Geometry and Topology Seminar

Thursday May 2, 15:00, room MS.04

Mark Brittenham (Nebraska-Lincoln)

Unknotting moves and group quotients

Abstract: From one point of view, a "knot" can be thought of as an equivalence class of knot diagrams under the Reidemeister moves: "local" moves on diagrams that change the picture of the knot without changing the underlying knot. From this perspective an unknotting operation is an additional move on diagrams so that with this move knots form a single equivalence class: that is, every knot can be transformed to the unknot by the extra move and by Reidemeister moves. There is a long history of studying such local moves to determine if they are unknotting; the knot group, the fundamental group of the complement of the knot, has often played a central role in constructing invariants of a potential unknotting move. In this talk we will mostly discuss a particular move, the 4-move, which has defied attempts to prove or disprove its status as an unknotting operation for more than 30 years, and discuss information we can obtain about the 4-move from studying invariant quotients of the knot group. This work is joint with Susan Hermiller and Robb Todd.

Thursday May 9, 15:00, room MS.04

Melanie DeVries (Nebraska-Lincoln)

Unknotting Moves and Virtual Knots

Abstract: Virtual knot theory is a generalization of classical knot theory that gives a way to study knots in spaces of higher genus and opens up avenues to apply combinatorial and computational methods to the field. Unknotting moves - local moves that can change any knot into the trivial knot - have been a subject of interest in classical knot theory as they give insight into a knot's structure and can be used to help create and calculate invariants. This talk will look at unknotting moves of virtual knots and the relationship between classical unknotting moves and virtual unknotting moves.

Thursday May 16, 15:00, room MS.04

Yasushi Yamasita (Nara Women's University)

The link volume of 3-manifolds

Abstract: We view closed orientable 3-manifolds as covers of S^3 branched over hyperbolic links. To a $p$-fold cover $M \to S^3$, branched over a hyperbolic link $L$ we assign the complexity $p \cdot \Vol(S^3 - L)$. We define an invariant of $3$-manifolds, called the link volume, by assigning to $M$ the infimum of this quantity as the cover varies, always requiring that the branch set is a hyperbolic link. The link volume measures how efficiently $M$ can be represented as a cover of $S^3$. We study the basic properties of the link volume. This is joint work with Yo'av Rieck.

Thursday May 16, 16:00, room MS.04

Gaven Martin (Massey)

The solution to Siegel's problem

Abstract: We outline the history and the proof of the identification of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter group extended by the involution preserving the symmetry of this diagram.

This solves (in three dimensions) the problem posed by Siegel in 1945

Siegel solved this problem in two dimensions by deriving the signature formula identifying the (2,3,7)-triangle group as having minimal co-area. There are strong connections with arithmetic hyperbolic geometry in the proof and the result has applications identifying three-dimensional analogues of Hurwitz's 84(g-1) theorem as Siegel's result do.

Thursday May 23, 15:00, room MS.04

Ser Peow Tan (NU Singapore)

A dilogarithm identity on moduli spaces of curves

Abstract: We will talk about a new identity for closed hyperbolic surfaces which involves the dilogarithm of the lengths of simple closed geodesics on the surface, and also relate it to some previously known identities by Basmajian, McShane and Bridgeman. This is joint work with Feng Luo.

Thursday May 30, 15:00, room MS.04

Fionntan Roukema (Sheffield)

The minimally twisted 5-chain link

Abstract: We will start by describing the program of enumerating cusped hyperbolic 3-manifolds with pairs of non-hyperbolic fillings at maximal distance. We will then introduce the minimally twisted 5-chain link and explain how this link is relevant to the program of enumerating exceptional pairs at maximal distance. The exceptional surgeries on the 5-chain link has a particularly simple classification which will shall describe. We will then see how this classification can be used to generate examples of cusped manifolds with exceptional pairs at maximal distance.

Thursday May 30, 16:00, room MS.04

Susan Hermiller (Nebraska-Lincoln)

Conjugacy growth series and languages in groups

Abstract:The growth of elements and of conjugacy classes for finitely generated groups have been widely studied. However, the associated series (generating functions) for conjugacy growth functions are not yet well understood. The conjugacy growth series has unexpected properties; in particular, Rivin shows that the growth series is not rational, for free groups with respect to a free basis. In this talk I will introduce the notions of geodesic conjugacy growth functions and series, and discuss the effects of various group constructions on rationality of both the geodesic conjugacy and (usual) conjugacy growth series, as well as on regularity of the geodesic and spherical conjugacy languages whose growth is measured by these functions. (No prior knowledge of growth of groups or languages will be assumed.) This is joint work with Laura Ciobanu.

Thursday June 6, 15:00, room MS.04

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Abstract: None

Thursday June 13, 15:00, room MS.04

Mark Bell (Warwick)

Canonical curve systems are small

Abstract: Let $h$ be the isotopy class of a homeomorphism of a punctured surface. Its canonical curve system, $\sigma(h)$, is the intersection of all maximal $h$--invariant multicurves and is the natural choice of multicurve to decompose $h$ along.

We will discuss an algorithm for producing $\sigma(h)$. This leads to an elementary proof of Koberda and Mangahas's result of an upper bound for the size of the $\sigma(h)$ in terms of the size of $h$, with respect to suitable norms.

Thursday June 20, 15:00, room MS.04

Richard Webb (Warwick)

Short proof of the Bounded Geodesic Image Theorem

Abstract: The curve graph of a surface $S$ is a combinatorial object: its vertices are curves on $S$, edges between disjoint pairs, and its geometry is defined by shortest length paths in the graph. It is a Gromov hyperbolic space and has applications (not limited to) mapping class groups, Teichmueller theory and infinite volume hyperbolic structures on $3$-manifolds. Distance in the curve graph is a way of measuring how mixed two curves are in the whole surface. To measure how mixed two curves are in subsurfaces requires the notion of subsurface projection. The Bounded Geodesic Image Theorem states that if two curves $a$ and $b$ are mixed enough on some subsurface (a large diameter subsurface projection), then any geodesic in the curve graph between $a$ and $b$ must have some vertex (i.e. curve) that misses the subsurface. This fact is crucial in some modern applications of the curve graph, and we shall give a combinatorial proof with a universal bound.

Thursday June 20, 16:00, room MS.04

Caroline Series (Warwick)

Growth functions for Fuchsian groups

Abstract: It is well known that the growth function of a Fuchsian group is rational, and that this property extends to much wider classes of groups. In this talk I will discuss a simplified proof of some further old results of W.Floyd and S.Plotnick: growth functions of Fuchsian groups are reciprocal and take a special value at $1$. These finer results are definitely not in general true.

Thursday June 27, 15:00, room MS.04

Simon Brain (University of Trieste)

The noncommutative geometry of self-dual gauge fields

Abstract: The moduli space of instantons (gauge fields with self-dual curvature) on a closed Riemannian four-manifold is an important invariant of its differential structure. In this talk I will survey some recent progress in understanding the extent to which moduli spaces of instantons can detect the differential structure of four-manifolds in noncommutative geometry.