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Low-Dimensional Topology

Submissions closed on 2025-06-15 11:59PM [Central Time (US & Canada)].

Talks related to knot theory and low-dimensional topology. Organizers: Carmen Caprau and Christine Lee.

Talks related to knot theory and low-dimensional topology, which have seen recent invigoration through the interaction of different techniques from algebra, geometry, combinatorics, and representation theory. Organizers: Carmen Caprau and Christine Lee.

Accepted Submissions:

Geometric triangulations of the complement of double twist knots — Dionne Ibarra

In this talk we will present and explain the construction of two different geometric triangulations of the complements of double twist knots of the form Kp,q obtained by Dehn filling the crossing circles of the Borromean rings. This is joint work with D. V. Mathews and J. S. Purcell.

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A minimality criterion for SL(4) skeins — Anup Poudel

We provide a 4-valent ribbon model for SL(4) skein category by working with a category with object an oriented marking and morphisms generated by tagged and untagged 4-valent vertices. The category is defined combinatorially in terms of diagrammatic generators and relations. We use linear algebraic and skein theoretic methods to explore topological invariants coming from such a category. As a consequence, we show that a specialization of our parameters provides a 4-valent category that is equivalent to the SL(4) representation category. We further provide a topological evaluation algorithm of closed webs providing a (topological) criterion for reducible webs. We also show that certain HOMFLY relations exist in our category. Our evaluation algorithm works at a very abstract level and doesn’t use any algebraic constraints coming from the representation theory. This is a joint work with Giovanni Ferrer and Jiaqi Lu. 

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Another proof of functoriality for odd Khovanov homology — Dean Spyropoulos

In 2024, Migdail and Wehrli proved that odd Khovanov homology is functorial with respect to link cobordism (up to sign). Unlike Khovanov's proof that the original theory is functorial, Migdail-Wehrli's is interesting in that it does not depend on any of the recent extensions of odd Khovanov homology to tangles. In recent ongoing work, we adapt Khovanov's original argument to one of these tangle theories to get a proof that Naisse-Putyra's odd tangle invariant is functorial with respect to tangle cobordisms (up to unit). This approach motivates a few novel constructions, including a new generalization of Hochschild (co)homology.

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Holonomicity from a Heegaard-Floer perspective — Ben Cooper

I'll discuss Sr-colored knot Floer homologies and categorified recurrence relations that they satisfy. The associated Euler characteristic implies q-holonomicity of the corresponding sequence of colored Alexander polynomials, inspired by the AJ conjecture for colored Jones polynomials.

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Ideal Triangulations and Once-Punctured Surface Bundles — Birch Bryant

A well-known result of Walsh states that if T∗ is an ideal triangulation of an atoroidal, acylindrical, irreducible, compact 3-manifold with torus boundary components and T∗ has essential edges, then every properly embedded, two-sided, incompressible surface S is isotopic to a spun-normal surface in T∗ unless S is isotopic to a fiber or virtual fiber. For a given manifold M that fibers over S1, it was previously unknown whether there exists an ideal triangulation in which the fiber appears as a spun-normal surface. We prove that such a triangulation exists and give an algorithm to construct the ideal triangulation provided M has a single boundary component.

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In search for homology of Bol-Moufang quasigroups — Jozef Przytycki

0ur goal is to initiate (co)homology theory for quasigroups of Bol-Moufang. Our approach which has its roots in the work of Eilenberg and his coauthors (MacLane, Cartan) is to analyze extensions of a quasigroup (X,∗X) by an affine quasigroup (A,∗A) of the same type. We study these extensions not to classify them but to have the first glimpse at their homology via their second and third boundary operation, ∂2(x,y) and ∂3(x,y,z) respectively. We compute the second homology groups for all distinguishing examples of Bol-Moufang quasigroups described by Phillips and Vojtechovsky. We specualte about use of homology of Bol-Moufang quasigroups in Knot Theory. It is a joint work with Anthony Christiana, Ben Clingenpeel, Huizheng Guo, Jinseok Oh, and Anna Zamojska-Dzienio.

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Investigations in Knot Positivity — Lizzie Buchanan

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.

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Lie superalgebras and the minimal genus of virtual links — Micah Chrisman

For links L⊂Σ×[0,1], where Σ is a closed orientable surface, we define a Uq(gl(1,1)) Reshetikhin-Turaev invariant with coefficients in Z[H1(Σ)]. This is well-defined up to multiples of the quantum supergroup variable q. This invariant turns out to be equivalent to an infinite cyclic version of the Carter-Silver-Williams (CSW) polynomial. The importance of the CSW polynomial is that half its symplectic rank gives strong lower bounds on the virtual genus. Recall that given a virtual link type L, the virtual genus of L is the smallest genus of all closed orientable surfaces Σ on which L can be represented by a diagram D on Σ. The main objective of this paper is to extend the CSW bound on the virtual genus to all Lie superalgebras Uq(gl(m,n)) with n>0. For links in thickened once-punctured surfaces Σ, we define a Uq(gl(m,n)) Reshetikhin-Turaev invariant with coefficients in Z[H1(Σ)]. We show that half its symplectic rank is also a lower bound on the virtual genus. Changing the value of the pair (m,n) can give lower bounds better than those available from other known methods. We compare the Uq(gl(m,n)) lower bounds to those coming from the CSW polynomial, the surface bracket, the arrow polynomial, hyperbolicity, and the Gordon-Litherland determinant test. As an application, we show that the Seifert genus of homologically trivial knots in thickened surfaces is not additive under the connected sum operation of virtual knots. This is joint work with Killian Davis and Anup Poudel.

