Virtual Braids

By Louis H. Kauffman, Sofia Lambropoulou. 2004

99 citations recorded by the source corpus.

Just as classical knots and links can be represented by the closures of braids, so can virtual knots and links be represented by the closures of virtual braids [17]. Virtual braids have a group structure that can be described by generators and relations, generalizing the generators and relations of the classical braid group. This structure of virtual braids is worth study for its own sake. The virtual braid group is an extension of the classical braid group by the symmetric group. In [13] a Markov Theorem is proved for virtual braids, giving a set of moves on virtual braids that generate the same equivalence classes as the virtual link types of their closures. Such theorems are important for understanding the structure and classification of virtual knots and links.

Subfields: Computational Mechanics, Computer Graphics and Computer-Aided Design, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/adb4a90db6dc7346fd18b9bb9e88d25bd1c2e0fd

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Louis H. Kauffman
2. Jenny Wilson
3. Sofia Lambropoulou
4. H. Dye

In Computer Graphics and Computer-Aided Design

1. Louis H. Kauffman
2. Jenny Wilson
3. Sofia Lambropoulou
4. Dror Bar-Natan
5. Zhiyun Cheng
6. Hongzhu Gao

In Geometry and Topology

1. Louis H. Kauffman
2. Sofia Lambropoulou
3. H. Dye
4. Jenny Wilson
5. Jacob Rasmussen
6. Zsuzsanna Dancso
7. Dror Bar-Natan
8. C. Damiani

Citations in the corpus

• FI_W-modules and stability criteria for representations of the classical Weyl groups (2013, background): “For details see for example Kauffman [Kau99, Kau00], Vershinin [Ver01], Kauffman– Lambropoulou [KL04], Bardakov [Bar04], and Bartholdi–Enriquez–Etingof– Rains [BEER06].”
• VIRTUAL BRAIDS AND THE L-MOVE (2005, methodology): “In [21] we gave, for example, a new braiding algorithm, which is applicable, in fact, to all the categories in which braids are constructed.”
• Virtual knot theory-unsolved problems (2005, unknown): “Is there a birack such that its action on virtual braids is faithful? Is the invariant of virtual braids in [87] (see also [74, 3, 57]) faithful? The action defined by linear biquandles is not faithful.”
• Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial (2014, background): “Our main object of study for this section, w-braids, are best viewed as “virtual braids” [24; 55; 25], or v-braids, modulo one additional relation; hence, we start with v-braids.”
• Virtual Crossing Number and the Arrow Polynomial (2008, background): “Note that for any virtual or classical link, we can construct an equivalent link diagram that is a braid [8], [9].”