RESEARCH STATEMENT

THOMAS KINDRED

My approach to math research is highly visual, explicitly constructive, often combinatorial, sometimes com- putational: ask intuitive but challenging questions about deep ideas; use elementary methods, especially constructive, visual, and/or combinatorial ones, to attack these questions; persist; engage fully with the writing and revision process, refining insights to a level of clarity where they seem obvious; use this clarity to look for further insights, perhaps computational; share; ask more questions; continue in this way. My goal is not just to prove, but to understand and convey. My research stems, at least indirectly, from a fascination with spanning surfaces in S3, which occupy a perfect intersectional space, relative to my strengths and interests, from which to reach out to diverse areas:

• Spanning surfaces are versatile: nearly every aspect of low-dimensional topology can be understood in terms of and links in S3, and spanning surfaces offer a visual, constructive, geometric approach to nearly every part of theory. The basic idea is that the space around a knotted circle carries a surprising wealth of mathematical data, and spanning surfaces help parse this data. • Spanning surfaces invite explicitly constructive approaches, often involving elementary methods, often combinatorial in nature. These all play to my relative strengths as a researcher. • Spanning surfaces are fun! You draw them. You can hold them in your hands. Their visual appeal conceals their profound mathematical relevance in a way that reminds me, by analogy, of a quotation:

Music is the pleasure the human mind experiences from counting without being aware that it is counting – Gottfried Liebniz

Organization: Section 1 surveys the mathematical and historical foundations of my research. I hope that mathematicians from all backgrounds find it accessible. It can be treated as a self-contained survey of the main themes of my research. Section 2 surveys my eight completed papers, and my ninth paper, in preparation, and describes ways that I plan to extend them. Section 3 describes other likely directions for my future research. First, though, I want to highlight a recent result which conveys the overall flavor of my research. As I will describe further in §§1-2, in 1993 Menasco and Thistlethwaite used the to give the first proof of Tait’s 1898 flyping conjecture. I recently gave the first purely geometric proof of their theorem: Flyping Theorem: Any two reduced alternating diagrams of a given prime nonsplit are related by a sequence of flype moves. My proof stems from the insight that a flype move on a diagram corresponds to a geometric operation, which I call re-plumbing, on one of that diagrams’ chessboard surfaces (and an isotopy of the other chessboard ). Figure 1 illustrates this correspondence. This particular insight and the approach to the flyping theorem that grew out of it (and less directly all of my research) all grew out of my first week conducting undergraduate research, with Colin Adams at Williams’ REU SMALL in 2005. 1 2 THOMAS KINDRED

T T 2 2 2 T

T T 1 1

Figure 1. A flype move (top) on a link diagram corresponds to a re-plumbing move (bot- tom) on one of its chessboard surfaces (here, the black surface) and an isotopy of the other surface (here, the white surface).

1. Mathematical and historical background

A mathematical knot is an embedded circle in 3-dimensional space S3; a knot consisting of more than one circle is called a link. Most of what follows works for links, but I will focus on knots, where the definitions and theorems tend to be more straightforward.

The first knot theorist, Gauss, described how to rep- resent knots diagrammatically, as in Figure 1. Gauss also described how to represent any knot diagram by a sequence of positive integers: pick a basepoint and orient the knot; then walk along the knot, recording each new crossing with the next unused integer and each repeated crossing with the corresponding inte- 1 ger. To this day, the interplay between the combina- Figure 2. An dia- torics of knot diagrams and the topology of knots and gram. Using the indicated basepoint manifolds often reveals new deep and surprising con- and orientation, the Gauss code is 2 nections (e.g. see §2.7), and Gauss codes are perfect (1, 2, 3, 1, 4, 5, 6, 3, 2, 4, 7, 6, 5, 7). for computing (see §2.5 for examples).

The first systematic treatment of came almost a century later, when in 1898 Tait constructed a table of all alternating knots through ten crossings. Tait justified his tabulation entirely on intuition (the hard part was proving that knots that seem to be different actually are different). Tait conjectured [64]:

• Any reduced3 alternating diagram realizes its knot’s (minimal) crossing number. • All reduced alternating diagrams of a given alternating knot have the same .4

1In general, one uses the over/under information at the crossings to attache signs to these integers; when the diagram alternates between over- and under-crossings, this is unnecessary, subject to the convention that the first crossing is an over-crossing. 2and DT codes, which are half the length but carry the same information 3A knot diagram is reduced if it has no “nugatory” crossing: or more generally . 4Orient a knot diagram D arbitrarily. Then every crossing looks like or . The writhe of D is | | − | |. RESEARCH STATEMENT 3

• All reduced alternating diagrams of any prime5 alternating knot are related by flypes (see Fig. 1).

