All Questions
40 questions
29
votes
5
answers
7k
views
Proof of the Reidemeister theorem
While preparing for my introduction to topology course, I've realized that I don't know where to find a detailed proof of the Reidemeister theorem (two link diagrams give isotopic links, iff they can ...
- 23.5k
26
votes
5
answers
2k
views
Complexity of random knot with vertices on sphere
Connect $n$ random points on a sphere in a cycle of
segments between succesive points:
I would like to know the growth rate, with respect to $n$, of the crossing number
(the minimal number of ...
- 151k
18
votes
3
answers
1k
views
Classification of knots by geometrization theorem
I read this interview with Ian Agol, where he says:
"...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots."
My question is:
How does this ...
- 1,314
17
votes
3
answers
2k
views
What is the state of the art for algorithmic knot simplification?
Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, ...
- 22.1k
14
votes
4
answers
1k
views
Obtain any 3-manifold from repeating surgeries on knots in $S^3$
In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) ...
- 755
13
votes
3
answers
1k
views
Random Reidemeister moves to unknot
Suppose one has a link diagram of the unknot, and applies random Reidemeister moves
until the unknot is reached.
Surely it requires an exponential number of moves, exponential in, say, the crossing ...
- 151k
12
votes
4
answers
1k
views
Elementary proof that knot complements are path-connected
The complement of any (topological) knot is path-connected. More precisely, if $K$ is a subset of $\mathbb{R}^3$ (or $S^3$) homeomorphic to $S^1$, then $\mathbb{R}^3\setminus K$ (or $S^3\setminus K$) ...
- 35.9k
12
votes
4
answers
3k
views
hyperbolic structure on Figure–8 knot complement
I was trying to understand the proof of the fact that there is a hyperbolic structure on Figure–8 knot complement initially from Thurston's notes and then from some online notes; but unfortunately I ...
- 789
11
votes
4
answers
1k
views
Distance between two knots
Are there some well-studied functions defining natural distance measures between two knots? One can imagine a function that counts, say, the minimum number of
moves, each of which passes one strand of ...
- 151k
11
votes
2
answers
518
views
Knots having the same Alexander module which are not S-equivalent
As is discussed in this answer, S-equivalence class can be regarded as the Alexander module plus Blachfield pairing, an isomorphism of some duals of Alexander modules.
There are examples of knots ...
- 548
10
votes
1
answer
1k
views
Introductory article of knot Heegaard Floer Homology
I am looking for some article that gives an introduction to Heegaard Floer homology of knot.
I heard that it is very useful to determine the unknotting number of a knot, but I couldn't find any ...
- 171
10
votes
2
answers
2k
views
Is there a table of (fibred knot) monodromies?
Background/motivation
I'm working on contact topology (in dimension three): a fundamental theorem of Giroux gives us a bijection between contact structures (up to isotopy) and open books (up to ...
- 10.9k
10
votes
3
answers
2k
views
A Reference for Schubert's Theorem
Schubert's Theorem in Knot Theory says that any knot can be uniquely decomposed as the connected sum of prime knots.
Unfortunately the original paper is in German.
Does anyone know a good english ...
- 2,441
9
votes
1
answer
637
views
Reference request: A knot is tame if and only if it has a tubular neighbourhood
Definitions:
A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal).
Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...
- 93
8
votes
7
answers
1k
views
Knot theory without planar diagrams?
I remember reading somewhere that there are just a few contributions to knot theory which do not involve knot diagrams. Hence my question:
Does anybody know about papers concerning knot theory which ...
- 785
8
votes
1
answer
427
views
Is there a combinatorial version of PL ambient isotopy in dimension $>3$?
The classical PL Reidemeister Theorem reads:
Reidemeister Theorem: Two knots in $S^3$ are PL ambient isotopic if and only if any diagram of one can be transformed into a diagram of the other by ...
- 22.1k
8
votes
1
answer
704
views
Is this knot invariant already treated somewhere in the literature?
Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$
We can turn $Y_K$ into a metric space by considering the distance induced by ...
- 197
8
votes
0
answers
222
views
references on categorification of knot invariants
I am extremely sorry if this is not the right place for this kind of question.
I have studied some knot theory, quantum invariants and would like to study more about categorification of knot ...
- 81
7
votes
2
answers
270
views
What are the "correct" references for the Vassiliev invariant?
Is there a good survey paper which describes the general ideas of
Vassiliev's invariant? I am not an expert on knot theory, many references are too technical for me.
Could Vassiliev's invariants be ...
- 91
7
votes
1
answer
636
views
Knots: locally flat, PL and smooth
In the classical dimension (knots in $S^3$), it is considered standard (I think?) that the following sets are in bijective correspondence:
locally flat knots up to ambient isotopy;
PL-knots up to PL ...
