# Postcard from a Geometric Topologist

### These postcards are ungrouped by topic and represent a range of beautiful ideas -- we hope you enjoy!

Image description: The figure-eight knot (shown as a blue curve) embedded in the three-page book (shown as three red discs attached to a black binding circle). Caption: Consider the three-page book $T\subset S^3$, which is built by attaching three distinct discs to a binding circle. Instead of projecting knots in $S^3$ to a $2$-sphere, another way to combinatorially study knots is to \emph{embed} them in $T$. For example, the picture on the left shows an embedding of the figure-eight knot in $T$. We can generalise this approach to knots in an \emph{arbitrary} closed $3$-manifold $\mathcal{M}$ by replacing $T$ with a $2$-dimensional \textbf{spine} of $\mathcal{M}$. One key feature of a spine $S$ is that we lose no information if we forget about how $S$ is embedded in $\mathcal{M}$: we can recover this information by thickening $S$ to get a $3$-manifold with spherical boundary components, and then filling these sphere boundaries with balls (you can check that this works for $T$ in $S^3$). After putting an embedding of a knot $K$ in a spine in ``general position'', we get an \textbf{arc diagram} for $K$. The first natural problem is to find a small collection of local moves that relates any two topologically-equivalent arc diagrams on a fixed spine. This has only been done very recently, in \texttt{arXiv:2202.02007}, which really highlights the fact that the idea of projecting knots to spines is still largely untapped. With all that being said, the real reason I like this so much is simple: it's an excellent excuse to draw lots of pretty pictures!

Text: Dear Reader, Have you heard the following remarkable result of Matveev? Theorem: Two n-component links in the 3-sphere have the same pairwise linking numbers if and only if they are related by a sequence of Borromean clasper moves. Whatever will they think of next!? Yours in GT, A. N. Miller. P.S. Rumor has it that one can realize this operation by a certain regluing of the standard Heegaard splitting of the 3-sphere by an element of the Torelli group--delightful if true! Image description: Three collections of arcs are tied into the Borromean rings (left), three examples of Borromean clasper moves on the 3-component unlink (right).

The graphic is a digital version of a chalk board. An infinite-type surface is shown with four curves; alpha, f of alpha, f squared of alpha, and f cubed of alpha, in different colors. The curves are disjoint from one another and wind around the topology of the surface. Handwritten text reads: Given an infinite-type surface S and an end-periodic homeomorphism f on S, iterates of a separating curve converge to an f-invariant lamination on the surface. Field, Kim, Leininger, Loving '21 showed that if f is an irreducible end-periodic homeomorphism, then its mapping torus M_f admits a hyperbolic metric. In a thought bubble reads: What might these laminations say about the geometry of M_f?.

Image Description: Montesionos knot with four rational tangles with two 4-punctured spheres drawn in red and blue. Caption: Montesinos links are made by connecting rational tangles with parallel arcs in a cyclic fashion. Oertel gave a complete classification of the closed essential (possibly disconnected) surfaces in the complement of such links via certain 4-punctured spheres (shown in red and blue) and tubes. For a certain infinite class of Montesinos knots, we count the exact number of connected such surfaces, up to isotopy, and show that there are 12 genus two surfaces and 8\phi(g-1) genus g surfaces, where \phi(g-1) is the Euler totient function.

Image description: A graph immersed in a nonagon. Caption: Let $S$ be a closed surface with a negatively curved Riemannian metric and consider its unit tangent bundle $T^1 S$. The \textit{geodesic flow} on $T^1 S$ is the flow $\phi^t$ that takes the time-$0$ tangent of each unit-speed geodesic $\gamma$ to its time-$t$ tangent at time $t$, i.e. $\phi^t(\gamma'(0))=\gamma'(t)$. In particular, the orbits of the geodesic flow are in one-to-one correspondence with the geodesics on $S$. From the general theory of \textit{Anosov flows}, it is known that the geodesic flow admits a \textit{Markov partition}. These can be thought of as directed graphs $\Phi$ that `remember' the dynamics of the flow. For example, the closed orbits of the flow are in correspondence with the cycles of $\Phi$. One can then attempt to study the closed geodesics on $S$ by just studying such a graph $\Phi$. However, one difficulty with this approach was that explicit examples of such Markov partitions were hard to come by. In a recent preprint, I reduced this barrier by constructing a Markov partition given any multicurve that cuts $S$ up into $n \geq 4$-gons. The corresponding graph is determined by its projection in each $n$-gon, for which a recipe can be given. The picture on the left is such a projected graph for $n=9$.

