Planar knots and related groups

Mahender Singh (IISER)

Study of stable isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces can be thought of as a planar analogue of virtual knot theory where the sphere case corresponds to the classical knot theory. It is intriguing to know which class of groups serves the purpose that Artin braid groups serve in classical knot theory. Khovanov proved that twin groups, a class of right angled Coxeter groups with only far commutativity relations, serves the purpose for the sphere case. In a recent work we showed that an appropriate class of groups called virtual twin groups fits into a virtual analogue of the planar knot theory. The talk will give an overview of some recent topological and group theoretic developments along these lines.

Virtual braid groups, virtual twin groups and crystallographic groups

Oscar Ocampo (Universidade Federal da Bahia)

Let n\ge 2. Let VB_n (resp. VP_n) be the virtual braid group (resp. the pure virtual braid group), and let VT_n (resp. PVT_n) be the virtual twin group (resp. the pure virtual twin group). Let \Pi be one of the following quotients: VB_n/\Gamma_2(VP_n) or VT_n/\Gamma_2(PVT_n) where \Gamma_2(H) is the commutator subgroup of H.

In this talk, we show that \Pi is a crystallographic group and then and then we explore some of its properties, such as: characterization of finite order elements and its conjugacy classes, and also the realization of some Bieberbach groups and infinite virtually cyclic groups. Finally, we also consider other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) module the respective commutator subgroup in each case.

Joint work with Paulo Cesar Cerqueira dos Santos Júnior (arXiv: 2110.02392)

Slides for talk!

PSL(2,C) Representations of Knot Groups

Kate Petersen (University of Minnesota Duluth)

I will discuss a method of producing defining equations for representation varieties of the canonical component of a knot group into PSL(2,C). This method uses only a knot diagram satisfying a mild restriction and is based upon the underlying geometry of the knot complement. In particular, it does not involve any polyhedral decomposition or triangulation of the link complement. This results in a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. This is joint work with Anastasiia Tsvietkova.

Geometry of alternating links on surfaces

Jessica Purcell (Monash University)

It is typically hard to relate the geometry of a knot complement to a diagram of the knot, but over many years mathematicians have been able to relate geometric properties of classical alternating knots to their diagrams. Recently, we have modified these techniques to investigate geometry of a much wider class of knots, namely alternating knots with diagrams on general surfaces embedded in general 3-manifolds. This has resulted in lower bounds on volumes, information on the geometry of checkerboard surfaces, restrictions on exceptional Dehn fillings, and other geometric properties. However, we were unable to extend upper volume bounds broadly. In fact, recently we showed an upper bound must depend on the 3-manifold in which the knot is embedded: We find upper bounds for virtual knots, but not for other families. We will discuss this work, and some remaining open questions. This is joint in part with Josh Howie and in part with Effie Kalfagianni.

Knots in overtwisted contact manifolds

Rima Chaterjee (University of Cologne)

Knots associated to overtwisted manifolds are less explored. There are two types of knots in an overtwisted manifold – loose and non-loose. Non-loose knots are knots with tight complements where as
loose knots have overtwisted complements. While we understand loose knots, non-loose knots remain a mystery. The classification and structure problems of these knots vary greatly compared to the knots
in tight manifolds. In this talk, I’ll give a brief survey followed by some interesting recent work. Especially I’ll show how satellite operations on a knot in overtwisted manifold changes the geometric
property of the knot. I will discuss under what conditions cabling operation on a non-loose knot preserves non-looseness. The ”recent part” of this talk is based on joint work  with Etnyre, Min and Mukherjee.

The Jones Polynomial from a Goeritz Matrix

Joe Boninger (CUNY)

The Jones polynomial holds a central place in knot theory, but its topological meaning is not well understood—it remains an open problem, posed by Atiyah, to give a three-dimensional interpretation of the polynomial. In this talk, we’ll share an original construction of the Jones polynomial from a Goeritz matrix, a combinatorial object with topological significance. In the process we extend the Kauffman bracket to any symmetric, integer matrix, with applications to links in thickened surfaces. Matroid theory plays a role.

Khomotopy type via simplicial complexes and presimplicial sets

Marithania Silvero (Universidad de Sevilla)

At the end of the past century, Mikhail Khovanov introduced the first homological invariant, now known as Khovanov homology, as a categorification of Jones polynomial. It is a bigraded homology supported in homological and quantum gradings. Given a link diagram, we refer to the maximal (resp. second-to-maximal) quantum grading such that the associated Khovanov complex is non-trivial as extreme (resp. almost extreme) grading.

In this talk we present a new approach to the geometrization of Khovanov homology in terms of simplicial complexes and presimplicial sets, for the extreme and almost-extreme gradings, respectively. We also study the relations of these models with Khovanov spectra, introducec by Robert Lipshitz and Sucharit Sarkar as a spatial refinement of Khovanov homology.

Constructions toward topological applications of U(1) x U(1) equivariant Khovanov homology

Melissa Zhang (University of Georgia)

In 2018, Khovanov and Robert introduced a version of Khovanov homology over a larger ground ring, termed U(1)xU(1)-equivariant Khovanov homology. This theory was also studied extensively by Taketo Sano. Ross Akhmechet was able to construct an equivariant annular Khovanov homology theory using the U(1)xU(1)-equivariant theory, while the existence of a U(2)-equivariant annular construction is still unclear.

