nLab virtual knot theory

Idea

One fundamental tool in a knot theorist’s toolbox is the knot diagram. Classically, a knot is an embedding of the circle $S^1$ into Euclidean space $\mathbb{R}^3$, and knot diagrams arise by projecting back down to the plane $\mathbb{R}^2$ 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 $\mathbb{R}^3$, it could equally well be realized inside of a “thickened” sphere $S^2 \times [0,1]$, with its shadow embedded in $S^2$. From that perspective, virtual knot theory generalizes classical knot theory by considering knots as embeddings of circles into thickened orientable surfaces $X \times [0,1]$ of arbitrary genus. Abstractly, the shadow of such a knot is a 4-valent graph embedded in $X$ (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:

• As equivalence classes of virtual knot/link diagrams, modulo the “virtual” Reidemeister moves (Kauffman 1999).
• As equivalence classes of “abstract” knot/link diagrams embedded in orientable surfaces, modulo the Reidemeister moves (Kamada and Kamada 2000).
• As embeddings of circles into thickened orientable surfaces, considered up to isotopy (Kuperberg 2003).
• As equivalence classes of Gauss codes, modulo relations corresponding to the Reidemeister moves (Kauffman 1999).

Categorifications

The original work on virtual knot theory was not expressed in categorical language, but a first attempt at categorifying the virtual braid group $VB_n$ 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 $V$ equipped with an invertible morphism $\sigma : V\otimes V \to V\otimes V$ satisfying the Yang-Baxter equation

$$(\sigma \otimes V) \circ (V \otimes \sigma) \circ (\sigma \otimes V) = (V \otimes \sigma) \circ (\sigma \otimes V) \circ (V \otimes \sigma)$$

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

The intuition here is that the (symmetric) braiding of the ambient symmetric monoidal category represents virtual crossings “for free”, while the braiding $\sigma$ on the object $V$ 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.

Related concepts

• Gauss diagram
• embedded graph

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”.