Greetings from California! We're studying double twist knots and their nonorientable 4-genera. Here's one family: Image description: A double twist knot K = C(m,n) where m = 22+8k and n = 62+8k. A nonorientable surface F that has first Betti number 3 which has boundary K. Caption: This surface is as good as it gets! Theorem. Let m = 22+8k and n = 62+8k and k ≥ 0. Then \gamma_4(C(m,n)) = 3. See our arxiv paper for more. Cheers, Cornelia Van Cott, Pat Shanahan, and Jim Hoste P.S. Can you find \gamma_4(C(2,12))?

Image description(s): KH Table of T(2,3), Diagram of smoothings for T(2,3), Projection of T(2,3) . Caption(s): In Khovanov Homology, it can often be helpful to look at the homology classes of knots through a visual table. Here, I have included the right handed four crossing projection of the trefoil knot (T(2,3)). To create these tables, we start with a standard projection and smooth each crossing. At each crossing, we make a binary choice of either creating an “A” smoothing or a “B” smoothing. Once all crossings have been smoothed, we are left with 2c(K) states (where c(K) is the crossing number of our given knot). Once we have these states, we enhance them with either green or blue colorings to particular state circles (pieces of the state). These tables can provide a visual guide to where our homology classes are, and how these visuals may differ between different projections. Our non-zero classes’ location correspond with the Jones Polynomial for the given knot.

Image: Bottom: a round hyperbolic plane maps to (covers) a hyperbolic surface, which maps totally geodesically into a hyperbolic 3-manifold. Top: a distorted hyperbolic plane, built from geodesic ideal triangles which are glued with some bending, covers a hyperbolic surface, which maps into a hyperbolic 3-manifold as a pleated surface. (Note: The "bending" doesn't really make sense except in a larger ambient space, e.g. hyperbolic 3-space), Caption: A pleated surface is a hyperbolizable surface S that is mapped into a hyperbolic 3-manifold M geodesically except along some collection \lambda of simple geodesics in S, which are still mapped to geodesics in M. Informally, the surface is "pleated" or "bent" along the "folds" given by the geodesics in \lambda. Under mild conditions on \lambda, a pleated surface is uniquely associated to a homomorphism from the fundamental group of S to Isom^+(H^3). Using this, pleated surfaces can be used to parametrize representations of surface groups into Isom^+(H^3).

Exotic behavior in dimension-four is ``unstable''. For example, a classic result of Wall states that simply connected, oriented, exotic four-manifolds become diffeomorphic after sufficiently many stabilizations, i.e. taking connected sums with $S^2\times S^2$. There is an analogue for exotic surfaces, which states that two exotic surfaces will become smoothly isotopic after some number of stabilizations, which in this context means attaching tubes. Conjecturally, a single stabilization should always be enough to dissolve the exotic behavior.

However, this need not be true in the relative case. The knot on the left bounds a pair of exotic disks (one is shown, and the second is obtained by rotating the knot and bands by 180 degrees). The two disks remain exotic until the surfaces are stabilized three times, as in the figure.

Image description: the knot shown bounds an exotic pair of disks. A handle decomposition of one of the disks is shown, together with a minimal collection of tubes which dissolves the exotic behavior.

# Hong Chang

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Image description: Possible ways to identify edge $\alpha$ on the polygon when $g=3$. (See caption for explanation) Caption: M\'enage Problem is asked by \`Edouard Lucas and Peter Tait and Tait's expression was ``the numbers of arrangements such that there are of n letters, when A cannot be in the first or second place, B not in the second or third'' (such arrangements can be regarded as permutations called \emph{m\'enage permutation}). Gilbert, E. N. modified Tait's problem further as the the number of conjugacy classes of m\'enage permutations up to a cyclic permutation. Let $\alpha$ and $\beta$ be two minimal filling coherent pairs (origami pairs) on $S_g$. When $g\ge 3$, $S_g\backslash\alpha\cup\beta$ will consist of only one component, which is a $4(2g-1)$-gon with 2 copies of $\alpha$ and 2 copies of $\beta$ as the sides. For two identified edges of $\alpha$ on the $4(2g-1)$-gon, we can prove their distance (on the polygon, ignoring edges of $\beta$) is always odd but not 1. Since we can rotate the polygon and relabel the numbers, counting the ways to identify edges is just the Gilbert's modified m\'enage problem. The figure left is showing possible ways to identify $\alpha$ when $g=3$ (there is also a case where all edges are identified with its opposite but this is also impossible). Edges belong to $\beta$ is not shown in the figure so it's a decagon instead of a 20-gon. Further research shows only the bottom-left case will really yield an origami pair so there is only one origami pair for $S_3$.