Interestingly, Tait was motivated by an erroneous belief in the vortex theory of atoms, which held that atoms were knotted circles in the ether; Tait hoped that his table might correspond to a table of the elements. Before long, physics abandoned the theory of the ether, in favor of Einstein’s general theory of relativity, and the vortex theory of atoms, in favor of quantum mechanics. Still, knot theory soon found new motivation in Dehn’s insight (circa 1910) that, by drilling out (thickened) knotted circles and gluing them back in “another way,” one could construct many different 3-manifolds. In 1960-62, Lickorish and Wallace used Heegaard splittings and mapping class groups of surfaces to prove that every closed orientable 3-manifold can be obtained via such Dehn surgery, thus implying that the study of 3-manifolds corresponds to the study of integer-weighted links in S3; the integer tells one how to glue the thickened knot back in. The advent of Kirby calculus in 1978 created a similar correspondence between closed oriented smooth 4-manifolds and integer-weighted links, this time in connected sums of S1 ×S2. The connection here is that performing Dehn surgery on S3 = ∂D4 corresponds to attaching a 2-handle to D4. (More on 4-manifolds at the end of §1.) Despite these fantastic reasons to study knots, Tait’s conjectures all remained unproven into the mid-80’s. This was especially striking, given that his tabulation was verified using discoveries from the 1920’s and 30’s:

• Reidemeister described three types of moves that relate all diagrams of a given knot. These moves provide a rigorous framework for developing knot invariants: numbers, polynomials, groups, etc. that one assigns to a knot in a way that doesn’t depend on the diagram. • Seifert proved algorithmically that every knot K is the boundary of a connected orientable (i.e. 2-sided) surface F embedded in S3. Such F is now called a for K. The minimum genus among all Seifert surfaces for K is called its classical genus, denoted g(K). • Alexander and Briggs discovered the first polynomial invariant for knots. One can compute the ∆(K) directly from a knot diagram, but ∆(K) has a geometric interpreta- tion: construct the infinite cyclic cover X of S3 \L by cutting along a Seifert surface, taking infinitely −1 many copies, and gluing them together; then H1(X) is isomorphic to Z[t, t ]/∆(K). In addition to verifying Tait’s tabulation, these discoveries opened up, and in some cases solved, new ques- tions. For example, in the 1950’s, Crowell and Murasugi independently used ∆(K) to prove that applying Seifert’s algorithm to any reduced alternating knot diagram yields a surface that realizes g(K) [18, 55]. Meanwhile, the two theories that had displaced the theory of the ether and atomic vortices had come into conflict, creating what to this day remains the biggest open problem in physics: how to “quantize gravity”? This would unify quantum mechanics and relativity, the theories which respectively govern the cosmically large and the immeasurably small. In the context of this conundrum and Tait’s still-unproven conjectures, a pair of revolutionary discoveries further ignited interest in the theory of knots. First, in the late 1970’s, Thurston realized that “most” knot complements admit a (unique) hyperbolic structure. Hyperbolic geometry, arises naturally in relativity as a “slice of the future,” the set of all points with a given time-like separation from, say, here and now. Second, Jones’ work in operator algebras (a vital tool in quantum mechanics) led in 1985 to the discovery of a new polynomial invariant for knots [39].6 The Jones polynomial quickly led to several proofs of Tait’s conjectures about crossing number and writhe [65, 42, 57, 68]. The proofs were surprisingly straightforward,

5A knot K is prime if it cannot be written as a , i.e. if, whenever an embedded sphere Q ⊂ S3 intersects K transversally in two points, one of the two arcs of K \ Q is unknotted. 6Jones made this discovery during a year studying operator algebras at MSRI; the other workshop that year at MSRI was on braids. This put Jones in close contact with Birman; their conversations played a crucial role in this seminal discovery. 4 THOMAS KINDRED

given that they resolved open problems of nearly 100 years; the Jones polynomial was strikingly and mys- teriously powerful! Shortly thereafter, Menasco and Thistlethwaite used geometric techniques together with new polynomial invariants to give the first proof of Tait’s flyping conjecture [54]. They noted:

The question remains open as to whether there exist purely geometric proofs of this and other results that have been obtained with the help of new polynomial invariants.

The value that our field places on these types of questions testifies to mathematicians’ drive not just to prove but to understand. (The root “mathematikos” means “one inclined to learn.”) In this spirit, Gabai gave an elementary geometric proof in 1986 of Crowell–Murasugi’s theorem about Seifert’s algorithm and alternating knots [18, 55, 26], and, in the 1950’s, Fox posed the following question:

What [geometrically] is an alternating knot? – Ralph Fox

In 2017, Greene and Howie [30, 34] independently answered Fox’s question by characterizing alternating knots in terms of the topology of their spanning surfaces, which are the same as Seifert surfaces, except they can be 1- or 2-sided. Given a knot diagram D ⊂ S2, one can construct two chessboard surfaces by coloring the disks of S2 \ D black and white in chessboard fashion and joining the disks of each color with a half-twist band at each crossing (see Figure 3). To explain their characterizations, we need some mathematical background (see Figure 4). Assume that F is a spanning surface for a knot K ⊂ S3. First, given a simple closed curve (“circle”) γ in F , the neighborhood of γ in F is a band—either an annulus or a mobius band. The framing of γ in F measures how much this band twists; more precisely it equals half the between the core and boundary of this band (with all curves co-oriented).7 The framing of K in F is called the boundary slope of F , denoted s(F ). Seifert surfaces have slope 0.