Observing that the U(1)xU(1) complex admits two symmetric algebraic gradings, those familiar with knot Floer homology over the ring F[U,V] may naturally ask if these filtrations allow for algebraic constructions already seen in the knot Floer context, such as Ozsváth-Stipsicz-Szabó’s Upsilon. In this talk, I will describe the construction and properties of such an invariant. I will also discuss some ideas on how future research might use the U(1)xU(1) framework to identify invariants similar to those constructed from knot Floer homology over F[U,V], and speculate on the topological information these constructions might illuminate.

This is based on joint work with Ross Akhmechet.

Khovanov Homology for Virtual Links

Ben McCarty (University of Memphis)

Shortly after Khovanov homology for classical knots and links was developed in the late 90s, Bar-Natan produced a Mathematica program for computing it from a planar diagram. Yet when Manturov defined Khovanov homology for virtual links in 2007, a program for computing it did not appear until Tubbenhauer’s 2012 paper. Even then, the theoretical framework used was quite different from the one Manturov described. In this talk, we describe a theoretical framework for computing the Khovanov homology of a virtual link, synthesized from work by Manturov, Dye-Kaestner-Kauffman and others. We also show how to use this framework to create a program for computing the Khovanov homology of a virtual link, which is directly based upon Bar-Natan’s original program for classical links.  This program is joint work with Scott Baldridge, Heather Dye, Aaron Kaestner, and Lou Kauffman.

Meridional rank and bridge numbers of knotted surfaces and welded knots

Jason Joseph (Rice University)

The meridional rank conjecture (MRC) posits that the meridional rank of a classical knot is equal to its bridge number. In this talk we investigate whether or not this is a reasonable conjecture for knotted surfaces and welded knots. In particular, we find criteria to establish the equality of these values for several large families. On the flip side, we examine the behavior of meridional rank of knotted spheres under connected sum, and, using examples first studied by Kanenobu, show that any value between the theoretical limits can be achieved. This means that either the MRC is false for knotted spheres, or that their bridge number fails to be (-1)-additive. This is joint work with Puttipong Pongtanapaisan.

Flattening Knotted Surfaces

Eva Horvat (University of Ljubljana)

A knotted surface \mathcal{K} in the 4-sphere admits a projection to a 2-sphere, whose set of critical points coincides with a hyperbolic diagram of \mathcal{K}. We apply such projections, called flattenings, to define three invariants of embedded surfaces: the width, the trunk and the partition number. These invariants are studied for some families of embedded surfaces.

Welded Tangles and the Kashiwara-Vergne Groups

Zsuzsanna Dancso (The University of Sydney)

In this talk I will explain a general method of “translating” between a certain type of problem in topology, and solving equations in graded spaces in (quantum) algebra. I’ll talk through several applications of this method from the 90’s to today: Drinfel’d associators and parenthesised braids, Grothendieck-Teichmuller groups, welded tangles and the Alekseev-Enriquez-Torossian formulation of the Kashiwara-Vergne equations, and most recently, a topological description of the Kashiwara-Vergne groups. The “recent” portion of the talk is based on joint work with Iva Halacheva and Marcy Robertson (arXiv: 2106.02373), and joint work with Dror Bar-Natan (arXiv: 1405.1955).

Cocycle Enhancements of Psyquandle Counting Invariants

Jose Ceniceros (Hamilton College)

In this talk we discuss pseudoknots and singular knots. Specifically, we discuss psyquandles and their application to oriented pseudoknots and oriented singular knots. Additionally, we bring cocycle enhancement theory to the case of psyquandles to define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function. As an application, we define a single-variable polynomial invariant of both oriented pseudoknots and oriented singular knots.

Chord Index type invariants of virtual knots

Zhiyun Cheng (Beijing Normal University)

As an extension of classical knot theory, virtual knot theory studies the embeddings of one sphere in thickened surfaces up to stable equivalence. Roughly speaking, there are two kinds of virtual knot invariants, the first kind comes from knot invariants of classical knots but the second kind usually vanishes on classical knots. Most of the second kind of virtual knot invariants are defined by using the chord parity or chord index. In this talk, I will report some recent progress on virtual knot invariants derived from various chord indices.

Minimal crossing number implies minimal supporting genus

Will Rushworth (McMaster University)

We prove that a minimal crossing virtual link diagram is a minimal genus diagram. That is, the genus of its Carter surface realizes the supporting genus of the virtual link represented by the diagram. The result is obtained by introducing a new parity theory for virtual links. This answers a basic question in virtual knot theory, and recovers the corresponding result of Manturov in the case of virtual knots.
Joint work with Hans Boden.

