Schedule for: 25w5497 - Interactions of Geometric and Quantum Topology focused on Links in Thickened Surfaces

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The composition of any two nontrivial classical knots is a satellite knot, and thus, by work of Thurston, is not hyperbolic. Here, we explore the composition of virtual knots, which correspond to knots in $S \times I$ where S is a closed orientable surface. We prove that for any two hyperbolic virtual knots, there is a composition that is hyperbolic. We then obtain strong lower bounds on the volume of the composition using information from the original virtual knots. We further consider nonorientable closed surfaces and surfaces with boundary.

There are 352 million prime knots with at most 19 crossings. I will discuss ongoing work with Sherry Gong showing that 1.6 million of these knots are slice (in fact ribbon) and nearly all of rest are not. Here "nearly all" means we have less than 15,000 knots (0.004%) whose slice status is unknown. Because neither slice nor ribbon is known to be algorithmically decidable, we use a mix of partial techniques. I'll mention some potential applications to the slice versus ribbon conjecture and to finding exotic 4-spheres.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

There are two equivalent definitions of the bridge index of classical knots: one is based on the minimal number of overpasses, and the other on the maximal number of points with respect to a height function. Nakanishi and Satoh asked whether these definitions can differ significantly in the virtual knot setting. Using biquandles, we show that they indeed can. If time permits, I will also discuss how these two versions of the virtual bridge numbers can be used to estimate hyperbolic volumes of knots in thickened surfaces.

The cosmetic crossing conjecture posits that switching a non-trivial crossing in a knot diagram always changes the knot type. This question is closely related to cosmetic surgery problems for three-manifolds, and has seen significant progress in recent years. We will discuss the conjecture, and present new obstructions to cosmetic crossing changes for a family of links that includes all alternating knots.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The meridian length conjecture posits that the meridian, when measured on a maximal cusp in a hyperbolic knot complement, has length at most 4. We impose conditions on an m-almost alternating knot diagram which guarantee that both checkerboard surfaces are algebraically essential in the knot exterior. This suffices to prove that such knots satisfy the meridian length conjecture and the Neuwirth conjecture.

Given a knot $K$ with a Seifert surface $\Sigma$, I dream that the well-known Seifert linking form $Q,$ a quadratic form on $H_1(\Sigma)$, has plenty docile local perturbations Pϵ such that the formal Gaussian integrals of $\exp(Q+P\epsilon)$ are invariants of $K$. In my talk I will explain what the above means, why this dream is oh so sweet, and why it is in fact closer to a plan than to a delusion. Joint with Roland van der Veen.

The family of right-angled tiling links consists of links built from regular 4-valent tilings of constant-curvature surfaces that contain one or two types of tiles. The complements of these links admit complete hyperbolic structures and contain two totally geodesic checkerboard surfaces that meet at right angles. In this talk, I will describe a complete characterization of which right-angled tiling links are arithmetic, and which are pairwise commensurable. The arithmeticity classification exploits symmetry arguments and the combinatorial geometry of Coxeter polyhedra. The commensurability classification relies on identifying the canonical decompositions of the link complements. This is joint work with Rose Kaplan-Kelly.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Menasco-Reid conjecture states that hyperbolic knot complements do not contain closed, embedded totally geodesic surfaces. We construct examples of knot complements that violate the conjecture. The knot complements are also interesting in that they are new examples of knot complements with hidden symmetries. This is joint work with Jason Deblois and Arshia Gharagozlou. It extends results from Gharagozlou's thesis work.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Given a rational homology 3-sphere, we introduce a triple linking form on its first homology, defined when the classical torsion linking pairing of three homology classes vanishes pairwise. We use these methods to formulate an embedding obstruction for rational homology spheres in $S^4$, extending a 1938 theorem of Hantzsche. (Based on work in progress with Michael Freedman)

I will discuss the number of isotopy classes of connected essential surfaces embedded in many cusped 3-manifolds and their Dehn fillings. We obtain polynomial bounds that are universal, in the sense that we obtain the same explicit formula for all 3-manifolds that we consider, with the formula dependent on the Euler characteristic of the surface and similar numerical quantities encoding topology of the ambient 3-manifold. Universal and polynomial bounds have been obtained previously for classical alternating links in the 3-sphere and their Dehn fillings, but only for surfaces that are closed or spanning. Here, we consider much broader classes of 3-manifolds and all topological types of surfaces. The 3-manifolds are called weakly generalized alternating links; they include, for example, many links that are not classically alternating and/or do not lie in the 3-sphere, many virtual links and toroidally alternating links. This is joint work with Anastasiia Tsvietkova.

Abstract: We define a family of Turaev-Viro type invariants of hyperbolic 3-manifolds with totally geodesic boundary from the 6j-symbols of the modular double of Uqsl(2;R), and prove that these invariants decay exponentially with the rate the hyperbolic volume of the manifolds and with the 1-loop term the adjoint twisted Reidemeister torsion of the double of the manifolds.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We describe a new method of producing equations for the canonical component of representation variety of a knot group into $PSL(2, C)$. Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of the knot complement, and uses only a knot diagram satisfying a few mild restrictions. This gives a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. The algorithm additionally yields an explicit description for the hyperbolic structures (complete or incomplete) that correspond to geometric representations of a hyperbolic knot. This is joint work with Kathleen Petersen. This was inspired by ideas from joint work with Thistlethwaite in 2012. In it, we developed an alternative method for computing the complete hyperbolic structure of a link in 3-sphere. The ideas from 2012 were recently extended to link complements in thickened torus by Kwon, Park and Tham. This poses some natural questions about potential extensions of our current work on varieties to links in thickened surfaces.