Second, β1(F ) = rank H1(F ) measures the complexity of F . Basically, β1(F ) counts the number of holes in F . More precisely, one can reduce F to a disk by cutting along some collection of disjoint properly embedded arcs; while there are many such collections, all have the same order, which equals β1(F ). Thus, β1(F ) is the number of cuts which reduce F to a disk.

Figure 4. The core circle of the Figure 3. Chessboard surfaces black surface B has framing 3/2; B has slope s(B) = 6 and β1(B) = 1.

3 A knot K ⊂ S is alternating if and only if it has spanning surfaces F+ and F− where:

• Howie: 2 (β1(F+) + β1(F−)) = s(F+) − s(F−) [34].

7If κ, λ ⊂ S3 are disjoint oriented circles, then κ bounds a Seifert surface S; the linking number of κ and λ equals the algebraic intersection number of S with λ, describing how far each curve is from being nullhomologous in the complement of the other. RESEARCH STATEMENT 5

• Greene: F+ is positive-definite, meaning that every circle γ ⊂ F+ either is positively framed in F+ or bounds an orientable subsurface of F+, and F− is negative-definite, defined analogously [30]. Greene proved, moreover, that a given pair of spanning surfaces for K can be realized as chessboard sur- faces for an alternating diagram if and only if they are definite of opposite signs. Greene then applied his characterization to give the first entirely geometric proof of part of Tait’s conjectures. This year, I applied both characterizations, along with other techniques, to give the first entirely geometric proof of Tait’s flyping conjecture; §2.7 has more detail about this result and related problems that remain open. Foremost among those other techniques are the notions of essential surfaces, generalized plumbing, and Menasco’s crossing ball structures. Here are the basic ideas.

Every knot K has many spanning surfaces F , but the most interesting are the simplest, with small β1(F ). Thus, one often requires F to be geometrically essential, admitting neither of these local simplifications:

3 A stronger condition requires F to be π1-essential, so that inclusion F,→ S \ L induces an injection of fundamental groups.8 The two conditions are equivalent for Seifert surfaces. Part of my doctoral thesis extends both conditions and compares their behavior, especially under plumbing (see §2.9).

Given spanning surfaces F1 for K1 and F2 for K2, there are many ways to glue F1 to F2. The sim- # = plest example is the boundary connected sum F1\F2, which spans the connected sum K1#K2; one can then decompose F1\F2 into its factors by cutting along a sphere. More generally, the operation of (generalized) plumbing, also called Murasugi sum, glues F1 to F2 along a disk in such a way that one can decompose (“de-plumb”) the resulting surface F = F1 ∗ F2 into 9 F1 and F2 by cutting along a sphere. ∗ = Figure 5 shows these operations and the related re- plumbing move, which replaces the disk in F1 ∗ F2 along which F1 and F2 were glued with another disk. Surfaces related by re-plumbing moves are generally non-isotopic in S3, but they are always related by proper isotopy through D4, where S3 = ∂D4.10 The central idea of my geometric proof of the flyping the- orem is that flype moves on alternating diagrams cor- Figure 5. Boundary connected respond to re-plumbing and isotopy moves on the di- sum, plumbing, and re-plumbing. agrams’ chessboard surfaces.

8As a technical note, one must also require that F is not a mobius band spanning the . 9Unlike boundary connected sum, this sphere cannot be transverse to F . 10When the plumbing is a boundary connected sum, re-plumbing is called spinning. Repeatedly spinning creates infinitely many surfaces; viewed as surfaces with boundary in S3, they are all equivalent (related by ambient isotopy) but viewed as properly embedded surfaces in the , the surfaces are generally all distinct. 6 THOMAS KINDRED

The other main technique I use in [7] translates topological problems about embedded surfaces in knot complements into combinatorial problems about curves on spheres. Menasco introduced this crossing ball technique and used it to verify many nice properties of alternating knots, including that all except the torus knots T2,q are hyperbolic and, with Thistlethwiate, that alternating knots satisfy the cabling conjecture (which is still open in general); this is also the main technique that Menasco-Thistlethwaite use (with an assist from the Jones polynomial) in their proof of the flyping theorem. Crossing ball structures are also central to two earlier papers of mine. My first paper, with Colin Adams, my undergrad thesis advisor at Williams College, introduces a way to construct many surfaces spanning any given knot and proves a classification theorem in the alternating case [1]. In particular, we determine the (the 1-sided analog of the classical genus) of all alternating knots and links. See §2.1. My paper about representativity, proves that any closed surface containing an alternating link must admit a compressing disk which intersects the knot in at most two points. See §2.3. To close §1, we return (separately) to two topics mentioned earlier: the Jones polynomial and 4-manifolds. What does the Jones polynomial mean? The Alexander polynomial has a straightforward interpretation in terms of branched covering spaces; does the Jones polynomial have a geometric interpretation vis a vis the topology and geometry of knot complements? Is there a nontrivial knot with the same Jones polynomial as the unknot? These questions remain open, as do the volume conjecture, which relates evaluations of colored Jones polynomials at roots of unity to hyperbolic geometry, and the slopes conjecture, which relates the degrees of “colored” Jones polynomials to boundary slopes of essential surfaces. I pursued the latter connection in [4], proving that essential spanning surfaces correspond to nonzero classes in the “categorified Jones polynomial” called ; see §2.4.11 Gay and Kirby introduced trisections of smooth 4-manifolds, which are decompositions into three 4-dimensional 1-handlebodies (boundary connected sums of S1 × D3’s) which nice intersection properties [28]. Using this construction, one can describe a closed 4-manifold by drawing three sets of curves on a surface, and one can describe any knotted surface in 4-space S4 by drawing three diagrams. Section 3 describes some topics in 4-manifold topology to which I hope to contribute in the near future. My most recent paper ad- dresses the more general notion of smooth multisections in arbitrary dimension. In particular, I prove that smooth multisections correspond to nice handle decompositions, and that many manifolds of even dimension n ≥ 6 admit no smooth multisection whatsoever: the obstruction is any nonzero homology group in odd dimension i 6= 1, n − 1. I also provide a detailed description of symmetric, efficient, smooth multisections of all odd-dimensional tori. See §2.8 for more details.

