**35**

votes

**2**answers

900 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 ...

**1**

vote

**1**answer

157 views

### Does knot Floer homology detect knot genus in rational homology spheres?

My question is the following:
Does knot Floer homology detect the genus of null-homologous knot in rational homology spheres?
If the answer is yes, I would like to have a reference for the ...

**6**

votes

**2**answers

412 views

### Reference for a fact (?) on homeomorphic knot complements

Does somebody have a reference (or an argument why it should be true) for the following statement?
“Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 ...

**8**

votes

**0**answers

104 views

### What fraction of knots are ribbon knots?

Pardon the naive question.
Perhaps there are several different variants of my question, but
essentially I am seeking to know whether ribbon knots are
vanishingly rare among all knots, or a positive ...

**2**

votes

**0**answers

55 views

### Mutant pairs distinguished by [2,1]-colored HOMFLY polynomials

Morton and Cromwell showed that the famous mutant pair, Kinoshita-Terasaka and Conway knots, can be distinguished by HOMFLY polynomials colored by [2,1] Young diagram. Are there any other mutant pairs ...

**6**

votes

**0**answers

108 views

### Stable, tame A-infinity isomorphism?

Linear duality provides a correspondence between differentials on free tensor algebras and A-infinity algebras. Under this correspondence, what is the A-infinity counterpart of "stable, tame ...

**9**

votes

**3**answers

316 views

### Unknotting number of knot diagrams

Define the "diagram unknotting number" of a knot diagram $D$ as the minimal number of crossings that need to be changed in $D$ in order to get a diagram of the trivial knot (the usual unknotting ...

**3**

votes

**1**answer

235 views

### Unknotting number and crossing number

It is well known that if $c(K)=2n+1$, then $u(K)$ is less than $n+1$. It can not be sharper because of the trefoil knot. On the other hand, if $c(K)=2n$, then similarly we have $u(K)$ is less than ...

**8**

votes

**1**answer

318 views

### Is more alternating always better?

While thinking about this interesting question asked by Dylan Thurston, it occured to me that in every case that I know, the closer a knot diagram is to being alternating, the better its properties. ...

**12**

votes

**1**answer

208 views

### Can infinitely many alternating knots have the same Alexander polynomial?

There exist many constructions of infinite families of knots with the same Alexander polynomial. However, alternating knots seem very special. While there are also many result on restricting the form ...

**6**

votes

**2**answers

291 views

### Vassilliev invariants of knots and their cables

The following is perhaps a standard question, but i could not find a plain enough answer
by simply searching online.
Q: Given a knot $K$ and its $(p,q)$-cable $K_{p,q}$ what is a relation
between ...

**23**

votes

**1**answer

224 views

### The Jones polynomial at specific values of $t$.

I've been calculating some Jones polynomials lately and I was just curious if there was a physical meaning to evaluating the Jones polynomial at a particular value of $t$.
For example, if I take the ...

**6**

votes

**1**answer

300 views

### Proving that the Jones polynomial is q-holonomic

The Jones polynomial is known to have many different interpretations or definitions, by now. There are connections with QFT, quantum groups, Hilbert schemes, Cherednik algebras, etc.
My question is ...

**1**

vote

**0**answers

62 views

### Ozsvath-Szabo orientation convention for Seifert fibred spaces

I am confused by the orientation convention that Ozsvath and Szabo use in On Heegaard Floer homology and Seifert Fibered Surgeries and would appreciate if someone clarifies this for me. On page 15 ...

**2**

votes

**2**answers

225 views

### Gauss Codes that produce classical knots as opposed to virtual knots

I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that
If $K$ is a virtual knot whose underlying Gauss ...

**80**

votes

**10**answers

8k views

### Are there any very hard unknots?

Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...

**1**

vote

**1**answer

36 views

### Question about general torus knot lengths [closed]

Can anyone send me a reference about calculating the length of a thin string wound on a p,q torus with major radius R and minor radius r in terms of p, q, R and r? This may appear trivial but I have ...

**7**

votes

**1**answer

217 views

### Does the limit in the Volume conjecture converge?

The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...

**18**

votes

**5**answers

2k 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 ...

**6**

votes

**6**answers

294 views

### Reference on representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations).
"Knots" by Burde and Zieschang discusses some material but it is not entirely ...

**8**

votes

**2**answers

887 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 ...

**3**

votes

**1**answer

75 views

### Quandle colorings under Reidemeister moves

Let $D$ be a knot diagram and $Q$ a quandle. We use $c$ to denote a fixed coloring of $D$ with $Q$. If $D'$ is another knot diagram of the same knot, and $R_1$ is a sequence of Reidemeister moves ...

**19**

votes

**0**answers

242 views

### Topological description of inverting a knot

The first figure shows an offset overhand knot. To tie it, you simply bring the two cords together and make an overhand knot in them. It's more secure than it looks, and several climbers have been ...

**0**

votes

**0**answers

141 views

### Understanding a program for computing Khovanov homology

I would like to understand how a computer program for computing Khovanov homology works. The particular program I have in mind is by John Baldwin: https://web.math.princeton.edu/~baldwinj/Kh.cpp
The ...

**16**

votes

**2**answers

316 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 ...

**5**

votes

**1**answer

253 views

### Kauffman's state model for the Alexander polynomial, via representation theory

I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander ...

**3**

votes

**1**answer

132 views

### Parity of knot signatures

I recently came upon a recursive formula for the (ordinary) signatures of torus knots. The formula, which I found in Murasugi's book "Knot Theory and Applications" (Springer, 2007), originally ...

