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.

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Is this approach to the combinatorics of knots well known?

I am teaching a course on knots for the first time, and this led me to play with an approach which I have not seen in the literature. I would be surprised if no one had used it before, so I am ...
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2answers
99 views

Minimal genus of Seifert surface of torus knot

Let $(p,q)$ be a pair of coprime (positive) integers. Consider the torus knot $T_{p,q}$. What is the minimal genus of an (embedded) oriented Seifert surface for this knot? It is not had to convince ...
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46 views

link which is the closure of some power of a braid

Recently I have stumbled upon links which are closures of braids, of the form $\sigma = \tau^{n}$. Such links generalize torus links. Are there any papers studying such links? In particular I am ...
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1answer
102 views

Oriented knot complement conjecture for fibered knots

Suppose I have two inequivalent fibered knots in a homology sphere. When I say 'inequivalent', I mean that there is no orientation-preserving homeomorphism of the space that takes one to the other. ...
7
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1answer
193 views

Can two fibered knots have the same exterior?

Suppose I have two distinct fibered knots in a homology sphere. Is it possible for them to have (orientation-preservingly) homeomorphic exteriors? See Oriented knot complement conjecture for ...
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1answer
129 views

Two-bridge knots and CW-complex

The fundamental group of any two-bridge knot K in $\mathbb{S}^3$ has a presentation with two generators and one relation. On the other hand, it's possible to provide a CW-complex with only one 0-cell ...
6
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1answer
95 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 ...
7
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147 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 ...
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2answers
239 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 $M$, ...
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1answer
358 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 ...
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1answer
94 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 ...
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1answer
241 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 ...
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117 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 ...
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127 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
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0answers
64 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 ...
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1answer
191 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 ...
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121 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
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3answers
367 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 ...
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1answer
234 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 ...
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1answer
411 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. ...
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2answers
442 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 ...
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0answers
65 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 ...
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2answers
247 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 ...
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2answers
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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 ...
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1answer
47 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 ...
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1answer
241 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 $$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$ ...
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6answers
308 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 ...
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1answer
340 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
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1answer
85 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 ...
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263 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 ...
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158 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
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2answers
324 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 ...
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1answer
104 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 ...
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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 ...
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2answers
505 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
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1answer
219 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 ...
23
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1answer
256 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
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2answers
351 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?
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1answer
121 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 ...
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292 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 ...
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3answers
296 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
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2answers
349 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
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1answer
141 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 ...
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0answers
109 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
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128 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 ...
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42 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 ...
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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
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1answer
115 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
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2answers
537 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
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1answer
138 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, ...