2. Papers

2.1. A classification of spanning surfaces for alternating links (with Colin Adams) [1]. My first research project began in 2005, at Williams’ REU, SMALL, where Colin Adams and I generalized Seifert’s algorithm in order to construct many spanning surfaces from a given knot diagram. At the time, the construction was new; the surfaces we constructed have since become known as state surfaces. We continued the project in an independent study, at SMALL 2006, and in my senior thesis. Using Menasco’s crossing ball technique, we classified spanning surfaces for alternating knots (and links) up to homeomorphism type and boundary slope. That is, we solved the geography problem in the alternating case: given a knot K, the problem is to list all pairs (s(F ), β1(F )) realized by spanning surfaces F for K. The geography problem remains open for most classes of knots and is a fertile area for future research.

11Witten’s interpretation of the Jones polynomial in terms of Feynmann path integrals only deepens the mystery. RESEARCH STATEMENT 7

As an immediate corollary, our classification gives the crosscap number and overall genus12 for all alternating links; previously, these were known only for 2-bridge links and certain pretzel knots. We also gave the first proof that all spanning surfaces for alternating links are connected and the first example of a knot whose overall genus can be realized by a 1-sided surface or by 2-sided surface. We explicitly computed the crosscap numbers of all alternating knots through 10 crossings and described an algorithm for computing the overall genus of an arbitrary alternating link. Kalfagianni and Lee applied this last algorithm to prove a 2-sided inequality relating the crosscap number of a knot to the degree span and penultimate coefficients of that knot’s Jones polynomial [40]. They used their bounds to compute the crosscap numbers of many 11- and 12-crossing alternating knots. See §2.5. Adams and I had tried to prove that certain adequate state surfaces were essential. While teaching high school math in Mississippi, I discovered how to prove this by showing that plumbing π1-essential surfaces always yields a π1-essential surface, but when I started grad school at Iowa, I found out that Ozawa had already published the same proof. Independently, Futer, Kalfagianni, and Purcell also proved that such state surfaces are essential, in a monograph in which they discovered deep structural relationships between state surfaces, hyperbolic geometry, and the Jones polynomial [22]. We had also tried to prove that every essential spanning surface for an alternating link is isotopic to a state surface. In fact this is false: Howie showed in his doctoral thesis that many alternating knots K have essential 0 0 0 0 13 spanning surfaces F+, F− for which s(F+) − s(F−) > 2c(K). Garoufalidis first noted the existence of such surfaces in the context of the slopes conjecture: the knots’ Jones slopes exceed their chessboard slopes[27].

2.2. Heegaard diagrams corresponding to Turaev surfaces (with Cody Armond and Nathan Druivenga) [2]. One builds a Turaev surface from a link diagram D on S2 ⊂ S3 by pushing the all-A and all-B states of D to opposite sides of S2, joining them with a cobordism whose saddle points occur precisely at the crossings of D, and capping off each state circle with a disk. Dasbach, Futer, Kalfagianni, Lee, and Stolzfus show that the resulting surface Σ is a Heegaard surface for S3, on which D forms an alternating diagram [19]. Our main theorem proves a converse to this fact, providing a correspondence between Turaev surfaces and certain link-adapted Heegaard diagrams [2]. Armond and Lowrance, and independently Kim, later applied this correspondence to characterize link dia- grams with low genus Turaev surfaces [10, 44]. Kalfagianni also applied our correspondence to characterize adequate knots in terms of their Jones slopes [41].

The Turaev genus gT (K), obtained by minimizing g(ΣD) across all diagrams D of K, measures “how far K is from being alternating” [49]; indeed, Turaev first used this construction to prove Tait’s first conjecture. The construction is related in an obvious way to the state sum formulation of the Jones polynomial, and gT (K) is related to this and other knot invariants, including Knot Floer homology and Khovanov homology [11, 15, 20, 21, 48]. The following problem remains open: Problem 2.1. Prove that the Turaev genus is additive under connected sum.

