Virtual Knots

Planar knots and related groups

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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.

Geometry of alternating links on surfaces

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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.

Chord Index type invariants of virtual knots

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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.

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

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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:

GL-pairing-talk

 

Classifying small virtual skein theories

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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.

Knot groups and virtual knots

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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.

References:

H. A. Dye and A. Kaestner, Virtual parity Alexander polynomials, https://arxiv.org/abs/1907.08709

 

Some algebraic structures for flat virtual links

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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.

Rotational Virtual Links, Parity Polynomials and Quantum Link Invariants

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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:

RotationalVirtualKnotsKeynote.key

References

Kauffman, Louis H, Rotational virtual knots and quantum link invariants, https://arxiv.org/abs/1509.00578

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

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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.

References:

Boden, H.U., Chrisman, M., “Virtual concordance and the generalized Alexander polynomial”, https://arxiv.org/pdf/1903.08737.pdf

Chrisman, M., “Milnor’s concordance invariants for knots on surfaces”, https://arxiv.org/pdf/2002.01505.pdf

 

Hyperbolicity and Turaev hyperbolicity of virtual knots

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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.

References:

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, https://arxiv.org/pdf/1912.09435.pdf

Adams, C., Eiseberg, E., Greenberg, J. Kapoor, K., Liang, Z., O’Connor, K., Pacheco-Tallaj, N., Wang, Y., Tg-Hyperbolicity of Virtual Links, https://arxiv.org/pdf/1904.06385.pdf