The Gordon-Litherland pairing for knots and links in thickened surfaces

Hans Boden (McMaster University)

We introduce the Gordon-Litherland pairing for knots and links in thickened surfaces that bound unoriented spanning surfaces. Using the GL pairing, we define signature and determinant invariants for such links. We relate the invariants to those derived from the Tait graph and Goeritz matrices. These invariants depend only on the $S^*$ equivalence class of the spanning surface, and the determinants give a simple criterion to check if the knot or link is minimal genus. This is joint work with M. Chrisman and H. Karimi. In further joint work with H. Karimi, we apply the GL pairing to give a topological characterization of alternating links in thickened surfaces, extending the results of J. Greene and J. Howie. 

Slides for talk:



Two-bridge knots and their crosscap numbers

Cornelia Van Cott (University of San Francisco)

Begin with two knots K and J. Simon conjectured that if the knot group of K surjects onto that of J, then the genera of the orientable surfaces that the two knots bound are constrained. Specifically, he conjectured g(K) \geq g(J), where g(K) denotes the genus of K. This conjecture has been proved for alternating knots and can be strengthened to an even stronger result in the case of two-bridge knots. In this talk, we consider the same sorts of questions, but in the world of nonorientable surfaces. We focus on two-bridge knots and find relationships among their crosscap numbers. This is joint work with Jim Hoste and Pat Shanahan.


Hoste, J., Shanahan, P., Van Cott, C., “Crosscap number and the partial order on two-bridge knots”,

Further Reading:

The geometric content of Tait’s conjectures

Thomas Kindred (University of Nebraska-Lincoln)

In 1898, Tait asserted several properties of alternating knot diagrams, which remained unproven until the discovery of the Jones polynomial in 1985. During that time, Fox asked, “What geometrically is an alternating knot?” By 1993, the Jones polynomial had led to proofs of all of Tait’s conjectures, but the geometric content of these new results remained mysterious. In 2017, Howie and Greene independently answered Fox’s question, and Greene used his characterization to give the first purely geometric proof of part of Tait’s conjectures. Recently, I used Greene and Howie’s characterizations, among other techniques, to give the first entirely geometric proof of Tait’s flyping conjecture (first proven in 1993 by Menasco and Thistlethwaite). I will describe these recent developments and sketch approaches to other parts of Tait’s conjectures, and related facts about tangles and adequate knots, which remain unproven by purely geometric means.


Kindred, Thomas, “A geometric proof of the flyping theorem”,

Slides for talk:


Classifying small virtual skein theories

Joshua Edge (Denison University)

A skein theory for the virtual Jones polynomial can be obtained from its original version with the addition of a virtual crossing that satisfies the virtual Reidemeister moves as well as a naturality condition. In general, though, knot polynomials will not have virtual counterparts. In this talk, we classify all skein-theoretic virtual knot polynomials with certain smallness conditions. In particular, we classify all virtual knot polynomials giving non-trivial invariants strictly smaller than the one given by the Higman-Sims spin model by classifying the planar algebras associated with them. This classification includes a family of skein theories coming from \text{Rep}(O(2)) with an interesting braiding. This talk is given in memory of Vaughan Jones.

Topological Deep Learning

Mustafa Hajij (Santa Clara University)

In this work, we introduce topological deep learning, a formalism that is aimed at two goals (1) introducing topological language to deep learning for the purpose of utilizing the minimal mathematical structures to formalize problems that arise in a generic deep learning problem and (2) augment, enhance and create novel deep learning models utilizing tools available in topology. To this end, we define and study the classification problem in machine learning in a topological setting. Using this topological framework, we show that the classification problem in machine learning is always solvable under very mild conditions. Furthermore, we show that a softmax classification network acts on an input topological space by a finite sequence of topological moves to achieve the classification task.  To demonstrate these results, we provide example datasets and show how they are acted upon by neural nets from this topological perspective.