**2**

votes

**1**answer

94 views

### Computable link invariants

I am interested in the following situation: given a braid $B$, it induces a link $L$ in a pretty straightforward way ("glue" the endpoints, like here). For a braid $B$, we know how to represent it in ...

**12**

votes

**0**answers

169 views

### What do tangles teach us about braids?

A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times ...

**10**

votes

**2**answers

493 views

### Is every knot unavoidable in the embeddings of some graph?

Is it the case that, for any given knot $K$,
there exists some graph $G$ whose every embedding into $\mathbb{R}^3$
(or into $\mathbb{S}^3$)
contains a cycle that realizes $K$?
I know the ...

**2**

votes

**3**answers

260 views

### Heegaard Floer Homology of double branched cover

The question is the following:
Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: ...

**4**

votes

**1**answer

208 views

### Knot invariants in 3-manifolds that are not $\mathbb{R}^3$ or $S^3$ or $B^3$?

This is just a reference request; I have no sharp mathematical question.
Inspired by the $(3+)$-year old MO question,
In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?,
I would ...

**4**

votes

**2**answers

344 views

### How to tell if two or more knots are linked

Given a number of knots, I would like to know if they are linked. I know that the linking number can tell if two knots are linked.
There is any method that completely solves this problem?

**3**

votes

**1**answer

115 views

### Khovanov $sl_2$ homology of a connected sum of some torus knots

Let $T_{p,q}$ be the (p,q) torus knot. Could anybody possibly compute either unreduced or reduced Khovanov $\mathfrak{sl}(2)$ homology of the connected sum $T_{2,3} \sharp T_{3,4}$ of the (2,3) and ...

**-2**

votes

**1**answer

281 views

### Do homeomorphic complements give homotopic knots?

Maybe this question is too trivial for a research site but there are so many notions of equivalence of knots that I am lost in literature. The question that interests me is the following.
A knot is a ...

**4**

votes

**2**answers

340 views

### Jones polynomial of the concatenation of two braids

Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings.
Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$?
Here, ...

**5**

votes

**0**answers

118 views

### Crossing bound implies Reidemeister move bound?

In 1998 Galatolo established an upper bound on the number of Reidemeister moves needed to convert a diagram $D$ of the unknot into a trivial loop diagram. The upper bound is a function of $n$, the ...

**0**

votes

**0**answers

97 views

### Alexander Invariant of Torus knot

I am very interested in knot theory, especially in knot groups and knot polynomials. As is well known, it is easy to calculate the Alexander polynomial from the fundamental group $\pi_{1}(K)$ of a ...

**7**

votes

**3**answers

199 views

### Measures of entangledness of an open curve

Let $\gamma$ be a simple (non-self-intersecting) open curve in $\mathbb{R}^3$.
I am seeking a measure of its degree of "entangledness," some measure that accords
with the intuition one senses with a ...

**5**

votes

**0**answers

122 views

### Software for BMW algebra calculations?

Does software exist for computations in the BMW algebra?
For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...

**2**

votes

**0**answers

40 views

### Tait conjectures for alternating w-links

The Tait Conjectures are useful in knot tabulation. For alternating knots and links, two of them state:
Any reduced diagram of an alternating link has the fewest possible crossings.
Any two reduced ...

**9**

votes

**2**answers

504 views

### The Alexander polynomial of a slice knot, Reidemeister tosion, Whitehead group

My question is about the Alexander polynomial of a slice knot.
For a slice knot $K$,
Fox-Millnor and Terasaka proved that
$$ \Delta_{K}(t) \doteq f(t) f(t^{-1})$$
for some polynomial $f(t) \in ...

**5**

votes

**1**answer

107 views

### Closed formula or program for computing Tristram-Levine signatures of torus knots?

I need to compute a series of Tristram-Levine signatures for a family of torus knots. I was wondering if this has already been done or whether there is a good way to streamline the computation.
I am ...

**7**

votes

**1**answer

125 views

### Relation between the Alexander module of a link and intermediate free abelian covers

I'm working through McMullen's paper "The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology" and have a question concerning the following setup:
Given a link complement $(X, ...

**17**

votes

**9**answers

8k views

### Applications of knot theory

An answer of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments.
I regularly teach a knot theory class. ...

**1**

vote

**0**answers

121 views

### The relations between some 3-components links and trefoil knots [closed]

It is intuitive to see that the 3-components links (under Alexander–Briggs notations) $6^3_1, 6^3_2, 6^3_3$ are closely related to each other; in a sense by doing a cut-gluing or sew-gluing surgery, ...

**3**

votes

**1**answer

145 views

### One question about the quandle

Given a finite quandle $Q$, for any knot $K$ one can associate an invariant, i.e. the number of proper colorings $p(K)$. Let us consider the inverse $K^{-1}$ and mirror image $K'$ of $K$. My queston ...

**1**

vote

**2**answers

195 views

### Freely Periodic map of $(S^{3} , K) $ and a fixed loop in the induced isomorphism of $\pi_{1} ( S^{3} \backslash K )$

Let $K$ be a link in $S^{3}$ and $f: S^{3} \rightarrow S^{3} $ a freely periodic map of order $n$ with $f(K) = K$. Let $\psi_{f} : \pi_{1} ( S^{3} \backslash K ) \rightarrow \pi_{1} ( S^{3} ...

**3**

votes

**2**answers

126 views

### Why is a braided left autonomous category also right autonomous?

In Selinger, P. A survey of graphical languages for monoidal categories (New Structures for Physics, Springer, 2011, 813, 289-233), it is stated that:
Lemma 4.17 ([23, Prop. 7.2]). A braided ...

**2**

votes

**1**answer

114 views

### Practical application of lattice knots

I am looking for examples of practical applications of lattice knots. Any help?