manifolds. Annie Dillard couldn’t untangle the knotted snakeskin she found at

Tinker Creek. Part of the skin had been turned inside-out, “like a peeled sock,” and a portion of the inside-out part had been pulled right-side out again, leaving “no edges to untie.” Montaigne had a different approach to knots: “I do not attempt to disentangle, moreover they have no end to take hold of; I often cut them, as Alexander did his knot.”

!

An amphicheiral knot, one to which Maxwell likens his soul, is a knot that is equivalent to its mirror image. In my summer research, I found the notion of reflection exploited over and

66 over. Mirrors are nice in that, by the Agol-Storm-Thurston Theorem (2007), if we cut a hyperbolic manifold along reflection surfaces and reglue the pieces into another hyperbolic manifold, the volume increases or stays the same. This theorem has implications in the trade of necklace knots, which are tangle-beads strung along a number of strands. Here is a four-bead example. It’s kind of simple: to find the volume of an individual bead in this necklace, a new four-bead necklace is strung with copies of the bead and its reflection, alternating. The new necklace is fed into SnapPy, whose output, divided by four, gives the bead’s volume.

Say I own a bead shop, with an inventory of beads. If you came in and handed me a necklace, any necklace, the theory says that I could just look up in my books the volume of each bead in your necklace, add the numbers up, and produce for you a lower limit on the necklace’s volume! But of course that was probably not what you were looking for.

That was my job over the summer, cutting up necklace knots, reflecting the beads, finding their volume, summing them together. I was Professor Adams’s knot correspondent. In

The Ashley Book, R. L. of Tiffany and Company is the knot correspondent for “the pearl knot”:

There will be two, three, or four strands of silk, according to the size of the pearls and

their holes. . . . At both ends, where the clasp and click are, the silk is knotted back

through the last pearl, so that the end of the silk is finally knotted between pearls and not

tied simply to the clasp and click. Larger knots often have to be made if the hole in the

67 pearl is too large for a single knot. These larger knots are made by knotting only two

strands of the three.

For Emerson, the tinted bits of glass that we call life are strung on something stronger than silk.

“Temperament is the iron wire on which the beads are strung.” Who cares what sensibility or discrimination I sometimes show, if I fall asleep on the cushion? Yet I’m beginning to find that even sitting perfectly straight is not a way out. The spine is too rigid, or, as Emerson goes on to say, temperament, an iron loop, “also enters fully into the system of illusions and shuts us in a prison of glass which we cannot see.” There is no end to māyā. “All things swim and glitter.”

But if knot theory has taught me anything, it’s that we may envision what we cannot see. In

Huayan Buddhism, the great god Indra expresses the shape of the universe with an infinite net.

At every node of the net, a jewel is sewn in, each jewel an individual entity, sparkling as every other jewel is reflected therein. Zooming in on one such reflection, we find a mirror image of the first jewel, and a complex causal relation begins to manifest that we can only glimpse sidelong.

A jewel is thus nothing but the labyrinthine conditions of pratītyasamutpāda, no self to speak of.

And the net itself is both twine and holes, form and emptiness coming together in Indra’s knotted jewel-worlds. In Professor Adams’s class we put on 3D glasses to watch a video simulation of flying through a hyperbolic universe. In the infinite crystalline structure, there’s a copy of myself in every cell. It’s like that experience of looking out a window only to see the back of your own head. Gwendolen Harleth of Daniel Deronda loves looking at herself in the mirror. The day she becomes the new mistress of Ryelands, she spills the casket of family diamonds onto the floor of her anteroom. Exiled from her sphere of existence, “she could not see the reflections of herself then; they were like so many women petrified white.” A merchant unrolls his carpet of baubles on 47th street, at my feet. My eyes gorge on the lode of light I cannot reach. I’m out of breath.

68 Acknowledgements

I would like to thank my advisor, Professor Cassandra Cleghorn, who showed me what creative nonfiction is all about two years ago, who planted the idea of a creative thesis in my head, and who had faith in this project when I didn’t. Dear Cass, thank you for encouraging me to write bravely and truly, for indulging the math, and for the infinite patience you have shown my sentences. You knead my shaggy dough together and save my bread. Thank you, also, for

Roget’s, for drafts grown hefty with sticky notes, for the night of rhubarb crumble and blood orange tea.

I thank Professor Colin Adams, from whom I lifted almost all of the knot and math factoids present in this writing, without having met whom I never would have studied math, let alone knots. Dear Professor Adams, I am terrified of sharing a copy of this with you tomorrow.

Thank you for coming in 8:30 in the morning as Mel Slugbate. The stars you sometimes draw on homeworks and the thrice-underscored “Yay!!!” you write on the board at the end of proofs are highlights of my day. Thank you for your kindness and your humor. Your tireless dedication and enthusiasm move me on a daily basis.

To Professor Bernie Rhie, many thanks for teaching me how to sit. Your Tuesday night series has become a raft, a staple. Thank you for pointing me to Jess Row’s Buddhist Essay class and, for always having encouraged me to write.

For helping me land on this project when I somehow submitted two proposals, I thank

Professor John Limon. For laughs and support, I thank my friends, Wendy and company.

This thesis, or indeed everything I do, is dedicated to my parents.

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