Mustafa Hajij, Kyle Istvan, “A topological framework for deep learning”,

Mustafa Hajij, Kyle Istvan, Ghada Zamzmi, “Cell complex neural networks”,

Unknotting with a single twist

Samantha Allen (Dartmouth)

Ohyama showed that any knot can be unknotted by performing two full twists, each on a set of parallel strands. We consider the question of whether or not a given knot can be unknotted with a single full twist, and if so, what are the possible linking numbers associated to such a twist. It is observed that if a knot can be unknotted with a single twist, then some surgery on the knot bounds a rational homology ball. Using tools such as classical invariants and invariants arising from Heegaard Floer theory, we give obstructions for a knot to be unknotted with a single twist of a given linking number. In this talk, I will discuss some of these obstructions, their implications (especially for alternating knots), many examples, and some unanswered questions. This talk is based on joint work with Charles Livingston.

A generalization of the Alexander polynomial from higher rank quantum groups

Matthew Harper (The Ohio State University)

Murakami and Ohtsuki have shown that the Alexander polynomial is an R-matrix invariant associated with representations V(t) of unrolled restricted quantum \mathfrak{sl}_2 at a fourth root of unity. In this context, the highest weight t\in\mathbb{C}^\times of the representation determines the polynomial variable. In this talk, we discuss an extension of their construction to a link invariant \Delta_{\mathfrak{g}}, which takes values in n-variable Laurent polynomials, where n is the rank of \mathfrak{g}.

We begin with an overview of computing quantum invariants and of the \mathfrak{sl}_2 case. Our focus will then shift to \mathfrak{g}=\mathfrak{sl}_3. After going through the construction, we briefly sketch the proof of the following theorem: For any knot K, evaluating \Delta_{\mathfrak{sl}_3} at {t_1=\pm1}, {t_2=\pm1}, or {t_2=\pm it_1^{-1}} recovers the Alexander polynomial of K. We also compare \Delta_{\mathfrak{sl}_3} with other invariants by giving specific examples. In particular, this invariant can detect mutation and is non-trivial on Whitehead doubles.


Harper, Matthew, “Verma Modules over Restricted \mathfrak{sl}_3 at a fourth root of unity”,

Harper, Matthew, “A generalization of the Alexander Polynomial from higher rank quantum groups”,

Why unoriented Khovanov homology is useful in graph theory

Scott Baldridge (LSU)

The Jones polynomial and Khovanov homology of a classical link are oriented link invariants—they depend upon an initial choice of orientation for the link. In this talk, we describe a Jones polynomial and Khovanov homology theory for unoriented virtual links. We then show how to transfer these unoriented versions over to graph theory to construct the Tait polynomial of a trivalent graph. This invariant polynomial counts the number of 3-edge colorings of a graph when evaluated at 1. If this count is nonzero for all bridgeless planar trivalent graphs, then the famous four color theorem is true. Thus, we show how topological ideas can be used to have an impact in graph theory. This is joint work with Lou Kauffman and Ben McCarty.


Baldridge, S., “A cohomology theory for planar trivalent graphs with perfect matchings”,

Baldridge, S., Lowrance, A., McCarty, B., “The 2-factor polynomial detects even perfect matchings”,

Baldridge, S., Kauffman, L. H., McCarty, B., “Unoriented Khovanov homology”,


Knot groups and virtual knots

H. A. Dye (Mckendree U) and A. Kaestner (North Park U)

In the paper, Virtual parity Alexander polynomials, we defined a virtual knot group that used information about the parity of the classical crossings. This virtual knot group was defined using ad-hoc methods. In the paper, Virtual knot groups and almost classical knots, Boden et al describe several different knot groups obtained from virtual knots. These knot groups are related and specializations lead to the classical knot group. Here, we construct a formal structure for virtual knot groups and examine specializations and extensions of the groups.


H. A. Dye and A. Kaestner, Virtual parity Alexander polynomials,


Triple linking and Heegaard Floer homology

Allison Moore (Virginia Commonwealth University)

We will describe several appearances of Milnor’s link invariants in the link Floer complex. This will include a formula that expresses the Milnor triple linking number in terms of the h-function. We will also show that the triple linking number is involved in a structural property of the d-invariants of surgery on certain algebraically split links. We will apply the above properties toward new detection results for the Borromean and Whitehead links. This is joint work with Gorsky, Lidman and Liu.


