# Postcard from a Geometric Topologist

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

Image description: a sketch of CV_2. It looks like a disc split into triangles, and each edge has a "vertical" triangle coming out of it (coloured green). Some of the triangles have graphs with two or three edges drawn. Around the outside are graphs representing points of CV_2: three are roses, labelled {a,b}, {a,ba} and {b,ab}. Another is a theta graph, with two cycles labelled a and b; a fifth has two loops and a separating edge, with one loop labelled a, and a dotted line indicating b as along the separating edge, round the other loop, and back. A red dashed arrow labelled "collapse" goes from the theta graph to one of the roses. Caption: CV_n is (almost) a simplicial complex, whose geometry lets us investigate Out(F_n) Points in CV_n correspond to free actions on trees, which are associated to marked graphs Collapsing edges moves you into a face (but not all faces exist) See "What is Outer Space" (Karen Vogtmann) for more! (https://www.ams.org/notices/200807/tx080700784p.pdf)

# Michelle Chu

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In 1903, Hurwitz proved that a hyperbolic surface has at most 84(g-1) automorphisms. A Hurwitz surface is a hyperbolic surface with exactly 84(g-1) automorphisms. These surfaces are realized as quotients of H^2 by normal finite-index torsion-free subgroups of the (2,3,7)-triangle group. Hurwitz surfaces only exists in some genus g. But sometimes there can be multiple with the same g. The first instance is when g=14: there are three. The Z-prime 13 splits into a product of three distinct Z[cos(2pi/7)]-primes. The three associated principal congruence covers correspond to the three genus 14 Hurwitz surfaces.

Image description: A red "normal arc" that joins two edges of a triangle (top left). A red "root arc" that joins a vertex of a triangle to the opposite edge (top middle). A red "tripod" in a triangle; the tripod has three arcs that join the three edges of the triangle to a single vertex that lies at the centre of the triangle (top right). A red graph embedded in a one-vertex triangulation of the genus-2 surface, with edge weights written in blue (bottom). Caption: We can encode a normal curve in a triangulated surface using edge weights (i.e., non-negative integers that tell us how many times the normal curve intersects each edge of the triangulation), provided these edge weights satisfy some combinatorial conditions. What happens if we allow these combinatorial conditions to fail? It turns out that any collection of edge weights can be sensibly interpreted as an embedding of a union of curves and graphs. Such an embedded object meets each face of our triangulated surface in a collection of normal arcs, root arcs and/or tripods (as illustrated on the left). Possible application. This might give a useful way to encode boundary patterns for 3-manifolds. Another application. Given a one-vertex triangulation of an oriented 3-manifold with boundary, consider a complete set of attaching circles on the boundary surface. These circles give us "instructions" for building a closed 3-manifold M by attaching a handlebody. If we isotope these circles to meet tangentially at the vertex, we get an embedded graph that we can encode using edge weights. There is a not-too-complicated algorithm that uses these edge weights to attach a handlebody, and hence build a one-vertex triangulation of M. This algorithm is work-in-progress with James Morgan and Em Thompson.

We would like to present an algorithm for constructing Kirby diagrams of 4-dimensional open books. It follows that if an open book was constructed with trivial monodromy, then it admits a punctured handlebody as page. We also give a Kirby diagram proof of the fact (due to Pao, Plotnick, Meier) that the spin of a lens space L(p,q) is independent of q. It will be on arXiv soon.

Text: This colored graph encodes the singularities of a Legendrian front. The Lagrangian projection yields an exact Lagrangian filling that can also be used to obtain one of infinitely many exact Lagrangian fillings of a particular twist-spun Legendrian torus in standard contact R^5. -James Hughes, UC Davis (joint with Agniva Roy) Image description: A bicolored graph with pentagonal and hexagonal faces arranged so that the colors alternate around the boundary. Caption: A rotationally symmetric exact Lagrangian filling of the (max-tb) Legendrian torus link T(3,6).

Image description: A picture of a mapping torus of several genus two surfaces glued along a loop. Caption: Consider a mapping torus $M_f$ of a $K(G,1)$ where $G$ is either -hyperbolic; -toral relatively hyperbolic; -or a right-angled Coxeter group. Suppose $\Gamma=\pi_1 M_f$ is residually finite. If $f$ grows polynomially, then for every Farber sequence $(\Gamma_k)_{k\in \mathbb{N}}$ and $j\geq0$ we have $\lim_{k\to\infty} \frac{|H_j(\Gamma_k;\mathbb{Z})_{tors}|}{|\Gamma:\Gamma_k|}=0$. -- Sam Hughes Based on joint work with Naomi Andrew, Yassine Guerch, and Monika Kudlinska Note: The last equation states that the torsion homology growth of $\Gamma$ vanishes independent of the chain.