# Kailey Perry

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Image description: A sequence of folds on a subdivided 3-rose. Caption: An example of a train track: Let φ be the map F3 → F3 that sends a → b → c → ca^{−1}b^2. We can represent φ as a map from the 3-rose to itself, as in the top left. This is not a train track, as φ^2(c) = ca^{−1}b^2b^{−1}c^2, which is not reduced. But, since the map is irreducible, there is an algorithm we can use to make a train track that represents φ, φ′ given by a → b → c_1a, c_1 → c_1b, as in the bottom left. For more, see "Train Tracks and Automorphisms of Free Groups" by Mladin Bestvina and Michael Handel. Kailey Perry, University of Arkansas

Image description: A stick figure fights a hydra labeled by a2 a3 a4 a1 a3 a2 a3. It cuts off its head, removing the first letter, a2. Then the hydra regrows the rest of its remaining letters, now being labeled a3 a2 a4 a3 a1 a3 a2 a2 a1 a3 a2. Caption: A hydra, as defined by Dison and Riley, is a finite-length word on the alphabet a_1, a_2, ..., a_n. A hydra is defeated by having all of its letters struck off, one by one, from left to right. But every time its left-most letter is removed (its head is cut off), the rest of the letters regrow: a_i becomes a_ia_{i-1}, except for a_1, which stays the same. All hydra are eventually defeated, but the recursive nature of its regrowing heads makes it a fearsome foe. Realized as groups, the hydra groups have presentation G_k = < a_1, ..., a_k, t | t^{-1}a_1t = a_1, t^{-1}a_i t = a_i a_{i-1} for all i>1 >. G_k is free-by-cyclic, CAT(0), and biautomatic, yet its subgroups H_k = < a_1 t, ... , a_k t> are free of rank k with wild distortion: comparable to A_k, where A_k are Ackermann's functions. For more info, see ``Hydra Groups" by Dison and Riley, or ``Hyperbolic Hydra" by Brady, Dison, and Riley for the hyperbolic version!

The postcard shows a picture of the universal cover of a strictly hyperbolized manifold. (The image looks like a tree like structure, with a dual square complex overlapped on top of it.) These manifolds are negatively curved, but in general they can be quite strange, and not actually hyperbolic. For instance, there is one in every cobordism class... However, their fundamental groups are very nice (= specially cubulated). In particular, they are linear and often even algebraically fibered. So, they feel like hyperbolic 3-manifold groups! Have a look at my preprint with J. Lafont: arXiv:2206.03620

These 1,215,034 points are the zeroes of the Jones polynomial for knots (up to 15 crossing-number) in the complex plane within the unit half-disk. It is known that eventually for all knots the points will be dense in this half-disk.

“A knot cobordism in R^3 x I can be generically projected to R^2 x I and given crossing information to create a broken sheet diagram. The singular set of these diagrams consists of double points, triple points, and branch points. At each double point the sheet that lifts below the other, based on the projection, is locally broken. Quandles can be used on these diagrams to calculate how many Reidemeister III moves needed to unknot trivial link diagrams.”

