Questions tagged [virtual-knots]

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6
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0answers
104 views

Is there a natural, purely group-theoretic definition of the virtual braid group?

The braid group $B_n$ has the well-known presentation $$\left<\sigma_i,i=1\ldots n-1\, \left| \begin{cases}\sigma_i\sigma_j=\sigma_j\sigma_i & |i-j|>1\\\sigma_i\sigma_j\sigma_i=\sigma_j\...
1
vote
0answers
58 views

Common invariants of virtual knots?

When I google invariants of virtual knots (links), there's a bunch of polynomial (and other) invariants, but it's very hard to distinguish which of these are considered as somewhat classical. In the ...
4
votes
1answer
193 views

Presentations of the monoidal categories of virtual tangles and of welded tangles by generators and relations

Reidemeister theorem implies, without too much fuss, that the monoidal categories of tangles, and of oriented tangles, can be presented by generators and relations. This is done for example in a) ...
6
votes
1answer
125 views

What is the original reference for disorientations on tangle diagrams?

There are several invariants whose "natural" domain is a category of disoriented tangles, that is tangles which are piecewise-oriented, but which contain points called `disorientations' at which the ...
2
votes
3answers
635 views

Gauss Codes that produce classical knots as opposed to virtual knots

I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that If $K$ is a virtual knot whose underlying Gauss ...
2
votes
0answers
117 views

Tait conjectures for alternating w-links

The Tait Conjectures are useful in knot tabulation. For alternating knots and links, two of them state: Any reduced diagram of an alternating link has the fewest possible crossings. Any two reduced ...
1
vote
2answers
329 views

Commutativity in the Fundamental Group and Knot Theory

Let $M$ be a connected $3$-manifold and let $\alpha$ and $\beta$ be elements in $\pi_1(M)$. Then $\alpha$ and $\beta$ can be represented by two knots $a$ and $b$ in $M$. We may further require that ...
8
votes
1answer
851 views

Diagrammatic proof of unique prime decomposition of knots

Consider a knot to be a diagram in a plane--- i.e. a drawing of a finite connected planar graph (loops and multiple edges allowed) whose vertices are 4-valent with cyclic ordering for the incident ...
2
votes
1answer
219 views

Application of a quandle cocycle invariant for virtual knots

In knot theory, a quandle cocycle invariant was defined. Moreover, to virtual knot theory it was generalized by avoiding for virtual crossings. Question Are there many application of a quandle ...
0
votes
1answer
318 views

Validity of generalized Reidemeister moves for a virtual knot

I am studying virtual knot theory. A virtual knot is a knot diagram with real or virtual crossing information. The equivalence relation includes generalized Reidemeister moves. There are premitted ...
23
votes
4answers
2k views

Utility of virtual knot theory?

Virtual knot theory is an interesting generalization of knot theory in which ``virtual" crossings are allowed. See Kauffman's Virtual Knot Theory for an introduction. Greg Kuperberg gave a nice ...