**3**

votes

**1**answer

56 views

### Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?

Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by ...

**4**

votes

**2**answers

534 views

### What is the mathematical significance of the IHES logo?

The logo of the IHES
http://www.ihes.fr/jsp/site/Portal.jsp
(upper left) is lovely, but what exactly does represent mathematically?
(There's a slightly larger version at
...

**18**

votes

**2**answers

1k views

### Are there moves between Reidemeister moves?

Background
Knots are typically written in 2 dimensions as a loop in the plane with normal crossings. One then asks when two such diagrams describe the same knot. Two diagrams describe the same knot ...

**4**

votes

**1**answer

325 views

### Is a “knot knot” or “double knot” a thing in knot theory?

I apologize in advance for my rudimentary knowledge of knot theory, but I've been trying to find out about the significance (if any) of taking a knot (particularly a torus knot), cutting it, and ...

**13**

votes

**3**answers

579 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 ...

**10**

votes

**3**answers

488 views

### Algorithm for detecting ribbon or slice links?

A link in $S^3$ is said to be slice if it bounds a collection of flat disks into the $4$-ball. Here "flat," means that there is a (locally) trivial normal bundle. This condition can be strengthened to ...

**5**

votes

**1**answer

110 views

### Ascending surfaces in the 4-ball

Let the standard symplectic structure on $B^4$ (viewed in $\mathbb{R}^4$ or $\mathbb{C}^2$) be given by $\omega=(1/2) d \eta$, for
\begin{align*}
\eta &:=
x_1 \, dy_1 - y_1 \, dx_1 + x_2 \, ...

**24**

votes

**9**answers

11k 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

**1**answer

168 views

### On Knot Equivalence problem statement

How is the knot equivalence problem represented?
By this I mean I am looking for an analogy that compares with graph equivalence. For graph equivalence, we have two graphs $G_1$ and $G_2$ with ...

**11**

votes

**4**answers

707 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 ...

**7**

votes

**4**answers

380 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 ...

**7**

votes

**2**answers

190 views

### Can every large point set be connected to a given knot?

Let $K$ be a given knot, and
$P$ a set of points in $\mathbb{R}^3$ in general position,
general position in the sense that no three points are collinear
and no four coplanar.
Define the point-set ...

**2**

votes

**0**answers

56 views

### Knot invariants which arise as the solution of differential equations along the knot

Do there exist knot invariants which can be computed by solving some kind of differential equation along the knot?

**6**

votes

**0**answers

131 views

### What does the representation category of the knot group know?

Let $K, K'$ be knots in $S^3$, and $T, T'$ the boundaries of their tubular neighborhoods.
Recall that by theorems of Waldhausen, and Gordon and Luecke, one knows the following: an isomorphism ...

**6**

votes

**3**answers

202 views

### Polynomial invariants for unoriented links

I have seen that usually one finds polynomial invariants for oriented links (for example the Jones polynomial, the Hompfly polynomial). Does anyone know what polynomial invariants exist for ...

**1**

vote

**0**answers

85 views

### Unnormalized Kauffman homology of the unknot

Is the unnormalized Kauffman homology of the unknot known? The Poincare polynomial of HOMFLY homology of the unknot is known as
$$\frac{1+at}{1-q}.$$
Is the Poincare polynomials of Kauffman homology ...

**1**

vote

**1**answer

92 views

### Cancellation of 2-component links

Consider a two-component tame link in 3-space, consisting of an arc from $(-1,1,0)$ to $(1,1,0)$ and an arc from $(-1,-1,0)$ to $(1,-1,0)$, confined to the slab $-1 \leq x \leq 1$. Call such a link ...

**17**

votes

**3**answers

428 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 ...

**6**

votes

**1**answer

199 views

### Can one twist fibred knots and still get fibred knots?

Suppose we have a fibred knot $K$ with a fiber surface $F$ and let $c$ be an unknot disjoint from $F$ (but not homotopically trivial in the complement of $F$). Is it possible that every twist along ...

**2**

votes

**1**answer

52 views

### Calculating the Upper Bound on the Sphere Radius of Knotted Channel Surfaces

This question is motivated by trying to determine the upper bound on the thickness of a rope of fixed length (w.l.o.g. $2\pi$), with which a knot of given topology can be realized under the further ...