We investigate two invariants of a hyperbolic 3-manifold M, the triangulation complexity (the minimal number of tetrahedra needed to triangulate M) and the degree of the trace field. We show that there are families of closed 3-manifolds where both of these invariants have growth bounded linearly, and there are also families whose triangulation complexity grows linearly but where the degrees of the trace fields grow exponentially. A key ingredient in the proof is establishing bounds for the degrees of trace fields of Dehn fillings in terms of the height of the filling parameter using sparse polynomials that are specializations of A-polynomials. This is joint work with Paul Fili and Neil Hoffman

The Goldman-Turaev loop operations equip the vector space of free loops in a surface with boundary with the structure of a Lie bialgebra. In this talk we will show that - at least in genus zero - the Goldman-Turaev loop operations are naturally induced by tangle operations in the corresponding thickened surfaces. In turn, the Kontsevich integral of tangles induces universal quantum invariants for the Goldman-Turaev Lie bialgebra. This is particularly interesting in light of a seminal result of Alekseev, Kawazumi, Kuno and Naef stating that in genus zero universal quantum invariants for the Goldman-Turaev Lie bialgebra correspond to solutions to the Kahiwara-Vergne equations arising from convolutions on Lie groups, opening up rich connections to quantum algebra. Our results are parallel to earlier work of Massuyeau (in the context of braids) and Alekseev-Naef (in the context of the KZ connection). Joint work with Dror Bar-Natan, Jessica Liu, Tamara Hogan, and Nancy Scherich.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Vaughan Jones showed how to associate links in the 3-sphere to elements of Thompson’s group F and proved that gives rise to all link types. This talk will introduce Jones’s construction and discuss two recent extensions– the first is a method of building annular links from Thompson’s group T, which contains as a subgroup, and the second is a method of building (n,n)-tangles from Thompson’s group F. Annular links from T arise from Jones’s unitary representations of the Thompson group, and tangles from F give rise to an action of F on Khovanov’s chain complexes. This talk includes joint work with Slava Kruskhal and Yangxiao Luo.

The Links-Gould polynomials are 2-variable quantum invariants from the super quantum group $sl(2|1)$ which generalize the Alexander polynomial. We give formulas for the Link-Gould invariant of certain cablings of knots. These lead to an enhancement of existing genus bounds for the Links-Gould polynomial, which themselves improve on the Alexander polynomial bounds. The talk is based on in-progress work with Stavros Garoufalidis, Rinat Kashaev, Ben-Michael Kohli, and Emmanuel Wagner.

A knot is "positive" if it has a diagram in which all crossings are positive. How does having such a diagram force patterns and structure to appear in the Jones polynomial and Khovanov homology? When can these patterns distinguish positive knots from almost-positive knots? In this talk we discuss results from the last few years and ongoing work to understand the Jones polynomial and Khovanov homology of positive knots and links. Particular attention is paid to the class of fibered positive knots, which contains all braid positive knots. We conclude with a discussion of ongoing work to completely describe the homological grading 2 of the Khovanov homology of fibered positive links.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Given a plumbing tree and a spin-c structure, I will discuss how to construct a plumbed 3-manifold invariant in the form of a Laurent series twisted by a root lattice. Such a series is invariant under the Neumann moves on plumbing trees and the action of the Weyl group. For irreducible root lattices of rank at least 2, there are infinitely many such series, each depending on a combinatorial puzzle defined on the root lattice. These families of invariants generalize the Z-hat series of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park and Ri. They are motivated by the study of the WRT invariants, and work of Akhmechet-Johnson-Krushkal makes connections of a related series to lattice cohomology. Time permitting, I will also discuss a multivariable generalization of the root lattice-twisted series for knot complements and gluing formulas. This is joint work with N. Tarasca.

Patrick Kinnear constructed a line bundle over the character stack of a closed, oriented three-manifold using TQFT methods associated with an odd-order root of unity. He subsequently conjectured that the invariant sections of this line bundle constitute the Kauffman bracket skein module of the underlying manifold at that root of unity. Since a line bundle over the character stack is equivalent to an equivariant line bundle over the representation variety, one can reframe the problem accordingly. In this context, the stated skein module of a once-punctured closed three-manifold with a single marking at an odd-order root of unity is given by the global sections of a sheaf over the twisted representation variety of the three-manifold. However, this sheaf does not necessarily inherit an equivariant structure. By analyzing the quantum symmetries of the stated skein module, I identify a natural submodule of the stated skein module that forms an equivariant sheaf over the twisted representation variety, whose invariant sections recover the Kauffman bracket skein module of the closed manifold.

While proving that Tait's Flyping Conjecture extends to virtual alternating links, I learned many lessons, including (1) subtleties about virtual links, their diagrams, and their spanning surfaces, (2) how to think about, diagram, and count flypes, (3) how Gordon-Litherland's classical perspective involving double branched covers of the 4-ball adapts to this context, and (4) plenty of technical mumbo jumbo. I will pass several of these lessons on to you, with as little of (4) as possible.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.