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Results tagged with knot-theory
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user 5001
Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.
7
votes
Accepted
Links with same Jones polynomial
Probably the best way to produce infinite families of links with same Jones polynomial is by Conway mutation, this operation does not alter the HOMFLY polynomial either. A good example
of this is give …
- 587
6
votes
1
answer
480
views
Computations and applications of Khovanov's functor valued invariant of tangles
Soon after his famous paper "A categorification of the Jones polynomial", Khovanov
introduced a "bordered" version. His theory assingns to every oriented even tangle
a complex of (H_n,H_m)-bimodules, …
- 587
8
votes
Complexity of surfaces bounding knots in 4-ball and 3-sphere respectively
This is an answer to the first question.
Let's indicate with $r(K)$ the ribbon number of a knot, i.e.the minimum
number of ribbon singularities needed to realize a ribbon disc spanning $K$.
We have
$ …
- 587
6
votes
1
answer
2k
views
Boundary links and ribbon links.
This question is about the relation between the notions of boundary link and ribbon link.
For the definition of ribbon link see: ribbon links - counterexamples.
An n-component link $L=L_1\cup\dots\c …
- 587
6
votes
1
answer
1k
views
When is a connected sum of torus knots a slice knot?
This question is about the beaviour of 4-genus of knots with respect to connected sum.
Let us indicate with $T(k)$ a Torus knot of type $(2,k)$, $k$ is an odd integer.
Fix an orientation for every $T …
- 587
6
votes
1
answer
652
views
How to distinguish Pretzel links with the same coefficients?
Let $P:=P(a_1,\dots,a_n)$ be a Pretzel link ( https://en.wikipedia.org/wiki/Pretzel_link ).
For every permutation $\sigma\in S_n$ we can consider the link $$\sigma P:=P(a_{\sigma(1)},\dots,a_{\sigma(n …
- 587
3
votes
0
answers
641
views
ribbon links - counterexamples
An n-component link is said to be ribbon if it bounds a ribbon surface consisting of n discs.
(a ribbon surface is an immersed surface with only ribbon singularities, see http://en.wikipedia.org/wiki/ …
- 587
1
vote
2
answers
818
views
Trivial pretzel links
Is there a simple rule to check whether a pretzel link P(n_1,...,n_k) is a trivial link?
I am interested in the 2-component case but every information would be helpful.
- 587
2
votes
Who thought that the Alexander polynomial was the only knot invariant of its kind?
You can find some historical remarks on polynomial invariants in chapter 9 of the book
"Knots and links" by Cromwell, he also gives a lot of references.
- 587
2
votes
2
answers
1k
views
Counting knots with fixed number of crossings
How to obtain an upperbound for knots up to k crossings?
I think I've found something which involves the genus but I'm not sure.
- 587