2.3. Alternating links have representativity 2 [3]. Gromov defined the distortion δ(γ) of a rectifiable curve γ ⊂ R3 by considering how many times farther apart two points are along the curve than in space: d (p, q) δ(γ) = sup γ 3 p,q∈γ dR (p, q)

12 The crosscap number of K is cc(K) = min{β1(F ): F is 1-sided and spans K}, and the overall genus is Γ(K) = min{β1(F ): F spans K}. 13Though essential, Howie’s surfaces are related by re-plumbing moves to surfaces which are inessential. 8 THOMAS KINDRED

Gromov asked whether knots have arbitrarily large distortion [31]. To show that they do, Pardon established a lower bound for distortion, 160δ(L) ≥ r(L), in terms of what is now called the representativity of L [61]: r(L) = max min |∂X ∩ L|. F ∈FL X∈XF

Here, FL is the set of positive genus closed surfaces containing L, and XF is the set of compressing disks for F . Ozawa computed the representativity of certain pretzel links and all torus and 2-bridge links, and conjectured that alternating links have representativity 2 [60]. I use crossing balls to confirm Ozawa’s conjecture: Theorem 2.2. If L ⊂ S3 is an alternating link and F is a closed surface that contains L (without crossings), then F has a compressing disk whose boundary intersects F in at most two points. Corollary 2.3 (Pointed out by Colin Adams). If F is a closed surface and L ⊂ F is a hyperbolic alternating link, then F \ L is not totally geodesic. Corollary 2.4 (A new proof of a theorem of Menasco). The only alternating torus links are the 2-braids T2,q.

Blair, Campisi, Taylor and Tomova bound δ(K) in terms of and distance, then apply my theorem to show that their bound improves Pardon’s for alternating knots with large enough distance [14].

2.4. Plumbing essential states in Khovanov homology [4]. In Viro’s interpretation of Khovanov ho- mology in terms of enhanced states, it is straightforward to see that alternating chessboard states correspond to nonzero Khovanov homology classes. These chessboard states are also essential, in the sense that they de- scribe essential surfaces. Diagrammatically plumbing alternating chessboards produces the homogeneously adequate state surfaces, all of them essential [59]. Based on these facts, I conjectured that these states might also correspond to nonzero Khovanov homology classes. After working examples, I discovered how to construct these nonzero classes explicitly. (Plamenevskaya’s distinguished element is a special case [62].) Theorem 2.5. If x is a homogeneously adequate state, then x gives nonzero Khovanov homology classes over 14 Z/2Z in two gradings. If a certain graph GxA is bipartite , then this is also true with integer coefficients.

After proving the theorem, I wanted to better understand the role of plumbing in Khovanov homology, at least in this class of examples. Had this gluing operation led me to a naive conjecture, which just turned out to be true, or was plumbing structurally important? I sought to express the nonzero classes from my theorem in terms of plumbing. Developing plumbing into an operation in Khovanov homology required a new rule of “trumps,” (to indicate when the label on one circle overrides the label on another) with which plumbing behaves roughly like interior multiplication composed with an exterior product,

| |x d(X ∗ Y ) = dX ♦∗ Y + (−1) X ∗♦ dY, leading to an alternate, inductive proof of the main theorem. This proof is much more complicated than the direct, constructive proof, but it does explain the main theorem in terms of the original, geometric motivation. I also pose the following open questions:

Question 2.6. If x is an essential state, does the associated Khovanov chain group, CF2 (x), always contain a (representative of a) nonzero homology class?

Question 2.7. If x1, x2 are essential states and both of CF2 (xi) contain nonzero homology classes, then must CF2 (x1 ∗ x2) also contain a nonzero homology class? Question 2.8. Does every link have a diagram with an essential state? A homogeneously adequate state?

14This corresponds to the A-smoothed part of x being a Seifert state. RESEARCH STATEMENT 9

I hope to extend these results in two directions. One is to investigate whether certain non-alternating essential chessboard states (such as those obtained by replacing each half-twist band in a chessboard state with a full-twist band15) also represent nonzero classes in Khovanov homology, and if so whether this fact extends via plumbing. The initial parts of the problem are computational and well-suited for collaboration with an undergraduate. (Lowrance led an undergraduate research project based on my paper.) Another is to study plumbings of essential chessboard states that yield chessboard states, which always take a particularly nice form; in the alternating case, they look like the “apparent” plumbings in Figure 6.