Image description: a portion of the tiling of the plane by squares and octagons. Caption: Squares and octagons tile the plane! Imagine gluing together *two* tiled planes along a single square—then repeat for every square you see in a tree-like fashion. The resulting contractible space X admits a proper, cocompact action by a certain Coxeter group W—but also, Theorem: The outer automorphism group of G = Z/2Z * Z/2Z * Z acts properly and cocompactly on X. In fact, we have that W is a normal subgroup of Out(G) of index four. It turns out that Theorem: F = A * B * Z, with A and B nontrivial but finite are the only free products for which we have both that Z+Z < Out(F) and Out(F) acts on "tree-like" (infinitely ended) spaces in this way!

Image description: The image describes the structure of first Khovanov homology of positive links. Caption: 1st-Khovanov homology of positive links. Text: "Greetings Everyone, We*(M.Kegel, L. Mousseau, M. Silvero) developed a new obstruction for positivity. The first Khovanov homology of positive links is supported in a single quantum grading. Did you know that $Kh^{1,j}=0 \forall j$ iff $L$ if also fibered. Have fun using it:)!"

Image description: a funny-looking ruled cylinder intersected by two planes, with the potato-like intersections of these planes with the cylinder colored blue. The lines in the boundary ruling tangent to the blue slices are colored differently. Caption: Greetings from Atlanta! Left is my best attempt at drawing the domain of discontinuity for a PSL(4,R) Hitchin representation in RP3. Any such domain of discontinuity has a natural foliation by convex sets in copies of RP2 inside RP3 (two are drawn in blue left). I've been thinking about these leaves' shapes recently. They're weird. The map here to consider is from the circle (the foliation's leaf space) to the set of projective equivalence classes of properly convex subsets in RP2 (which is not T1), and is given by taking the projective class of each leaf. This map is continuous and constant on many dense subsets of the circle. I recently showed it is either constant the ellipse or not constant. The non-constant case is generic, which surprised me. Alex Nolte

Image description: A solid torus with blue faces decomposing into 3-manifolds with red and blue faces. Caption: A dynamic pair on a 3-manifold $M$ is, very roughly, a pair of branched surfaces $(B^s, B^u)$ along with a vector field $V$ such that the flow generated by $V$ is contracting along $B^s$ and expanding along $B^u$, along with some requirements on the topology of the complementary regions $M \cut (B^s \cup B^u)$. The figure shows an exploded view of how a component of $M \cut B^u$ -- a cusped solid torus -- can decompose as components of $M \cut (B^s \cup B^u)$. A dynamic pair imitates the dynamical behavior of a pseudo-Anosov flow. In fact, the stable and unstable branched surfaces $B^s$, $B^u$ of a dynamic pair must carry the stable and unstable foliations of a pseudo-Anosov flow. Dynamic pairs were invented by Mosher as a tool to show that every finite depth foliation admits an almost transverse pseudo-Anosov flow. Michael Landry and myself have been working on a project that aims to upgrade this result of Mosher (also due to Gabai) to characterize when the pseudo-Anosov flow has no perfect fits. We have recently completed the first part of the project; the paper is available at arXiv:2304.14481!

# Konstantinos Varvarezos

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Consider the knot $K=T_{2,5}\#T_{-2,3}.$ Infinitely many surgery slopes are \emph{non-characterizing} for this knot; that is, for infinitely many $r\in\mathbb{Q},$ there exists some other knot $K'$ such that $S^3_r(K)\cong S^3_r(K').$ In fact, this is true for all the knots $T_{2,2n+3}\#T_{-2,2n+1}$ [1]. This works because these knots admit a band surgery to the Hopf band (see figure). Can a this result be generalized to 2-bridge knots? Namely, if $J,K$ are 2-bridge knots which differ by a single crossing change, does $K\#\overline{J}$ have infinitely many non-characterizing slopes? Figure: the knot $T_{2,2n+3}\#T_{-2,2n+1}$ presented as a link with a band attached - the original link turns out to be the Hopf link.

Fact #1: pi_1(S^3 \ trefoil) = B_3, the braid group on 3 strands. This is easy enough to compute, but I can't really "see" it. Fact #2: pi_1(S^3 \ 3-component Hopf link) = PB_3, the subgroup of B_3 of only pure braids. This one is easier (for me) to see! Start with some isotopy: [Image: 3-component Hopf link with 120-degree rotational symmetry isotoped into a 2-component Hopf link with an additional loop around both strands.] pi_1(S^3 \ 3-component Hopf link) is generated by rho, beta, and lambda, with relations rho*lambda*beta = lambda*beta*rho = beta*rho*lambda. [Image: Each of the three generators illustrated in the link complement.] PB_3 is generated by T_12, T_13, and T_23 with the same relations! [Image: T_12 is illustrated as a swap between the first two strands of a braid, and T_13 and T_23 are defined analogously.] That the iso matches rho with T_12 and beta with T_13 is not too hard to see. The correspondence between lambda and T_23 takes a bit more effort. [Image: An example braid in PB_3 and its image in pi_1.]