**17**

votes

**3**answers

456 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

156 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

204 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

528 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

252 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

432 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

365 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

155 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

360 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

479 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: ...

**5**

votes

**2**answers

402 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

173 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

148 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

144 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

58 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

139 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

164 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

665 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

199 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

237 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, ...

**5**

votes

**2**answers

185 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

129 views

### Practical application of lattice knots

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

**10**

votes

**0**answers

228 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

212 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

268 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

162 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 $I^3$, ...

**0**

votes

**0**answers

122 views

### Jones polynomial of 2-knots

Question: is it possible to define the Jones polynomial for knotted surfaces (or $S^2$ for simplicity) in $R^4$?
Jones polynomial has several definitions (see How many definitions are there of the ...

**8**

votes

**1**answer

533 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 ...

**7**

votes

**2**answers

727 views

### Are Fourier series of length 2 'asymmetric enough' to generate all crossing patterns? - A reformulation of the Fourier-(1,1,2) knot question

Given $N$ pairs of distinct real numbers $t_i, t'_i \in [0,1]$, $i = 1,\ldots,N$, we ask if there is a function $f(x) = \cos(2\pi mx+\alpha) + \gamma\cdot \cos(2\pi nx+\beta)$, with $m, n \in ...

**3**

votes

**2**answers

208 views

### Handlebody decomposition of a 3-manifold adapted to a link

Given a compact connected 3-manifold $M$ with non-empty boundary, and a link $L \subset M$, is there a handlebody decomposition of $M = H^0 \cup (\cup_i H^1_i) \cup \{\text{2-handles}\}$ such that:
...

**5**

votes

**0**answers

118 views

### Ribbon knot presentations

Suppose $K$ is a $n$-knot, $n\geq 2$, which bounds two different ribbon disks $D_1, D_2$. These ribbon disks induce unique ribbon $(n+1)$-knots $K_1, K_2$ respectively. Is it known whether $K_1$ and ...

**12**

votes

**4**answers

992 views

### What properties of knots lead Lord Kelvin to hypothesize that atoms were knots in the ether?

I've often heard that Lord Kelvin was one of the first people to study knot theory, as he hypothesized that atoms were knots in the ether. I assume that he had some compelling evidence for this fact.
...

**2**

votes

**1**answer

145 views

### Is there a criterion for a link complement to have a hyperbolic structure with finite volume

For many links in $S^3$, the link complement can be equipped with a Riemannian structure which is complete, of constant sectional curvature -1, and has finite volume (i.e., a hyperbolic structure with ...

**1**

vote

**1**answer

132 views

### Bitangent locus of torus knots

Anyone know how to compute the bitangent locus of a space curve, e.g. a torus knot (pick whatever parametrization you like)? Specifically, what is the set of normal vectors (in the two-sphere) of ...

**1**

vote

**0**answers

133 views

### Entangled helical knots

Consider a pair of disjoint, congruent helices $H_1$ and $H_2$
passing through one another in the following sense.
(Caveat lector: This question is not of general interest! It is also long.)
$H_1$ is ...

**8**

votes

**3**answers

446 views

### What constant ensures hyperbolicity of Dehn surgery?

I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from ...

**1**

vote

**2**answers

198 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} ...

**5**

votes

**1**answer

400 views

### The cyclic branched covers of “simple” knots in $S^3$

Is there a convenient place in the literature where the geometric decompositions of cyclic branched covers of $S^3$ branched over "small" knots is recorded?
By small knots, I'm referring to things ...

**7**

votes

**1**answer

202 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 ...

**6**

votes

**1**answer

188 views

### Knots indistinguishable by HOMFLY

Is there any list (incomplete of course) of knots, that have similar HOMFLY polynomials? I am mainly interested in torus knots.

**4**

votes

**2**answers

283 views

### Getting surgery link from Heegaard splitting

From Lickorish-Wallace theorem, every 3-manifold is an integral surgery on a link in $S^3$. From its proof from Saveliev's book, it seems obvious that if I know the Heegaard splitting of a closed ...

**7**

votes

**2**answers

326 views

### How do I efficiently find a sequence of Reidemeister moves between equivalent link diagrams?

In knot theory, two link diagrams are equivalent if and only if they can be related by performing a finite number of Reidemeister moves. But sometimes it is so confusing that I don't know which type ...

**11**

votes

**1**answer

747 views

### Kontsevich integral : state of the art

The Kontsevich integral is known to be a universal Vassiliev invariant.
It is still an open question whether it is a complete knot invariant, i.e. whether it distinguishes a given knot from all other ...

**15**

votes

**3**answers

717 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, ...

**3**

votes

**0**answers

134 views

### More questions about high-dimensional knot invariants

In a question yesterday I asked about the existence of algebraic invariants for embeddings of n-manifolds into n+2-spheres. The answers in the positive dimension all made certain assumptions about ...

**9**

votes

**3**answers

372 views

### Invariants of high-dimensional knots

In the study of knots in three dimensions, it can be shown that the fundamental group together with a specification of a meridian and longitude form a complete invariant for knots. What is known about ...

**1**

vote

**2**answers

300 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 ...

**7**

votes

**3**answers

638 views

### Knots and Dynamics. Recent breakthroughs?

I recently started reading Étienne Ghys slides on knots and dynamics <http://www.umpa.ens-lyon.fr/~ghys/articles/icm.pdf> which seem very interesting. I know this approach to knots and dynamics is ...

**16**

votes

**3**answers

597 views

### Looking for “large knot” examples

This question is about knots and links in the 3-sphere. I want to find an example of a "large" knot or link with some special properties. I'm looking for some fairly specific examples, but I'm also ...

**12**

votes

**3**answers

636 views

### Representing SU(3) with 3 ropes in 3 dimensions

The short question is: how exactly is SU(3) realized with ropes?
The long question: There is this idea that deformations of a configuration of three infinitely long, flexible ropes that cross each ...