2.5. Crosscap numbers of alternating links via unknotting splices [5]. While my paper with Colin Adams determined the crosscap number and overall genus of every alternating link in theory, the problem remained open as to how best to compute these in practice. In 2018, Ito and Takimura defined a new diagrammatic u−(K) which I later called the splice- and proved that every knot satisfies cc(K) ≤ u−(K). For alternating knots K, I used tangle decompositions to prove the reverse inequality, giving cc(K) = u−(K) (Ito-Takimura independently proved this as well). Then I used this equality together with Gauss codes (an interesting complication arose while dealing with flype moves) to compute crosscap numbers of all alternating knots through 13 crossings (so far). Follow-up computational projects here are also well-suited for undergraduates. One idea is to compute the overall genus Γ(K) of alternating links K using another recent result of Ito-Takimura which holds that Γ(K) equals a related diagrammatic invariant Bl(K). Another idea is to study the space of all reduced alternating diagrams of a given knot, viewed as a graph with diagrams as vertices and flype moves as edges, especially in the context of splice-unknotting. I suspect that such analysis might help explain the emergence of knots whose Jones slopes exceed their chessboard slopes.

2.6. Nonorientable spanning surfaces for knots [6]. In this survey article, I proved a new version of the well-known fact that all Seifert surfaces for a knot are related by attaching and deleting tubes. Yasuhara later proved that all spanning surfaces for a knot (or link) are related by attaching and deleting tubes and crosscaps. Given a knot (or link) K, I proved that all chessboard surfaces from all diagrams of K are related by attaching and deleting crosscaps, and that all spanning surfaces for K with a given boundary slope are related by attaching and deleting tubes. This gives a fresh perspective on the geography problem for a given knot or link K: construct a graph whose vertices correspond to (isotopy classes of) spanning surfaces F for K, each labeled with (s(F ), β1(F )), and whose edges correspond to crosscapping and tubing moves. What does this graph look like? In the 2-bridge case, the answer follows directly from a paper of Hatcher-Thurston [32]. The alternating case seems challenging but approachable.

2.7. A geometric proof of the flyping theorem [7]. My proof of the flyping theorem is entirely geometric, not just in the formal sense that it does not use the Jones polynomial, but also in the more genuine sense that it conveys a geometric way of understanding why the flyping theorem is true. Namely, using Greene’s characterization, I leave the diagrammatic context for a geometric context, where I show that re-plumbing moves correspond to flypes (see Figures 1 and 6); then I prove that these moves are sufficiently robust. Specifically, suppose D and D0 are reduced alternating diagrams of a prime nonsplit link L with respective chessboard surfaces B,W and B0,W 0, where B and B0 are positive-definite. With this setup, I prove two theorems, which, together with Greene’s characterization, immediately imply the flyping theorem: Theorem 2.9. D and D0 are flype-related iff B and B0 are related by re-plumbing moves, as are W and W 0.

15The resulting state will be essential provided the original diagram admitting no n → 0 “pass move” htw. 10 THOMAS KINDRED

Theorem 2.10. Any essential positive-definite surface spanning L is related to B by re-plumbing moves.16

The second theorem is harder to prove than the first. The trick is to define a type of standard position, using crossing balls, that is asymmetric with respect to the projection sphere, so that an innermost circle on one side leads to a complexity-reducing re-plumbing move. To find the move, I repeatedly use a new lemma about the way that positive and negative-definite surfaces can intersect. The move itself is hard to visualize. Modified versions of some of these arguments hold in related settings: for alternating knots in thickened surfaces, and for other spanning surfaces for alternating knots in S3.

Figure 7. An irreducible alternat- Figure 6. If two positive-definite ing tangle diagram. Reflecting surfaces are related by re-plumbing across the sphere gives a reduced moves, then they are related by re- adequate diagram. Both diagrams plumbing moves of this form, which minimize crossings. correspond to flypes.

Related problems remain open. For example, while Greene proved that all reduced alternating diagrams of a given knot K have the same number of crossings (and the same writhe), he did not prove that they realize the crossing number c(K). All existing proofs of this fact use the Jones polynomial. Problem 2.11. Give a purely geometric proof that every reduced alternating diagram of a link K realizes c(K).

My basic approach is to understand a knot diagram as a geometric object, coming from a pair of intersecting spanning surfaces (as in Greene and Howie’s characterizations of alternating knots). The key is to impose some control on how the surfaces intersect, and to use the combinatorics of these intersections to understand other spanning surfaces for the knot. It also makes sense to consider Turaev surfaces in this context. Closely related open problems are: Problem 2.12. Give a purely geometric proof that every reduced adequate diagram of a link K realizes c(K). Problem 2.13. Give a purely geometric proof that every irreducible alternating tangle diagram minimizes crossings.

The same perspective also offers a sensible approach to the notoriously difficult: Problem 2.14. Prove that crossing number is additive under connected sum.

16Likewise for negative-definite surfaces and W . RESEARCH STATEMENT 11

2.8. Symmetric efficient smooth multisections of odd-dimensional tori [8]. Gay–Kirby’s trisections of 4-manifolds generalized the classical notion of Heegaard splittings of 3-manifolds, which are decompositions into two 3-dimensional handlebodies glued by a homeomorphism between their boundaries. Rubinstein and Tillmann further extended Gay–Kirby’s notion in the PL category by defining multisections of manifolds of arbitrary dimension. Lambert-Cole and Miller then showed that every smooth 5-manifold has a smooth trisection. My most recent paper extends all of this work by considering: Definition: Let X be a closed manifold of dimension n = 2k − 1 (resp. 2k − 2). A smooth multisection is a S T decomposition X = Xi, where, denoting XI = Xi for I ⊂ k: i∈Zk i∈I Z

• Each Xi is an n-dimensional 1-handlebody. • For I $ Zk with |I| ≥ 2, XI is an (n + 1 − |I|)-dimensional |I|- (resp. (|I| − 1)-) handlebody. • The central intersection XZk is a closed k- (resp. (k − 1)-) dimensional submanifold.

