SHORT STORIES

Wild Wild Whitehead Danny Calegari

Take a knot. Take a tube around the knot. Put a new knot result can be shrunk down to a point. The track of the knot in the tube, twisted around and clasping itself as in Fig- 퐾1 during this process sweeps out an immersed disk. ure 1. The new knot goes once around the tube, and then In terms of the fundamental group, a knot 퐾 in a space “doubles back” and clasps itself. The new knot is the White- 푋 determines a conjugacy class [퐾] in the fundamental head double of the old. group 휋1(푋). Now, 휋1(푁0) = ℤ, and since 퐾1 bounds an immersed disk, [퐾1] is trivial in 휋1(푁0). We can keep going. Let 푁1 be a tube around 퐾1, thin enough to fit in 푁0, and let 퐾2 ⊂ 푁1 be the Whitehead double of 퐾1. And so on. Each 퐾푖 bounds an embedded disk in 푆3, but each of these disks must go (many times!) all the way outside 푁0. The tubes get thinner and thinner as we go, and longer Figure 1. The green knot is the Whitehead double of the black and longer. Consequently, the knots must get longer and knot. longer too: Each 퐾푛 must wind back and forth at least 푛 2 times around 푁0, clasping itself in a complicated way at the end. The infinite intersection ⋂ 푁푖 is called the Let’s call the first knot 퐾 and its Whitehead double 퐾 , 푖 0 1 Whitehead continuum, which we write Wh, see Figure 2. The and let’s call the tube around 퐾 (actually a solid torus) 0 Whitehead continuum is connected but not path connected. 푁 . It makes sense to take the Whitehead double of any 0 It has an entangled dyadic Cantor set of “strands” that wind knot, but in Figure 1, 퐾0 is a trivial knot; i.e., it bounds 3 around 푁0. an embedded disk in 푆 . In this case, 퐾1 is trivial too: it 3 3 The complement 푆 −Wh is an open 3-manifold called bounds an embedded disk in 푆 . But 퐾 is knotted in 푁 . 3 1 0 the Whitehead manifold. It turns out that 푆 − Wh is con- Any embedded disk that 퐾 bounds must go outside 푁 . 1 0 tractible but not homeomorphic to a 3-ball. Let’s see why. On the other hand, 퐾1 is homotopically trivial in 푁0; i.e., 3 ′ The outside of 푁0 in 푆 is another solid torus 푁0, whose it bounds an immersed disk, one that crosses itself, but does ′ ′ core is a knot 퐾0 linking 퐾0 in a Hopf link. [퐾0] is the not cross 푁0. To see this, just push one of the clasps of 퐾1 3 ′ generator of 휋1(푆 − 푁0) = 휋1(푁0) = ℤ. The knots through the other one; this undoes the knotting, and the ′ 퐾0 and 퐾1 together form a 2-component link called the Whitehead link. This link is symmetric: we can isotope it Danny Calegari is a professor of mathematics at the University of Chicago. His 퐾′ 퐾 email address is [email protected]. around and interchange 0 and 1; see Figure 3. 3 ′ For permission to reprint this article, please contact: Since [퐾1] is trivial in 휋1(푁0) = 휋1(푆 − 푁0), it ′ 3 [email protected]. follows by symmetry that [퐾0] is trivial in 휋1(푆 − 푁1). 3 3 DOI: https://doi.org/10.1090/noti1837 Consequently the inclusion map 푆 − 푁0 → 푆 − 푁1

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Figure 2. Successive approximations converge to Wh.

Nudging adjusts points in 푁1 ×ℝ by sliding each point ×ℝ slightly backward or forward in time. Nudging ex- tends to a self-homeomorphism 휈 of 푁0 × ℝ, fixed on the boundary.

Figure 3. The Whitehead link is symmetric.

