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Results tagged with knot-theory
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user 2051
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
How much do homological knot invariants improve the classification problem of knots?
I don't think that quantum topology or Heegaard Floer homology are useful as knot tabulation tools. Geometric invariants (e.g. hyperbolic volume) and classical invariants (e.g. signature) have been th …
- 22.1k
5
votes
$\omega$-triviality of knots?
I would guess that you would need to understand the transfinite lower central series of the mapping class group. A knot complement which could not be distinguished from a solid torus by any finite low …
- 22.1k
4
votes
Knot invariants with skein relation of order 3 or 4
The Links-Gould invariant, a 2-variable polynomial invariant coming from a one-parameter family of representations of the quantum superalgebra $\mathcal{U}_q(\mathfrak{gl}(2|1))$, satisfies the follow …
- 22.1k
2
votes
Quandle colorings under Reidemeister moves
No.
For example, form a small loop with an R1 move on a strand coloured $x$, imagine the whole rest of the knot inside a small ball, and pass that ball on a loop-the-loop through the R1 loop. Then pe …
- 22.1k
6
votes
Measures of entangledness of an open curve
Peter Roegen works on this problem, with the practical goal of effectively identifying certain knotted proteins. His descriptors (not "invariants", because open curves are topologically unknotted) are …
- 22.1k
4
votes
Is there any analogs of Vassiliev invariants in higher dimensions?
This is a nice question. There is actually quite a bit of work which has been done along these lines, although we are a very long way from having a good understanding of how a theory of finite-type in …
- 22.1k
3
votes
Can you do surgery on framed tangles?
You can cut out a collection of 3-balls (a regular neighbourhood of the tangle), and glue them back in a different way. Such modifications are a part of the Montesinos trick. Montesinos uses them to p …
- 22.1k
6
votes
What is the metamathematical interpretation of knot diagrams?
Knot diagrams are a special sort of tangle diagrams, so I will reinterpret your question as being about tangle diagrams. Tangle diagrams are a "planar algebra" generated by $\{\text{overcrossing},\tex …
8
votes
Accepted
Rational homology spheres and knots
For Question 1, I believe that the answer follows from:
Montesinos, José M. Surgery on links and double branched covers of $S^3$. Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. …
- 22.1k
1
vote
Difference between Alexander polynomial and Blanchfield pairing
Clearly the signature, and more generally the Tristram-Levine signatures, would be needed.
- 22.1k
8
votes
2
answers
1k
views
Difference between Alexander polynomial and Blanchfield pairing
For a Seifert matrix $V$ of a knot $K$, the Alexander module has presentation matrix $V-tV^T$. The determinant of this matrix is the Alexander polynomial, which is the order of the Alexander module. I …
- 22.1k
4
votes
2
answers
235
views
Order of "one minus automorphism"
This is something I am stuck on (it might well be trivial- in which case this is an embarassing question):
Let V be a dimension r vector space over Fp, the field with p prime elements (I also care abo …
- 22.1k
8
votes
2
answers
858
views
Are there any very hard unlinks?
This question is closely related to a question of Gowers: Are there any very hard unknots? .
I'm thinking about how to create interesting knots from small numbers of local moves on unlinks. The "stand …
- 22.1k
6
votes
Best Computational Knot Invariants
If your polymer chains are open (embedded closed line segment in 3-space), then I wouldn't recommend using global knot invariants (Alexander polynomial, Jones polynomial) because they will not make an …
- 22.1k
23
votes
In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?
This is a nice question!
Knot theory is in fact knot-complement theory, and a knot complement in S3 is a compact 3-manifold, while a knot complement in R3 is an open 3-manifold. Compact (or closed) 3- …
- 22.1k