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virtual knot theory

Contents

Idea

One fundamental tool in a knot theorist’s toolbox is the knot diagram. Classically, a knot is an embedding of the circle S 1S^1 into Euclidean space 3\mathbb{R}^3, and knot diagrams arise by projecting back down to the plane 2\mathbb{R}^2 to obtain a 4-valent plane graph (referred to as the shadow of the knot), while keeping track of whether each vertex corresponds to an under-crossing or an over-crossing. Although the knot is traditionally seen as embedded in 3\mathbb{R}^3, it could equally well be realized inside of a “thickened” sphere S 2×[0,1]S^2 \times [0,1], with its shadow embedded in S 2S^2. From that perspective, virtual knot theory generalizes classical knot theory by considering knots as embeddings of circles into thickened orientable surfaces X×[0,1]X \times [0,1] of arbitrary genus. Abstractly, the shadow of such a knot is a 4-valent graph embedded in XX (i.e., a topological map). However, if one tries to project the knot onto the plane, the corresponding diagram might contain crossings that do not represent places where the knot passes over/under itself inside the thickened surface, but rather are artifacts of the knot’s non-planar shadow. So, in a virtual knot diagram such crossings are explicitly indicated as “virtual”, using a distinct notation from that for under/overcrossings.

Definitions

There are various equivalent definitions of virtual knots/links:

Categorifications

The original work on virtual knot theory was not expressed in categorical language, but a first attempt at categorifying the virtual braid group VB nVB_n was made by Kauffman and Lambropoulou. Victoria Lebed studied the question extensively in her thesis (Lebed 2012), and developed a categorical approach based on the notion of a braided object in a symmetric monoidal category, that is an object VV equipped with an invertible morphism σ:VVVV\sigma : V\otimes V \to V\otimes V satisfying the Yang-Baxter equation

(σV)(Vσ)(σV)=(Vσ)(σV)(Vσ)(\sigma \otimes V) \circ (V \otimes \sigma) \circ (\sigma \otimes V) = (V \otimes \sigma) \circ (\sigma \otimes V) \circ (V \otimes \sigma)

In particular, Lebed shows that the virtual braid group VB nVB_n is isomorphic to the group of endomorphisms End 𝒞 2br(V n)End_{\mathcal{C}_{2br}}(V^{\otimes n}), where 𝒞 2br\mathcal{C}_{2br} is the free symmetric monoidal category generated by a single braided object VV.

The intuition here is that the (symmetric) braiding of the ambient symmetric monoidal category represents virtual crossings “for free”, while the braiding σ\sigma on the object VV represents “real” over- and under-crossings. (Compare some remarks by John Baez, which are similar in spirit.)

Terminology

Warning: a virtual knot/link has a genus in the sense of the genus of the underlying thickened surface into which it embeds (or equivalently, the genus of its shadow as a topological map), but this is unrelated to the classical notion of knot genus, in the sense of the minimal genus of a Seifert surface whose boundary is the knot.

References

The original paper on virtual knot theory and some early followup work:

  • Louis Kauffman, Virtual Knot Theory, European Journal of Combinatorics (1999) 20, 663-691. pdf

  • Naoko Kamada and Seiichi Kamada, Abstract Link Diagrams and Virtual Knots, Journal of Knot Theory and its Ramifications, Vol. 9, No. 1 (2000), 93-106. doi

  • Greg Kuperberg, What is a virtual link?, Algebraic & Geometric Topology Volume 3 (2003), 587-591. pdf

Miscellaneous papers:

  • Louis Kauffman, Introduction to Virtual Knot Theory. July 2012. arXiv

  • Oleg Viro, Virtual Links, Orientations of Chord Diagrams and Khovanov Homology, Proceedings of 12th Gökova Geometry-Topology Conference, pp. 184–209, 2005. pdf

  • Vassily Olegovich Manturov and Denis Petrovich Ilyutko, Virtual Knots: The State of the Art. Series on Knots and Everything (vol. 51), World Scientific, 2013.

  • Louis Kauffman, Rotational Virtual Knots and Quantum Link Invariants, 14 Oct 2015. arxiv:1509.00578

On the question of categorifying virtual knot theory, see:

  • John Baez, comment at the n-Café following a post titled “Categorification in New Scientist”, October 2008.

  • Louis Kauffman and Sofia Lambropoulou, A Categorical Structure for the Virtual Braid Group, Comm. Algebra, 39(12):4679–4704, 2011. (pdf)

  • Victoria Lebed, Objets tressés: une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles, Thèse, Université Paris Diderot, 2012. (pdf) Note that the title is in French (“Braided objects: a unifying study of algebraic structures and a categorification of virtual braids”) but the main text of the thesis is in English.

  • Victoria Lebed, Categorical Aspects of Virtuality and Self-Distributivity, Journal of Knot Theory and its Ramifications, 22 (2013), no. 9, 1350045, 32 pp. (doi) According to the author, arXiv:1206.3916 is “an extended version of the above JKTR publication, containing in particular a chapter on free virtual shelves and quandles”.

Revised on December 22, 2016 10:04:07 by Noam Zeilberger (213.111.4.76)