UB Geometry and Topology Seminar 

Abstract: We define a handlebody decomposition of a closed orientable 3-manifold via 2-dimensional polyhedron, which generalizes both Heegaard splittings and Koenig trisections. We introduce stabilizations and destabilizations of handlebody decompositions, and show the stable equivalence theorem. We suggest a mathematical model of polycontinuous patterns, which are 3-periodic structures assembled by polymers. It turns out that the mathematical model, net-like patterns, corresponds to handlebody decompositions of the 3-dimensional torus. As an example of a tricontinuous pattern, we observe that the 3srs pattern can be destabilized to the hexagonal honeycomb pattern. This is joint work with Naoki Sakata, Ryosuke Mishina, Masaki Ogawa, Kai Ishihara, Yuya Koda, Koya Shimokawa.

Abstract: In this talk I will briefly describe link Floer homology toolbox and its usefulness. Then I will show how link Floer homology can detect links with small ranks, using a rank bound for fibered links by generalizing an existing result for knots, and some very useful spectral sequences. I will also show that stronger detection results can be obtained in a sense that the knot Floer homology can be shown to detect $T(2,8)$ and $T(2,10)$, and that link Floer homology detects $(2,2n)$-cables of trefoil and the figure eight knot. This talk is based on joint work with Fraser Binns.

Abstract: The notion of efficient geodesics in $\mathcal{C}(S_{g>1})$, the complex of curves of a closed orientable surface of genus $g$, was first introduced in "Efficient geodesics and an effective algorithm for distance in the complex of curves". There it was established that there exists (finitely many) efficient geodesics between any two vertices, $ v_{\alpha} , v_{\beta} \in \mathcal{C}(S_g)$, representing homotopy classes of simple closed curves, $\alpha , \beta \subset S_g$. The main tool for used in establishing the existence of efficient geodesic was a dot graph, a booking scheme for recording the intersection pattern of a reference arc, $\gamma \subset S_g$, with the simple closed curves associated with the vertices of geodesic path in the zero skeleton, $\mathcal{C}^0(S_g)$. In particular, it was shown that any curve corresponding to the vertex that is distance one from $v_\alpha$ in an efficient geodesic intersects any $\gamma$ at most $d -2$ times, when the distance between $v_\alpha$ and $v_\beta$ is $d \geq 3$. In this note we make a more expansive study of the characterizing ``shape'' of the dot graphs over the entire set of vertices in an efficient geodesic edge-path. The key take away of this study is that the shape of a dot graph for any efficient geodesic is contained within a spindle shape region.

Abstract: (joint work with A.Rasmussen and C.Abbott) Recent papers of Balasubramanya and Abbott-Rasmussen have classified the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace-Cornulier-Monod-Tessera. This machinery applies in particular to solvable groups with virtually cyclic abelianizations.

Abstract: (joint with Xin Ma) Boundaries of certain $CAT(0)$ spaces and group actions on them play important roles in geometric group theory. In this talk, we will talk about boundary actions of $CAT(0)$ spaces from a point of view of topological dynamics and $C^\ast$-algebras. In particular, we will describe the actions of right angled Coxeter groups and right angled Artin groups on certain boundaries. This provides some pure infiniteness results for reduced crossed product $C^\ast$-algebra of these actions. Next, we will talk about the action of fundamental groups of graph of groups on the visual boundaries of their Bass-Serre trees. This provides a new method in identifying $C^\ast$-simple generalized Baumslag-Solitar groups.

Abstract: Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinear-Morse boundary to more broadly capture these generic directions. In joint work with Abdul Zalloum, we develop the geometric foundations of sublinear-Morseness in the mapping class group and Teichmuller space. We prove that their sublinearly-Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph. Moreover, we completely characterize sublinear-Morseness in terms of the hierarchical structure on these spaces. Our techniques include developing tools for modeling sublinearly-Morse rays via CAT(0) cube complexes. Part of this analysis involves establishing a direct connection between the geometry of the curve graph and the combinatorics of hyperplanes in these cubical models.

Abstract: Two knots $K_0$ and $K_1$ are said to be smoothly concordant if the connected sum $K_0^{}\#m(K_1^r)$ bounds a disk smoothly embedded in the 4-ball. Smooth concordance is an equivalence relation, and the set $\mathcal{C}$ of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. Satellite operations, or the process of tying a given knot $P$ along another knot $K$ to produce a third knot $P(K)$, are powerful tools for studying the algebraic structure of the concordance group. In this talk I will describe conditions on the pattern $P$ that suffice to conclude that the function $P:\mathcal{C}\to \mathcal{C}$ is not a homomorphism. This is joint work with Tye Lidman and Allison Miller.

Abstract: Knot Floer homology (introduced by Ozsváth–Szabó and Rasmussen) is an invariant whose definition is based on symplectic geometry, and whose applications have transformed knot theory over the last two decades. Starting from a grid diagram of a knot, I will explain how to construct a spectrum whose homology is knot Floer homology. Conjecturally, the homotopy type of the spectrum is an invariant of the knot. The construction does not use symplectic geometry, but rather builds on a combinatorial definition of knot Floer homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.)