The Jones-Krushkal polynomial and minimal diagrams of surface links

Homayun Karimi (McMaster University)

We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating
links in thickened surfaces. It implies that any reduced alternating diagram of a link
in a thickened surface has minimal crossing number, and any two reduced alternating
diagrams of the same link have the same writhe. This result is proved more generally
for link diagrams that are adequate, and the proof involves a two-variable generalization
of the Jones polynomial for surface links defined by Krushkal. The main result is
used to establish the first and second Tait conjectures for links in thickened surfaces
and for virtual links.  This is joint work with Hans Boden.

Skein modules and framing changes for links in 3-manifolds

Rhea Palak Bakshi (George Washington University)

We show that the only way of changing the framing of a link by ambient isotopy in an oriented 3-manifold is when the manifold admits a properly embedded non-separating S^2. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough’s work on the mapping class groups of 3-manifolds. We also express our results in the language of skein modules. In particular, we relate our results to the presence of torsion in the framing skein module.

Some algebraic structures for flat virtual links

David Freund (Harvard University)

Flat virtual links are generalizations of curves on surfaces that have come out of Kauffman’s virtual knot theory. In particular, we consider the curves up to homotopy and allow the supporting surface to change via the addition or removal of empty handles. Under this equivalence, a flat virtual link obtains a completely combinatorial structure. Analogous to the classical problem of finding the minimal number of intersection points between two curves, we can ask for the minimal number of intersection points between components of a flat virtual link. By moving between geometric and combinatorial models, we develop generalizations of the Andersen-Mattes-Reshetikhin Poisson bracket and compute it for infinite families of two-component flat virtual links using a generalization of Henrich’s singular based matrix for flat virtual knots. Throughout the talk, we emphasize the motivation behind different constructions.

Diagrammatic Algebra, Part II

J. Scott Carter (University of South Alabama)

In this talk, I discuss replacing axioms in a Frobenius algebra with diagrams and constructing glyphs to represent those diagrams. The ideas are extended to considering isotopy classes of knots as a 4-category. Then we discuss braids, braided manifolds, braid movies, charts, chart movies, curtains, and curtain movies as methods of braiding simple branched covers in codimension 2. As usual, there are lots of diagrams.

What is a braidoid diagram?

Neslihan Gügümcü (Izmir Institute of Technology and University of Goettingen)

In this talk we first review the basics of the theory of knotoids introduced by Vladimir Turaev in 2012 [1]. A knotoid diagram is basically an open-ended knot diagram with two open endpoints that can lie in any local region complementary to the plane of the diagram. The theory of knotoids extends the classical knot theory and brings up some interesting problems and features such as the height problem [1,3] and parity notion and related invariants such as off writhe and parity bracket polynomial [4]. It was a curious problem to determine a “braid like object” corresponding to knotoid diagrams. The second part of this talk is devoted to the theory of braidoids, introduced by the author and Sofia Lambropoulou [2]. We present the notion of a braidoid and analogous theorems to the classical Alexander Theorem and the Markov Theorem, that relate knotoids/multi-knotoids in the plane to braidoids.

[1] V.Turaev, Knotoids, Osaka J.of Mathematics 49 (2012), 195–223.

[2] N.Gugumcu and S.Lambropoulou, Braidoids, to appear in Israel J.of Mathematics,

[3] N.Gugumcu and L. Kauffman, New Invariants of Knotoids, European J.of Combinatorics, (2017), 65C, 186-229,

[4] The Guassian parity and minimal diagrams of knot-type knotoids, submitted.