- 1,033
6
votes
1
answer
140
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 ...
- 22.1k
5
votes
3
answers
505
views
Embedded ribbons and regular isotopy
I'm reading Kauffman's 1990 paper "An Invariant of Regular Isotopy" about knots that are isotopic through only Reidemeister Type II and III moves, which is known as a regular isotopy. His ...
- 217
5
votes
2
answers
228
views
Reference for Cochran-Orr-Teichner's filtrations on knot concordance
I would be interested in recommendations for easily accessible/somehow simply readable texts to Cochran-Orr-Teichner's filtrations on knot concordance:
Tim D. Cochran, Kent E. Orr, and Peter Teichner....
user160180
5
votes
1
answer
192
views
Reference for a theorem on crossing changes of links
I've recently stumbled upon a paper of Scharlemann on crossing changes:
link text
In particular I am interested in understanding Theorem 2.2 (page 6):
"Theorem: If links A and B
are related by a ...
- 51
5
votes
2
answers
406
views
Unknotting tunnels in surface bundles
Given a once-punctured surface $F$ and an orientation preserving self homeomorphism $\phi$, let $M_\phi$ be the bundle over $S^1$ with fiber $F$ and monodromy $\phi$.
In Sakuma's survey article The ...
- 1,601
4
votes
1
answer
260
views
Bounds for the crossing number in terms of the braid index?
Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$?
For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for ...
- 9,114
4
votes
0
answers
88
views
What is the explicit relationship between the shape parameters and the holonomy of a hyperbolic ideal triangulation?
Let $K$ be a hyperbolic knot in $S^3$. One way to describe the hyperbolic structure is to give a discrete, faithful representation $\pi_1(S^3 \setminus K) \to \operatorname{PSL}_2(\mathbb C)$, the ...
- 2,533
3
votes
2
answers
950
views
hyperbolic 3-manifold of finite volume
Is there a complete description of hyperbolic 3-manifold of finite volume ?
Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume?
...
- 1,629
3
votes
1
answer
516
views
Ambiguity in the unoriented knot connected sum
It is well known that there might be ambiguity in the unoriented knot connected sum if the knots concerned are not invertible.
E.g., consider 8_17, the only knot with crossing number 8 which is non-...
3
votes
1
answer
212
views
Picturing twisting of strands explicitly after blow downs
In order to simplify Kirby calculus proofs, one can use a box notation which indicates a number of full twists up to sign. In the scenario of two strands with twist boxes, it is straightforward to ...
- 213
3
votes
1
answer
87
views
Reference for birational equivalence of $A$-polynomial curve and character variety
For $K$ a knot in $S^3$, the character variety $\mathfrak{X}_K$ parametrizes conjugacy classes of representations $\pi_1(S^3 \setminus K) \to \operatorname{SL}_2(\mathbb C)$.
Another object that does ...
- 2,533
3
votes
0
answers
209
views
Braids of fibered knots
There are some theorems saying that the closure of a braid of a specific form is fibered. For instance, a theorem of Stallings says that the closure of a homogeneous braid is a fibered knot.
I am ...
- 1,430
2
votes
1
answer
817
views
Is every quasipositive knot strongly quasipositive?
A link is called quasipositive if it has a special braid diagram, namely a product of conjugates of the positive standard generators of the braid group. If this product only contains words of the form ...
- 711
2
votes
1
answer
144
views
English version of a paper by Gusarov
I am looking for the english translation of the paper in russian Variations of knotted graphs, geometric technique of n-equivalence, St. Petersburg Math. J. 12-4 (2001) by Gusarov.
There is a .ps file ...
- 23
2
votes
1
answer
188
views
Surveys on unknotting number
Any knot diagram could be converted to an unknot by cross change.
The unknotting number of a knot diagram is the minimal number of cross changes needed.
A knot could have many different diagrams and ...
- 91
2
votes
0
answers
218
views
Skein relation, Braids, and Hecke algebra
Many knot invariants (e.g. Alexander polynomial, Jones Polynomial,etc) admit a recursive algorithm
based on the so-called skein relation
But why the skein relation is a natural thing?
People have been ...
- 21
1
vote
2
answers
350
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 ...
- 1,108
1
vote
1
answer
124
views
Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?
My question is in the tittle:
Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?
If the answer is yes, is there a reference for this.
- 11
1
vote
1
answer
326
views
Addition of two homology classes is zero in construction of Poincare Sphere
I ask here the question since it hasn't been answered in
Math Stack Exchange.
I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...
- 909
1
vote
1
answer
286
views
Jones polynomial of cable knots
Let $K_{p,q}$ be a $(p,q)$-cable of the non-trivial knot $K$ in $S^3$.
Is there a closed formula for the Jones polynomial for $K_{p,q}$ as in the case of Alexander polynomial or Seifert matrices?
user160180