induces the zero map on 휋1. Each 푁푖 is unknotted in 3 푆 , and each 푁푖+1 sits in 푁푖 the same way that 푁1 sits 3 in 푁0. So each 휋1(푆 − 푁푖) = ℤ, and each inclusion 3 3 푆 − 푁푛 → 푆 − 푁푛+1 induces the zero map on 휋1. Tak- 휋 (푆3 − ) ing a direct limit, 1 Wh itself is trivial. A similar Figure 4. Folding the clasps of each 푁1 back and forth in time 3 argument shows that all the homotopy groups of 푆 −Wh nudges the union of all 푁1’s into a collection of spirals. vanish, and it is contractible. 푆3 − − 푁′ = On the other hand, the complement Wh 0 By the way, there’s not just one spiral—there’s a circle’s 푁0 − Wh has an infinitely generated fundamental group; worth of them, filling the whole of 푁1 × ℝ. Two slices each 휋1(푁푖−푁푖+1) is complicated (it contains free groups 휈(푁1 × 푡), 휈(푁1 × 푠) are in the same spiral if and only if of every rank), and all of them include as subgroups of 푡 − 푠 휖/2 3 is an integer multiple of . 휋1(푁0 − Wh). This shows that 푆 − Wh is not a ball. 3 After nudging, the next move will straighten out this Now let’s compactify 푆 − Wh by adding a single point and every other spiral so that its projection to the 푆3 factor at infinity. This compact space can also be thought ofas < 휖/2 3 is small (let’s say for concreteness it has diameter ) the quotient space 푆 /Wh that we get by crushing Wh to a without affecting the projection to the ℝ factor. 3 3 single point. Because 푆 − Wh is not a ball, 푆 /Wh is not 3 The cylinder 퐾0 × ℝ ⊂ 푆 × ℝ has polar coordinates a manifold. However—remarkably—it is a manifold factor: (휃, 푡) where 휃 ∈ ℝ/ℤ. Extend these polar coordinates to a (푆3/ ) × ℝ 푆3 × ℝ the product Wh is homeomorphic to ! small tubular neighborhood of 퐾0 ×ℝ containing 푁1 ×ℝ, How can this possibly be??? with closure contained in the interior of 푁0 × ℝ. First, each 푁1 slice can be unknotted by a tiny perturba- We can “untwist” every spiral simultaneously by the map tion in 푁0 × ℝ. To distinguish the ℝ factor and, for the sake of brevity, we refer to it as the “time” coordinate (this (휃, 푡) → (휃 − 2푡/휖, 푡) is purely a notational convenience). In this language, we unclasp 푁1 from itself by nudging one clasp very slightly on our small tubular neighborhood. Twisting extends to forward into the future, and the other very slightly back a self-homeomorphism 휏 of 푁0 × ℝ, once again fixed on into the past. After the nudge, 푁1 will not clasp itself, but the boundary. it will clasp a “future” 푁1 on one side, and a “past” 푁1 In summary, first we nudge, then we twist. After doing on the other. Instead of 푁1 clasping itself in a circle, we this, every 휏휈(푁1) slice projects to subsets of diameter at 3 get a chain of successive 푁1’s, each clasping the next, in a most 휖/2 in both the ℝ and the 푆 directions. So 휏휈(푁1) slowly ascending spiral. Let’s let 휖/4 be the size of the per- has diameter at most 휖. turbations of each clasp in the time direction, so that the In other words, ℎ1 ∶= 휏휈 simultaneously shrinks all 3 projection of each 푁1 to the time coordinate after it’s been the 푁1 slices in 푆 × ℝ as small as we like, while keeping 3 nudged has total length 휖/2. (푆 − 푁0) × ℝ fixed pointwise.

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n interactive, dual-panel reading experience Figure 5. Screw the top and bottom in opposite directions like for journals published by the American you’re taking the lid off a pickle jar. A Mathematical Society. Take a sequence 휖푖 → 0, and repeat this operation for AMS MathViewer’s responsive HTML format each 푖 > 1 in place of 1 with 휖푖 in place of 휖. We get a se- perfectly renders mathematical content on 3 3 desktop, tablet, or phone. Click on embedded quence of self-homeomorphisms ℎ푖 ∶ 푆 × ℝ → 푆 × links to formulas, theorems, gures, and references ℝ, each supported in 푁 × ℝ, as a composition of a 푖−1 to view them in the second panel, in tandem with nudge-and-twist ℎ푖 ∶= 휏푖휈푖. Each 푁푖 slice gets smaller the original text. and smaller in diameter as we apply consecutive ℎ푖’s. Un- der application of successive ℎ푖’s, the orbit of every point is a Cauchy sequence, and the infinite composition 3 3 ℎ ∶= lim ℎ푖 ℎ푖−1 ⋯ ℎ1 ∶ 푆 × ℝ → 푆 × ℝ 푖→∞ is well-defined and continuous. For any compact subset 푋 of 푆3 − Wh the restriction of ℎ to 푋 is the composition of finitely many homeomor- phisms, so the restriction of ℎ to (푆3 −Wh)×ℝ is a home- omorphism. On the other hand, each Wh slice is succes- sively shrunk smaller and smaller by successive ℎ푖, so in the end ℎ crushes each Wh slice to a point, and ℎ factors as ℎ = 푔휋, 휋 푔 푆3 × ℝ −→(푆3/Wh) × ℝ −→푆3 × ℝ, where 휋 ∶ 푆3 × ℝ → (푆3/Wh) × ℝ is the quotient map, and 푔 is the homeomorphism we’ve been looking for.

AUTHOR’S NOTE. The main theorem in this article CURRENTLY AVAILABLE FOR: and its proof are both well-known, and not due to me! Transactions of the AMS—Series B They are due to J. Andrews and L. Rubin, Bull. Amer. Math. Soc. 71(1965), 675-677. Proceedings of the AMS—Series B Journal of the American Mathematical Society

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Credits All images are courtesy of Danny Calegari.

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