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Multiple Virtual Knot Theory — Lou Kauffman

This talk will discuss a generalization of virtual knot theory (stabilized embeddings of knots and links in thickened surfaces) that uses many types of virtual crossings. The theory is motivated by graph coloring problems and their analogs as bracket polynomials for multiple virtual knots. We discuss a number of invariants of virtuals, conjectures and open problems.

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On algebraic invariants of multi-virtual knots — Sujoy Mukherjee

Multi-virtual knot theory is a generalization of virtual knot theory that associates labels to the virtual crossings of a virtual knot. After discussing basic ideas in multi-virtual knot theory, I will talk about algebraic invariants of multi-virtual knots constructed using operator quandles. The talk is based on joint work with Louis H. Kauffman and Petr Vojtechovsky.

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On the neighborhood of knots in the space of open curves — Eleni Panagiotou

In this talk we will discuss a new framework for classifying knots by exploring the neighborhood of knot embeddings in the space of (collections of) simple open curves in 3-space with no constraints at their endpoints. The latter gives rise to a knotoid (or linkoid) spectrum of a knot that consists of a knot-type knotoid and pure knotoids. We will examine to what extent the pure knotoids of the knotoid spectrum determine the knot type. For example, we will prove that the pure knotoids in the knotoid spectra of a knot, which are individually agnostic of the knot type, can distinguish knots of Gordian distance greater than one. We will also prove that the open curve neighborhood of, at least some, embeddings of the unknot can be distinguished from any embedding of any non-trivial knot that satisfies the cosmetic crossing conjecture. Topological invariants of knots can be extended to their open curve neighborhood to define continuous functions in the neighborhood of knots. We will discuss their properties and prove that invariants in the neighborhood of knots may be able to distinguish more knots than their application to the knots themselves. For example, we will prove that an invariant of knots that fails to distinguish mutant knots (and mutant knotoids), can distinguish them by their neighborhoods, unless it also fails to distinguish non-mutant pure knotoids in their spectra. Studying the neighborhood of knots opens the possibility of answering questions, such as if an invariant can detect the unknot, via examining possibly easier questions, such as whether it can distinguish height one knotoids from the trivial knotoid.

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Skein Modules for 3-Manifolds and Their Structure — Rhea Bakshi

Skein modules were introduced by Przytycki and independently by Turaev as generalizations of the polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. Among these, the Kauffman bracket skein module (KBSM) has been studied most extensively. Recently, Gunningham, Jordan, and Safronov demonstrated that for any closed 3-manifold, the KBSM is finite-dimensional over Q(A); however, this finiteness does not extend to the KBSM over Z[A±1]. Moreover, computing the KBSM of a 3-manifold remains a notoriously challenging problem, especially over this ring. In this talk, we will survey these developments and explore several open questions concerning the structure of the KBSM over Z[A±1].

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Skein identities at roots of unity — Vijay Higgins

The skein algebra of an oriented surface is spanned by framed links in the thickened surface subject to the Kauffman bracket relations. Multiplication of links is given by stacking in the direction of the thickening. We will discuss special skein identities which hold when the quantum parameter q is specialized to a root of unity. The identities involve Jones-Wenzl projectors and are certain incarnations of special cases of Steinberg tensor product identities from the representation theory of Uq(sl2). We will discuss how the easiest such identity can be used to recover the Chebyshev-Frobenius homomorphism of Bonahon-Wong. This is joint work with Indraneel Tambe.

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Thompson's Groups, Annular Links, and Tangles — Louisa Liles

Vaughan Jones showed how to associate links in the 3-sphere to elements of Thompson’s group F and proved that F 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 F as a subgroup, and the second is a method of building (n,n)-tangles, which give rise to an action of F on Khovanov's chain complexes. This talk includes joint work with Slava Kruskhal and Yangxiao Luo.

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Triple point numbers of 2-twist spun knots — Scott Carter

The minimal triple point number of the 2-knot that is the 2-twist spin of the knot 52 is bounded between 8 and 12. The movie suggested by Fox's example 15 has triple point number 12. To improve this bound we follow Shin Satoh's work on triple point number and construct virtual surfaces that have small triple point number and for which Mochizuki's 3-cocycle vanishes. This is joint work with Seonmi Choi, Hongdae Kim, Sangsu Lee, and Seong Yeop Yang.

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