**8**

votes

**0**answers

181 views

### Alexander polynomial in branched covers

Suppose I am given a 3-manifold as a double branched cover over a link. Let a null-homologus knot in this space be given as a lift of an arc with endpoints on the link (it is automatically ...

**9**

votes

**2**answers

316 views

### How many knots are there with hyperbolic volume less than a given constant

Are there any known upper bounds on:
$$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$
? I expect this grows at least exponentially in $M$, ...

**5**

votes

**1**answer

381 views

### Are there spaces in which there are no fibered knots?

I am looking for orientable closed 3-manifolds in which there are no fibered knots. Although I know little about this, I think for links the answer to the question above is "no", and the result is ...

**4**

votes

**1**answer

108 views

### Is there a known Legendrian simple link?

Several knots like unknot, $4_1$, $3_1$ are known to be Legendrian simple, i.e., Thurston-Bennequin number and rotation number determine Legendrian type completely.
How about the same notion for link ...

**3**

votes

**1**answer

301 views

### Jones polynomial of tangles using Temperley-Lieb algbra?

The definition of the Jones polynomial of tangles (a la Reshetikhin and Turaev) uses the space of invariants for $U_q sl_2$ and R-matrices. It seems to me the same thing cane be done in terms of the ...

**2**

votes

**0**answers

128 views

### Jones Polynomial as an Equivariant Index

In Khovanov homology the Jones polynomial of a given link is computed as the graded Euler characteristic of a certain complex associated with the link. From other side the Atiyah-Singer theorem ...

**8**

votes

**0**answers

141 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

73 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 ...

**2**

votes

**1**answer

219 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

**0**answers

135 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

435 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 ...

**13**

votes

**1**answer

282 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 ...

**8**

votes

**2**answers

474 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. ...

**6**

votes

**2**answers

507 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 ...

**1**

vote

**0**answers

79 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

290 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 ...

**37**

votes

**2**answers

2k 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

61 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 ...

**8**

votes

**1**answer

264 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
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...

**6**

votes

**6**answers

354 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 ...

**6**

votes

**1**answer

382 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 ...

**3**

votes

**1**answer

103 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

303 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

170 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

350 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 ...

**2**

votes

**1**answer

118 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 ...

**13**

votes

**0**answers

190 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

514 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 ...

**4**

votes

**1**answer

243 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 ...

**24**

votes

**1**answer

373 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 ...

**4**

votes

**2**answers

358 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

138 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

331 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 ...

**3**

votes

**3**answers

384 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

**2**answers

372 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, ...

**3**

votes

**1**answer

157 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 ...

**0**

votes

**0**answers

133 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 ...

**5**

votes

**0**answers

136 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

48 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 ...

**5**

votes

**0**answers

130 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 ...

**5**

votes

**1**answer

129 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 ...

**9**

votes

**2**answers

618 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 ...

**7**

votes

**1**answer

161 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, ...

**1**

vote

**0**answers

220 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

**2**answers

159 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

123 views

### Practical application of lattice knots

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

**10**

votes

**0**answers

212 views

### The Markov trace via Bott-Samelson fibers?

Let $H_n$ be the Hecke algebra of GL(n), i.e., the algebra over $\mathbb{Q}(q)$ with generators $T_1,
\ldots, T_{n-1}$ which satisfy the braid relations and also $T^2 = (q-1) T + q$.
Recall the ...

**5**

votes

**1**answer

181 views

### Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?

Consider the following question:
If $K\subset S^{3}$ is a nontrivial knot. Let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ ...

**1**

vote

**1**answer

216 views

### Clarification of Gabai's exposition of Murasugi Sums in 'the Murasugi sum is a natural geometric operation'

Gabai states that the Murasugi sum of two hopf bands yields a spanning surface of either the figure eight knot, the trefoil knot or a link of three components. Figure one shows two oppositely twisted ...

**8**

votes

**0**answers

160 views

### Does the shortest path between two braids pass through string links?

One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links.
This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, ...