I establish the following general properties:

S Theorem 2.15. Let X = Xi be a smooth multisection of a closed manifold of dimension n = 2k − 1. i∈Zk Then X has a handle decomposition in which each Xi provides all the 2i- and (2i + 1)-handles.

S Theorem 2.16. Let k ≥ 3. If a closed (2k − 2)-manifold has a smooth multisection X = Xi, then i∈Zk X has a handle decomposition in which each Xi provides all the 2i-handles, and the only handles with odd-dimensional cores are the 1-handles from X0 and the (n − 1)-handles from Xk−1.

Therefore, in even dimension n ≥ 6, any nontrivial homology group in odd dimension i 6= 1, n − 1 obstructs S the existence of smooth multisections. It also follows that any smooth multisection X = Xi satisfies i∈Zk g(Xi) ≥ rank π1(X) for each i. I also investigate smooth multisections of odd-dimensional tori in detail. Roughly stated, the main result of that investigation is:

Theorem 2.17. Each n = (2k − 1)-torus admits an efficient smooth multisection which is symmetric with respect to the permutation action by Sn on the indices and the translation action by Zk along the main diagonal.

I also construct such a trisection of T 4, lift all symmetric multisections of tori to certain cubulated manifolds, and obtain combinatorial identities as corollaries.

2.9. Essence of a spanning surface [9]. The paper I’m currently writing, based largely on work from my thesis, extends the geometric and algebraic notions of essential surfaces by defining the essence ess(F ) (resp. the geometric essence essg(F )) of a spanning surface F in such a way that F is compressible iff ess(F ) = 0 and essential iff ess(F ) ≥ 2; the essence measures “how far” F is from being inessential. Extending a theorem of Gabai and later Ozawa, I prove that plumbing two π1-essential surfaces F1,F2 gives a surface F with ess(F ) ≥ mini=1,2 ess(Fi). I also prove that Gabai–Ozawa’s theorem does not extend from π1-essential surfaces to geometrically essential surfaces. Figure 8 shows a counterexample. isotopy

plumbing

12 THOMAS KINDRED

Figure 8. Plumbing geometrically essential surfaces can give a geometrically compressible surface.

I also consider the uniqueness of plumbing decompositions, proving that geometrically inessential surfaces almost never de-plumb uniquely, and likewise for many surfaces F with essg(F ) = 2. Conversely, I show that a state surface F from an alternating diagram de-plumbs uniquely if ess(F ) ≥ 4 (as do alternating chessboards, by [7]). I also use essg(F ) to describe a new, simple condition to improve the computation of crosscap numbers of alternating links.

3. Other future work

3.1. Plumbings of bands. Suppose a surface F spanning a K de-plumbs along disjoint spheres F Q = r Qr as a product of essential, unknotted annuli and mobius bands. Then K is called an arborescent link, and F is uniquely described by a directed tree with integer weights on the vertices [25]. In case this tree has only two leaves, K is a 2-bridge link, and all essential surfaces spanning K have this same form and are organized by paths in the Farey diagram [32]. Generic plumbings of these essential unknotted bands can be much messier. To wit: Question: Does every knot have a spanning surface obtained by plumbing essential unknotted bands? Conjecture: Every alternating knot is spanned by some plumbing of essential unknotted bands.

3.2. Free surfaces, chessboards, plumbing, and alternatingness. Every state surface can be realized as a chessboard surface, and every chessboard surface is free, meaning that its complement is a handlebody. I have promising approaches to each of the following problems: Conjecture 3.1. Every incompressible surface spanning an alternating link is free. Conjecture 3.2. Every spanning surface for an alternating link is related by re-plumbing moves to a surface which is either inessential or a state surface. Question 3.3. Which free surfaces can be realized as chessboard surfaces?

3.3. Spatial graphs. Menasco-Thistlethwaite define flypes as special pairwise homeomorphism (S3,K) → (S3,K). The flyping theorem implies that every such mapping for an alternating knot comes from flypes and symmetries of the diagrams. It follows that the mapping class group of any alternating knot complement is isomorphic to the group of pairwise homeomorphisms (S3, Γ) → (S3, Γ), where Γ is a spatial graph constructed from any reduced alternating diagram D of K by attaching a vertical arc at each crossing and attaching arcs in certain regions of S2 \D; each flype move exchanges one of the latter arcs with a vertical arc. In at least some cases, these groups are also isomorphic to the group of pairwise homeomorphisms (S3,S) → (S3,S) for a certain branched surface S. These correspondences rely in several ways on the alternating RESEARCH STATEMENT 13

condition. It is interesting to consider analogous structures coming from non-alternating diagrams. That investigation may tie in nicely to Problems 2.11-2.12, for example, by providing insight into the geometric meaning of crossings from knot diagrams.

