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Results tagged with knot-theory
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user 5010
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.
41
votes
What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov,...
To strengthen Sam Nead's answer, note that it is trivial to compute the Jones polynomial from the Khovanov homology. It is known that computing (or even approximating) the Jones polynomial is #P-hard: …
- 10.1k
24
votes
What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov,...
This isn't directly what you ask, but it's also worth noting that unknot detection is in $\text{NP} \cap \text{co-NP}$, that is, there are polynomial-checkable certificates that will show that either …
- 10.1k
18
votes
3
answers
630
views
Classification of tangles?
Has anybody done any work on making a classification of low-complexity tangles, analogous to the work for knots and links? I expect most of the small ones to be rational, and those that aren't rationa …
- 10.1k
49
votes
2
answers
4k
views
Can knot diagrams be monotonically simplified using under moves?
It is well known that knot diagrams cannot be monotonically simplified using Reidemeister moves. For instance, the Goeritz unknot cannot be directly simplified. On the other hand, there is a stronger …
- 10.1k
10
votes
2
answers
238
views
What are the possible linking matrices of a quasi-positive link?
I was surprised recently to come across a 3-component link where the linking number of two of the components was negative. For a while I thought I had made a mistake, then I thought a little more and …
- 10.1k
8
votes
What are the points of Spec(Vassiliev Invariants)?
The points of Spec $V$ live n a somewhat complicated completion of the space of knots (not formal linear combinations), but I think it's illuminating to think about the case of braids and their "Vassi …
- 10.1k
10
votes
HOMFLY and homology; also superalgebras
Lots of good answers above, but the one that seems to be missing is what belongs in the third column for unspecialized HOMFLY. (As Geordie Williamson points out, Khovanov and Rozansky wrote two relat …
- 10.1k
10
votes
2
answers
579
views
Is the Lisca-Matic bound (aka slice-Bennequin bound) strictly stronger than the Bennequin bo...
The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$,
$$\mathrm{sl}(K) \le - \chi(\Sigma)$$
for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-l …
- 10.1k
6
votes
Do the results of (1/n)-surgery determine the link?...
If the orginal link $U_1 \cup U_2$ was hyperbolic, the answer is yes. For large enough $n$, $S^3 \setminus K(n)$ will also be hyperbolic, and will approach $S^3 \setminus (U_1 \cup U_2)$ in the Grom …
- 10.1k
20
votes
Accepted
Does this knot invariant distinguish trefoil chiralities?
I'm very curious where this came up. In any case, the answer to the first question is yes, it does distinguish these trefoils; you found the minimal representatives.
Let $a_0,\dots,a_{N-1}$ be the r …
- 10.1k
12
votes
Accepted
Maslov index and Heegard Floer homology
Robert Lipshitz has the nicest formula, described in Corollary 4.3 of this paper:
Robert Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geometry & Topology 10 (2006) 955–1096, DOI: …
- 10.1k
25
votes
Accepted
Who thought that the Alexander polynomial was the only knot invariant of its kind?
The skein relation approach to knot invariants was not very popular before the Jones polynomial. The Alexander polynomial was thought of as coming from homology (of the cyclic branched cover); Conway …
- 10.1k