Invariants of framed links from cohomology of ternary self-distributive structures

Emanuele Zappala (University of South Florida)

In this talk I recall the definitions of shelf/rack/quandle and their cohomology theory. I also give the construction of cocycle invariant of links, due to Carter, Jelsovsky, Kamada, Langford and Saito (Trans. Amer. Math. Soc. 355 (2003), 3947-3989). Then, I introduce higher arity self-distributive structures and show that an appropriate diagrammatic interpretation of them is suitable to define a ternary version of the cocycle invariant for framed link invariants, via blackboard framings. I discuss the computation of cohomology of ternary structures, as composition of mutually distributive operations, and a cohomology theory of certain ternary quandles called group heaps. Furthermore I mention a categorical version of self-distributivity, along with examples from Lie algebras and Hopf algebras, and the construction of ribbon categories from ternary operations that provide a quantum interpretation of the (ternary) cocycle invariant. 



Elhamdadi, Mohamed, Masahico Saito, and Emanuele Zappala. “Higher Arity Self-Distributive Operations in Cascades and their Cohomology.” arXiv preprint arXiv:1905.00440 (2019).

Elhamdadi, Mohamed, Masahico Saito, and Emanuele Zappala. “Heap and Ternary Self-Distributive Cohomology.” arXiv preprint arXiv:1910.02877 (2019).

Zappala, Emanuele. “Non-Associative Algebraic Structures in Knot Theory.” (2020).

Lower bounds on the tunnel number of composite spatial theta graphs

Scott Taylor (Colby College)

The tunnel number of a graph embedded in a 3-dimensional manifold is the fewest number of arcs needed so that the union of the graph with the arcs has handlebody exterior. The behavior of tunnel number with respect to connected sum of knots can vary dramatically, depending on the knots involved. However, a classical theorem of Scharlemann and Schultens says that the tunnel number of a composite knot is at least the number of factors. For theta graphs, trivalent vertex sum is the operation which most closely resembles the connected sum of knots.The analogous theorem of Scharlemann and Schultens no longer holds, however. I will provide a sharp lower bound for the tunnel number of composite theta graphs,using recent work on a new knot invariant which is additive under connected sum and trivalent vertex sum. This is joint work with Maggy Tomova.

Rotational Virtual Links, Parity Polynomials and Quantum Link Invariants

Lou Kauffman (University of Illinois-Chicago & Novosibirsk State University)

This talk discusses virtual knot theory and rotational virtual knot theory. In virtual knot theory we introduce a virtual crossing in the diagrams along with over crossings and under crossings. Virtual crossings are artifacts of representing knots in higher genus surfaces as diagrams in the plane.

Virtual diagrammatic equivalence is the same as studying knots in thickened surfaces up to ambient isotopy, surface homeomorphisms and handle stabilization. At the diagrammatic level, virtual knot equivalence is generated by Reidemeister moves plus detour moves. In rotational virtual knot theory, the detour moves are restricted to regular homotopy of plane curves (the self-crossings are virtual). Rotational virtual knot theory has the property that all classical quantum link invariants extend to quantum invariants of rotational virtual knots and links. We explain this extension, and we consider the question of the power of quantum invariants in this context.

By considering first the bracket polynomial and its extension to a parity bracket polynomial for virtual knots (Manturov) and its further extension to a rotational parity bracket polynomial for knots and links (Kaestner and Kauffman), we give examples of links that are detected via the parity invariants that are not detectable by quantum invariants. In the course of the discussion we explain a functor from the rotational tangle category to the diagrammatic category of a quantum algebra. We delineate significant weaknesses in quantum invariants and how these gaps can be fulfilled by taking parity into account.

The slides for the talk are given below:



Kauffman, Louis H, Rotational virtual knots and quantum link invariants,

Diagrammatic Algebra, Part 1

J. Scott Carter (University of South Alabama)

This talk is based upon joint work with Seiichi Kamada. Abstract tensor notation for multi-linear maps uses boxes with in-coming and out-going strings to represent structure constants. A judicious choice of variables for such constants leads to the diagrammatic representation: boxes are replaced by glyphs. One of the most simple examples of a multi-category is the diagrammatic representation of embedded surfaces in 3-space that we all grew up learning. The standard drawing of a torus (an oval, a smile and a moustache) is a representation based upon drawings of surface singularities. We start from a two object category with a pair of arrows that relate them. Cups and caps are constructed easily. From these, births, deaths, saddles, forks, and cusps are created as triple arrows. At the top level, there are critical cancellations, lips, beak-to-beak, horizontal cusps, and swallow-tails. Interestingly, the structure extends inductively to describe many relations about handles in higher dimensions.