# Pier-Olivier Rodrigue

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Image description : Figure 1 : Triangles tessellation of the upper half plane by (2,4,6) triangles. Figure 2 : Tesselation of $\mathbb{RP}^3$ by the surface formed by three parts of hyperboloids. Text : Choose three positive integers $p,q,r$, such that $ \frac{1}{p}+\frac{1}{q}+\frac{1}{r} < 1$ to construct a triangle $T$ in $\mathbb{H}^2$ with angles $\frac{\pi}{p}$,$\frac{\pi}{q}$ and $\frac{\pi}{r}$. Let $R_1,R_2,R_3$ be the reflections in each side of $T$. By applying the reflections $R_i$ to $T$, new triangles $R_i(T)$ are created. Due to our choices of angles, the reflections satisfy $(R_1 R_2)^p = (R_2 R_3)^q = (R_1 R_3)^r = Id$. We can use this process to obtain a tessellation of $\mathbb{H}^2$ by triangles as in Figure \ref{fig: figure 1}. Now, identify $\mathbb{RP}^3$ with $T^1\mathbb{CP}^1$ to create a tridimensional analog of the above construction. Consider $\pi$ the natural projection of the unit tangent bundle and $\ell$ a geodesic in the upper-half plane model of $\mathbb{H}^2$. The preimage $\pi(\ell)^{-1}$ is a one sheeted hyperboloid in $\mathbb{RP}^3$. Each triangle will then correspond to a surface formed by three parts of hyperboloids. Consider the map $\xi : \mathbb{RP}^1 \rightarrow Lag(\mathbb{R}^4)$ defined by $\begin{bmatrix} t\\s\end{bmatrix} \mapsto \begin{pmatrix}t & 0\\0 & t\\s & 0\\0 & s\end{pmatrix}$. Let $\ell$ in $\mathbb{H}^2$ be a geodesic whose end points are $a,b \in \mathbb{RP}^1$. It defines the pair of Lagrangians $\xi(a),\xi(b)$. We use the symplectic form $dx_1 \wedge dx_3 + dx_2 \wedge dx_4$ for our Lagrangians. To the hyperbolic reflection in $\ell$, we associate the Lagrangian reflection $R_{\xi(x),\xi(y)}$ which preserves the corresponding hyperboloid. We then apply the reflections on each side of the surface and draw the results.

Image: an illustration of the complement of a closed ball in R^3, with a curve gamma along with a "null-homotopy" of gamma in the complement. Text: A space X-tilde is n-connected at infinity if for every compact set K contained in X-tilde, there exists a compact set C contained in K such that every map from the n-sphere into X-tilde minus K is nullhomotopic in X-tilde minus C. The figure illustrates the general fact that R^n is (n-2)-connected at infinity. If G is the fundamental group of X, where X is, say, a closed manifold with universal cover X-tilde = R^n, then we say that G is (n-2)-connected at infinity, and in our case the (co-)homology of G satisfies Poincaré duality. As it happens, G having cohomological dimension n and being (n-2)-connected at infinity implies that G satisfies a generalization of Poincaré duality due to Bieri and Eckmann. —Rylee Lyman, Rutgers University–Newark

Image description: The cartoon character from the Ikea instruction manuals stands next to two translation surfaces saying "So... how do I assemble this eigenform?" The first translation surface is slim and jagged, whereas the second is a filled-out L-shape. There is a yellow arrow pointing from the first plot to the second plot, indicating that one surface can be assembled into the other with the right choice of straight-line cut-and-pastes. The left polygon is the image of the right polygon under an element of its Veech group, the group of invertible matrices preserving the polygon up to cuting-and-pasting. The particular element is a B_2 butterfly move, an element we use in our forthcoming paper "Periodic points of Prym eigenforms" to classify the points on certain translation surfaces called Prym eigenforms that have finite orbit under their affine automorphism groups.

Image description: A flow chart of classification of infinite graphs, whose pure mapping class groups are CB or locally CB. Title: CB & locally CB classification of Pure Mapping Class Group of Infinite Graphs. (Joint with George Domat (Rice Univ.) and Hannah Hoganson (Univ of Maryland)) This work is posted on arXiv: https://arxiv.org/pdf/2201.02559.pdf. Caption: Let 𝛤 be a locally finite infinite graph. Algom-Kfir and Bestvina defined Maps(𝛤) as "Graph analogue of Big MCG of Surfaces" (Note it's not Out(F_\infty). e.g. see https://arxiv.org/abs/2207.12518). We completely classified graphs with coarsely bounded(CB) / locally CB PMaps(𝛤), the subgroup of Maps(𝛤) fixing the ends of 𝛤. Bottom right text: Sanghoon Kwak(University of Utah) For more, see Figure 1,2 of https://arxiv.org/pdf/2201.02559.pdf.

Figure upper right: A photo of a piece of cloth with perpendicular green (vertical) and purple (horizontal) threads. A giant pin at the lower left corner represents a "cusp". Figure lower left: A line art image of a flow box decomposition. The tops of the boxes are rectangles A and B. These (subdivided and hyperbolically distorted) reappear on the bottoms of the boxes. Text upper left: The figure-eight knot complement M is a punctured-torus bundle over the circle with monodromy F given by the matrix [2, 1; 1, 1]. That is, M has the form (T² - pt) × [0, 1] / (x, 1) ~ (F(x), 0). The intervals of the second factor descend to give the _suspension flow_. Text upper right: Saul Schleimer, University of Warwick, Henry Segerman, Oklahoma State University. Text mid right: The eigenfoliations of F give the universal cover of T² - pt the structure of a _loom space_ (as commemorated in our stamp). Text lower right: The rectangles A and B form a Markov partition for F; each flows upwards to give a _flow box_. The result is a _flow box decomposition_if M.