1 3.4. Surfaces in 3 2 dimensions. Spanning surfaces offer a natural gateway to 4-dimensional thinking. It starts with pushing the interior of a surface F into the interior of the 4-ball D4, while fixing the knot 3 4 K = ∂F in S = ∂D . Gordon-Litherland describe a symmetric bilinear pairing h·, ·i on H1(F ) [29]. When F ⊂ S3, the self-pairing h[α], [α]i of a circle α ⊂ F equals twice its framing; when F is properly embedded in 4 D , the Gordon-Litherland pairing h·, ·i describes the intersection form on MF , the double-branched cover of D4 over F . Thus, all information coming from h·, ·i, including the slope and definiteness of F and the signature and determinant of K, can be understood either in terms of the spanning surface F ⊂ S3 or the 4-manifold MF . Thus, one can use 4-dimensional techniques to study spanning surfaces (and vice-versa). For example, this perspective immediately reveals that re-plumbing a definite surface yields a definite surface. Also, Alex Zupan and I are currently investigating the following question about spanning surfaces in this 1 “3 2 -dimensional” setting: 3 4 Question 3.4. View S as the equatorial sphere in S . Let F1 and F2 be spanning surfaces for a given 3 4 link K ⊂ S . Push the interiors of Fi, while fixing ∂Fi = K, into opposite 4-ball hemispheres of S . Is the resulting closed surface F1 ∪K F2 always unknotted?

(A surface in S4 is unknotted if it bounds a 3-dimensional 1-handlebody.) Like Problems 2.11-2.12, Question 3 3.4 comes down to understanding the ways that F1 and F2 can intersect in S (before we push them into opposite 4-balls). The following problem remains open: Conjecture 3.5 (The slice-ribbon conjecture). Any knot K ⊂ S3 that bounds a smooth properly embedded disk in D4 also bounds a ribbon disk in S3.17

3.5. Khovanov homology in terms of nonorientable Turaev surfaces in 4-space. Modify the con- 3 struction of a Turaev surface by pushing the all-A and all-B states xA and xB to opposite sides of S in 4 2 3 S (rather than to opposite sides of S in S ); then join xA and xB to one another through a cobordism 3 that contains the link L ⊂ S , and cap off xA, xB with disks so as to obtain a closed surface Σ satisfying 3 3 4 3 3 3 3 Σ ∩ S = L. Parameterizing a bicollar of S in S as S × [−1, 1], so that L ⊂ S ≡ S × {0}, xA ⊂ S × {1}, 3 3 and xB ⊂ S × {−1}; and letting h : S × [−1, 1] → [−1, 1] be projection onto the second factor, this cobor- −1 1 dism can be constructed to have critical values precisely at 2 , 2 . Some, but not all, of the corresponding critical points on Σ will be saddles. The rest of the critical points will look like this:

(1)

In particular, these Turaev-type surfaces will always be nonorientable. By perturbing h to obtain a generic morse function hˆ, spanning each non-critical hˆ−1(t) with a state surface, and taking closures, one obtains a (2+1)-dimensional cobordism M with boundary equal to the (1+1)-dimensional nonorientable cobordism ∂M = Σ. Because M contains 2-sided, nonorientable surfaces, it is nonorientable. While the setup and rein- terpretation are basically tautological, they lend themselves more directly to interpretation. Thus Khovanov homology carries information about closed surfaces in 4-space that our 3-space slices in the given link.

17A ribbon disk is an immersed disk whose self-intersections all take a particular form. 14 THOMAS KINDRED

3.6. Knotted surfaces in S4. The 4-sphere S4 contains no knotted circles, but it does contain knotted surfaces. Meier-Zupan show that one can represent any surface in S4 by a triplane diagram, which consists of three tangle diagrams on D2 (with the same arbitrary number of strands) and gluing information on their boundaries, such that each tangle is trivial and each pair of tangles glue to give an unknot. Thus, one can study gluings of trivial tangles in order to understand knotted surfaces. In fact, one could, in theory, completely characterize knotted surfaces in 4-space by solving the following problem for arbitrary n; the problem is open for all n ≥ 4:

Problem 3.6. Let T1 and T2 be trivial tangles in 3-balls with n strands. Characterize the gluings between the boundaries of these 3-balls which yield .

Most of what we know about knotted surfaces in S4 was discovered prior to 1980. In particular, the intervening decades have revealed few new constructions outside of the slice-ribbon context, and so most of our examples still come from spinning knots in S3 and twisted or deformed versions of this construction. My group at this past summer’s trisectors workshop found that even starting a tabulation of knotted surfaces is quite challenging. Problem 3.7. Describe new ways to construct knotted surfaces in S4.

Current work of Meier, Miller, and Zupan shows that every knotted surface S ⊂ S4 bounds a Seifert solid X, which is an orientable 3-dimensional submanifold of S4 with ∂X = S. Problem 3.8. Develop a theory of spanning solids for knotted surfaces in S4.

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