Classification of welded links and welded string links up to w_q-concordance

Boris Colombari (Aix-Marseille Université)

The notion of w_q-concordance has been introduced by J-B. Meilhan and A. Yasuhara through the use of arrow calculus. It is a welded analogue of the C_k-concordance on classical links coming from the clasper calculus introduced by K. Habiro. In this talk I will present a classification of welded string links (resp. welded links) up to w_q-concordance by their Milnor invariants (resp. by their q-nilpotent peripheral system). I will compare these results to the classification of classical links up to C_k-concordance obtained by J. Conant, R. Schneiderman and P. Teichner before introducing the relevant invariants on welded objects. I will give elements of the proof of my results using a version of arrow calculus adapted to the representation of welded objects by Gauss diagrams.

Region coloring invariants for knots

Sam Nelson (Claremont McKenna University)

In this talk we will survey some region-coloring structures for knots (Niebrzydowski tribrackets, virtual tribrackets, multitribrackets and psybrackets) and related structures and see applications to counting invariants and enhancement as well as some applications to music.

Arrow calculus for welded links

Akira Yasuhara (Waseda University)

We develop a calculus for diagrams of knotted objects. We define arrow presentations, which encode the crossing information of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w–tree presentations, which can be seen as “higher-order Gauss diagrams”. This arrow calculus is used to develop an analogue of Habiro’s clasper theory for welded knotted objects, which contain classical link diagrams as a subset. This provides a “realization” of Polyak’s algebra of arrow diagrams at the welded level, and leads to a characterization of finite- type invariants of welded knots and long knots. This is a joint work with Jean-Baptiste Meilhan (University of Grenoble Alpes).

The reductivity of a knot projection

Ayaka Shimizu (NIT, Gunma College, Japan)

The reductivity of a knot projection is defined to be the minimum number of splices required to make the projection reducible, where the splices are applied to a knot projection resulting in another knot projection. It has been shown that the reductivity is four or less for any knot projection and shown that there are infinitely many knot projections with reductivity 0, 1, 2, and 3. The “reductivity problem” is a problem asking the existence of a knot projection whose reductivity is four. In this talk, we will discuss some strategies for the reductivity problem focusing on the region of a knot projection.

Slides for talk:


Characterization(s) of the Reduced Peripheral System

Jean-Baptiste Meilhan (Université Grenoble Alpes)

The reduced peripheral system was introduced by Milnor in the 50’s for the study of links up to link-homotopy, i.e. up to homotopies leaving distinct components disjoint. This invariant, however, fails to classify links up to link-homotopy for links of 4 or more components. The purpose of this paper is to show that the topological information which is detected by Milnor’s reduced peripheral system is actually 4-dimensional. We give a topological characterization in terms of ribbon solid tori in 4-space up to link-homotopy, using a version of Artin’s Spun construction. The proof relies heavily on an intermediate characterization, in terms of welded links up to self-virtualization, providing hence a purely topological application of the combinatorial theory of welded links.

Arf invariants in low dimensions

Michael Klug (UC-Berkeley)

I will briefly discuss some work in progress regarding a relationship between several different mod 2 invariants in dimensions 2, 3, and 4. In particular, I will relate the Arf invariant of a knot, the Arf invariant of a characteristic surface, the Rochlin invariant of a homology sphere, and the Kirby-Siebenmann invariant of a 4-manifold.

Pure Braids and Link Concordance

Miriam Kuzbary (Georgia Tech)

The knot concordance group can be contextualized as organizing problems about 3- and 4-dimensional spaces and the relationships between them. Every 3-manifold is surgery on some link, not necessarily a knot, and thus it is natural to ask about such a group for links. In 1988, Le Dimet constructed the string link concordance groups and in 1998, Habegger and Lin precisely characterized these groups as quotients of the link concordance sets using a group action. Notably, the knot concordance group is abelian while, for each n, the string link concordance group on n strands is non-abelian as it contains the pure braid group on n strands as a subgroup. In this talk, I will discuss my result the quotient of each string link concordance group by its pure braid subgroup is still non-abelian.

Movie moves for singular link cobordisms in 4-dimensional space

Carmen Caprau (California State University-Fresno)

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.

Slides for talk:

CKVK Seminar-March 1, 2021

Links in homology spheres are homotopic to slice links – an application of the relative Whitney trick.

Christopher Davis (U. Wisconsin-Eau Claire)

Generalizing the notion of sliceness for links in S^3, a link in a homology sphere is called slice if it bounds a disjoint union of locally flat embedded disks in a contractible 4-manifold. It is trivial to see that any link in S^3 can be changed by a homotopy to a slice link, indeed any link is homotopic to the unlink. We prove that the same is true for links in homology spheres. Our argument passes through a novel geometric construction which we call the relative Whitney trick. If time permits we will explore an application of the relative Whitney trick to the existence of Whitney towers.

Slides for talk:


A volumish theorem for alternating virtual links

Ilya Kofman (CUNY)

Dasbach and Lin proved a “volumish theorem” for alternating links. We prove the analogue for alternating link diagrams on surfaces, which provides bounds on the hyperbolic volume of a link in a thickened surface in terms of coefficients of its reduced Jones-Krushkal polynomial. Joint work with Abhijit Champanerkar.


Champanerkar, Abhijit and Kofman, Ilya, “A volumish theorem for alternating virtual links”,

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.