Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with knot-theory
Search options not deleted
user 1465
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.
3
votes
Probabilistic knot theory
I believe there are a few known "random knotting" type results out there. Not the kind of results the original poster requested, but related. Take n points in R^3 generated by a random walk, join t …
- 44.3k
3
votes
Why "Categorify"? Relating to link/knot homologies...
A slightly different answer than Francesco's.
Why would one want to categorify? I think it comes down to how you want to think of knots. By that I mean, the kind of invariants you study very muc …
- 44.3k
18
votes
Why is it so hard to implement Haken's Algorithm for knot theory?
Ben Burton has implemented Haken's algorithm in Regina. He (and Rubinstein and Tillman) have a functional algorithm to determine if a normal surface in a triangulated 3-manifold is (in)compressible. …
- 44.3k
1
vote
Accepted
Cancellation of 2-component links
What you are talking about are called string links. String links have a stacking monoid operation. It's non-commutative (provided you have two or more strands). The invertible elements are precisel …
- 44.3k
7
votes
Are knots determined by their complements within a homotopy class?
The answer is no.
It stems from the fact that links are not determined by their complements. If you take the Borromean rings (for example), and think of one component as being a knot in the exterio …
- 44.3k
13
votes
Accepted
Smooth proof of Reidemeister theorem
I don't believe it's written up anywhere.
edit: in the comments Charlie Frohman corrects me:
MR2128054 (2005m:57041)
Roseman, Dennis(1-IA)
Elementary moves for higher dimensional knots. (Eng …
- 44.3k
6
votes
Knots and Dynamics. Recent breakthroughs?
There are certainly plenty of connections between knot theory and dynamical systems. On the fairly physical end of things, there was a recent workshop you might find interesting:
http://www.kitp.uc …
- 44.3k
4
votes
Questions about knot (link) of surface in four dimension
Given two disjoint surfaces $\Sigma_1, \Sigma_2$ in $\mathbb R^4$ there are the linking invariants
$$ l_1 : H_1 \Sigma_1 \to \mathbb Z $$
and
$$ l_2 : H_1 \Sigma_2 \to \mathbb Z $$
$l_1$ of a cy …
- 44.3k
17
votes
Complete knot invariant?
You're looking at the right paper. His results apply to a broad class of 3-manifolds, which knot complements happen to be a part of. I don't have the paper here with me but I believe the class was t …
- 44.3k
4
votes
Space of all topological knots (tame and wild)
A "long topological knot" in $\mathbb R^n$ is a topological embedding $f : \mathbb R \to \mathbb R^n$ such that $f(x) = (x,0)$ for all $x \in \mathbb R \setminus (-1,1)$.
Let $K_n$ be the space of …
- 44.3k
10
votes
Accepted
When is a connected sum of torus knots a slice knot?
I believe the answer to your last two questions is yes, and it follows from Litherland's (1979) computation of the Tristram-Levine invariants of torus knots. See Kearton's survey here:
http://www.m …
- 44.3k
1
vote
Construction of Kirby-diagram for slice-complement
It would be wonderful if there were bounds on the complexity of a slice disc for a slice knot, in terms of some kind of measure of the complexity of a knot -- crossing-number, or perhaps the number of …
- 44.3k
4
votes
Does every knot contain all four vertices of an isosceles trapezoid?
In the configuration space of 4 points in $\mathbb R^3$, the subspace of isosceles trapezoids has codimension three. So on a generic knot, you'd expect a 1-parameter (possibly empty) family of isosce …
- 44.3k
7
votes
Difference between Alexander polynomial and Blanchfield pairing
The Blanchfield pairing has many formulations, I like to think of it as a sesquilinear form:
$$ A \otimes A \to \Lambda / \mathbb Z[t^\pm] $$
where $A$ is the Alexander module and $\Lambda$ is the f …
- 44.3k
3
votes
Accepted
Is there a known method for finding the minimum bridge index of a knot?
I imagine bridge index is readily computable, at least for "small enough" knots. Bridge index is a lot like Heegaard genus of a 3-manifold. Heegaard splitting surfaces can be found via (almost) norm …
- 44.3k