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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 ...
Tetsuya Ito's user avatar
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 ...
bd99's user avatar
  • 23
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 ...
Charles's user avatar
  • 9,114
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 ...
Terry Black's user avatar
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 ...
Anthony Conway's user avatar
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-...
user202107011110's user avatar
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 ...
Calvin McPhail-Snyder's user avatar
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 ...
Henry's user avatar
  • 1,430
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 ...
user3435656's user avatar
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 ...
user8749's user avatar
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 ...
Jernej Grlj's user avatar
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 ...
user8749's user avatar
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?
user avatar
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 ...
maxematician's user avatar
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 ...
Calvin McPhail-Snyder's user avatar
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....
user avatar
9 votes
1 answer
636 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 ...
Lilalas's user avatar
  • 93
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.
Christian's user avatar
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? ...
mathphys's user avatar
  • 1,629
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 $S^3\setminus K$) ...
Mark Grant's user avatar
  • 35.9k
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 ...
D1811994's user avatar
  • 909
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 ...
Marc Kegel's user avatar
  • 1,314
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 ...
Joseph O'Rourke's user avatar
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 ...
this_is_an_apple's user avatar
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 ...
Bingo's user avatar
  • 789
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) ...
miss-tery's user avatar
  • 755
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 ...
Daniel Moskovich's user avatar
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 ...
Daniele Celoria's user avatar
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 ...
Daniel Moskovich's user avatar
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, ...
Daniel Moskovich's user avatar
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 ...
Zuriel's user avatar
  • 1,108
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 ...
Springfield's user avatar
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 ...
Aru Ray's user avatar
  • 711
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 ...
Marius Buliga's user avatar
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 ...
Joseph O'Rourke's user avatar
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 ...
Marco Golla's user avatar
  • 10.9k
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 ...
Joseph O'Rourke's user avatar
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 ...
b b's user avatar
  • 1,601
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 ...
Owen Sizemore's user avatar
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 ...
algori's user avatar
  • 23.5k