Image description: A genus 3 surface with two transverse subsurfaces U and V. Second part shows a thin cross like region in the product of the curve complexes of U and V. Caption: Given two subsurfaces $\mathcal{U}$ and $\mathcal{V}$ in a hyperbolic surface, one can consider the \emph{subsurface projections}, $\pi_{\mathcal{U}}$ and $\pi_{\mathcal{V}}$, of any simple closed curve $\gamma$ to the curve complexes $\mathcal{CU}$ and $\mathcal{V}$. If $\mathcal{U}$ and $\mathcal{V}$ are not disjoint and not nested in one another, then $\pi_{\mathcal{U}}(\gamma)$ and $\pi_{\mathcal{V}}(\gamma)$ cannot simultaneously be too far from $b_{\mathcal{U}}$ and $b_{\mathcal{V}}$. This means that a geodesic triangle in the Teichmüller space which has one edge with large projection to $\mathcal{CU}$ and one edge with large projection to $\mathcal{CV}$ must lie entirely in the green region when projected to $\mathcal{CU} \times \mathcal{CV}$, and thus be thin. The thinness can be upgraded to thinness in Teichmüller space very often, providing enough hyperbolicity for negative curvature techniques to work in the setting of Teichmüller space.

Image descriptions: 60% of the postcard is covered by a code-generated Mondrian-style artwork depicting the Conway knot. The top right of the postcard has some New Zealand postage stamps. Text: "Hey! How's it going? This is a Conway knot in the style of Piet Mondrian. You can generate your own knot-art at the address github.com/sfushidahardy/grid_to_mondrian. Have fun! Shintaro"

Image description: (2,1)-cable of the figure-eight knot. Caption: The figure describes the (2,1)-cable of the figure-eight knot; Kawauchi asked in 1980 whether this is slice. This question is important because it serves as the simplest possible potential counterexample to the slice-ribbon conjecture. We prove that this knot is not smoothly slice.

Image description: The knot Floer homology of the right-handed trefoil (on the left) and the knot Floer homology of the (2,3)-cable of the right-handed trefoil (on the right).

Text on postcard:

Greetings from Atlanta!

$\sum_{m \in \mathbb{Z}} \dim \widehat{HFK}_m ((T_{2,3})_{2,3}, -2) > \sum_{m \in \mathbb{Z}} \dim \widehat{HFK}_m (T_{2,3}, -2)$

$\sum_{m \in \mathbb{Z}} \dim \widehat{HFK}_m ((T_{2,3})_{2,3}, -1) < \sum_{m \in \mathbb{Z}} \dim \widehat{HFK}_m (T_{2,3}, -1)$

$\sum_{m \in \mathbb{Z}} \dim \widehat{HFK}_m ((T_{2,3})_{2,3}, 0) = \sum_{m \in \mathbb{Z}} \dim \widehat{HFK}_m (T_{2,3}, 0)$

$\sum_{a \in \mathbb{Z}} \dim \widehat{HFK}_m ((T_{2,3})_{2,3}, a) > \sum_{a \in \mathbb{Z}} \dim \widehat{HFK}_m (T_{2,3}, a), \forall m \in \mathbb{Z}$

$\dim \widehat{HFK} ((T_{2,3})_{2,3}) > \dim \widehat{HFK} (T_{2,3})$

See arXiv:2207.01787 for a general result about (1,1)-satellites.

Weizhe Shen, Georgia Tech

A parallelogram with vertices labelled (1,1), (1,-i), (i,-\sqrt(i)), and (i,\sqrt(i). There are many colored lines running through the parallelogram and bouncing off the sides, some labeled E_8, E_7, E_6, F_4, O(8), G_2, PSL(3), PL(2), SOSp(1|2), and Triv. There are also dashed lines.

Text: Parameter space (v,w) for a conjectural 2-variable knot polynomial, similar to HOMFLY-PT or Kauffman polynomials. If it exists, it specializes to 1-variable polynomials related to each exceptional Lie algebra.

Dashed lines are places the theory is singular or degenrate.