CKVK* Recommendation: The talk has many beautiful figures that are best viewed on Youtube in HD.

Extending Milnor’s concordance invariants to virtual knots and welded links

Micah Chrisman (Ohio State University)

Milnor’s \bar{\mu}-invariants for links in the 3-sphere vanish on any link concordant to a boundary link.  In particular, they are are trivial for any classical knot. Here we define an analogue of Milnor’s concordance invariants for knots in thickened surfaces \Sigma \times [0,1], where \Sigma is closed and oriented. These invariants vanish on any knot concordant to a homologically trivial knot in \Sigma \times [0,1]. We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index polynomial, and generalized Alexander polynomial. Moreover, we complete the slice status classification of the 2564 virtual knots having at most five classical crossings and reduce to four (of 90235) the number of virtual knots with six classical crossings having unknown slice status. Furthermore, we prove that in contrast to the classical knot concordance group, the virtual knot concordance group is not abelian. As part of the construction of the extended \bar{\mu}-invariants, we also obtain a generalization of the \bar{\mu}-invariants of classical links in S^3 to ribbon torus links in S^4 and welded links.

Slides for the talk are available here.


Boden, H.U., Chrisman, M., “Virtual concordance and the generalized Alexander polynomial”,

Chrisman, M., “Milnor’s concordance invariants for knots on surfaces”,


Knots with critical bridge spheres

Puttipong Pongtanapaisan (University of Iowa)

David Bachman introduced the notion of critical surfaces and showed that they satisfy useful properties. In particular, they behave like incompressible surfaces and strongly irreducible surfaces. In this talk, I will review some techniques that have been used to study bridge spheres and give examples of nontrivial knots with critical bridge spheres. This is joint work with Daniel Rodman.


Pongatanpaisan, P., Rodman, D., Critical Bridge Spheres for Links with Arbitrarily Many Bridges,

Hyperbolicity and Turaev hyperbolicity of virtual knots

Colin Adams (Williams College)

Abstract: Hyperbolic volume has been a powerful invariant for classical knots. In this talk we extend it to virtual knots, including calculations of virtual knot volumes. We further introduce Turaev volume for knots and virtual knots and show that EVERY knot has a Turaev volume.


Adams, C., Eiseberg, E., Greenberg, J. Kapoor, K., Liang, Z., O’Connor, K., Pacheco-Tallaj, N., Wang, Y., Turaev Hyperbolicity of Classical and Virtual Knots,

Adams, C., Eiseberg, E., Greenberg, J. Kapoor, K., Liang, Z., O’Connor, K., Pacheco-Tallaj, N., Wang, Y., Tg-Hyperbolicity of Virtual Links,

Dehn surgery on links vs the Thurston norm

Maggie Miller (Princeton University)

Abstract: David Gabai showed that a minimum-genus surface in a knot complement remains minimum-genus when capped off into the zero-surgery on that knot, implying that surgery on a nontrivial knot can never yield S^1\times S^2 (thus proving the Property R conjecture). I study this problem for links. In particular, I show that if L is a 2-component link with nonzero linking number and nondegenerate Thurston norm on its complement, then there exists a finite set E\subset H_2(S^3\setminus\nu(L),\partial;\mathbb{R}) so that if S is norm-minimizing and not in E up to scalar multiplication, then \hat{S} is norm-minimizing in the 3-manifold obtained from S^3 by doing Dehn surgery on L according to \partial S. (The result generally holds for n>1-component links with E (n-2)-dimensional.) The proof involves constructing a taut foliation on S^3\setminus\nu(L) with nice boundary properties, motivated by Gabai’s proof of the Property R conjecture.


Miller, M., The effect of link Dehn surgery on the Thurston norm.