**21**

votes

**1**answer

388 views

### Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials?

The Jones polynomial can be computed from the representation theory of $\mathcal{U}_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of ...

**4**

votes

**1**answer

181 views

### Physical strength of a link [closed]

Assuming that we construct a link/chain using a collection of knots.
Is there a way to measure the physical strength of this chain?

**6**

votes

**2**answers

227 views

### Fibered example of topologically slice knots

Is there any known example of fibered knot which is topologically slice but not (expected to be) smoothly slice?

**8**

votes

**1**answer

178 views

### Questions on poincare homology spheres and branched covers

I have two questions:
Question 1. Suppose that $K$ is a knot in $S^3$. Let $\Sigma(K)$ be the double branched cover of $S^3$ branched along $K$. If $\Sigma(K)=\#_{i=1}^n\Sigma(2,3,5)$, then ...

**18**

votes

**0**answers

444 views

### Concordance and homology cobordism

If two knots $K_1$ and $K_2$ in $S^3$ are smoothly concordant, then for any rational number $r$, the $r$-surgeries $S^3_r(K_1)$ and $S^3_r(K_2)$ are homology cobordant. Is the converse true? What if ...

**5**

votes

**1**answer

1k views

### Understanding Penrose diagrammatical notation

I arrived to Penrose's paper Applications of negative dimensional Tensors after reading some bits of Baez's Prehistory (link) and the first two chapters of Turaev's Quantum invariants of knots and ...

**6**

votes

**1**answer

246 views

### Satellite knot example

Can someone provide me with an example of a satellite knot with symmetry group which is neither cyclic nor dihedral?

**12**

votes

**0**answers

330 views

### Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?

Let $K$ be a knot in the 3-sphere $S^3$.
Here we denote by $s(K)$ Rasmussen's s-invariant for $K$,
and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$
by attaching a ...

**11**

votes

**1**answer

440 views

### A fun game related to knot theory

I recently learned the following rather fun game: a group of people is standing up roughly in circle, facing each other. Then participants randomly join hands, in such a way that nobody holds its own ...

**6**

votes

**3**answers

213 views

### Surgery along an arc connecting the components of a $2$-component link gives the unknot

Math Overflow seems to have a dearth of low dimensional topology, but this seems like an interesting question. Let $L$ be a $2$-component link in $S^3$. Suppose that there is a framed arc joining the ...

**1**

vote

**1**answer

85 views

### Essential surfaces in the Exterior of Montesinos knots

Hatcher and Oertel computed the boundary slopes of essential surfaces of Montesinos knots in this paper. But they do not consider surfaces that do not intersect the boundary of the exterior. An ...

**22**

votes

**0**answers

213 views

### Metrics on the 3-sphere with knotted geodesics

According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic ...

**7**

votes

**2**answers

718 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 ...

**7**

votes

**2**answers

241 views

### What is the easiest way to compute Ozsváth-Szabó tau invariant of a knot?

Suppose that we have a knot $K$ with 40 crossings which is not a cable knot or an alternating knot.
Then, what is the easiest way to compute Ozsváth-Szabó's invariant $\tau(K)$?
Are there any ...

**1**

vote

**0**answers

65 views

### Distiguishing mutant knots

Can an invariant from a quantum Lie algebra ever distinguish mutant knots?
(Maybe if it is "chiral"...whatever that means :-)
(Note that Kauffman abstract tensors/skein equations CAN distinguish ...

**5**

votes

**0**answers

112 views

### Framed singular knots

I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let $D$ be ...

**8**

votes

**0**answers

201 views

### Union of knots and its Alexander polynomial

In 1957 Kinoshita and Terasaka in their article "On unions of knots" generalized the operation of connected sum; it's called $union$.
I have several questions :
1.) If the Alexander polynomial of a ...

**0**

votes

**0**answers

89 views

### Codimension one embeddings

For smooth knots in $S^3$ "isotopy" and "ambient isotopy" are equivalent (although this is not true in the topological category). I guess that therefore also for tori in $S^3$ "isotopy" and "ambient ...

**85**

votes

**10**answers

10k 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 ...

**6**

votes

**1**answer

418 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 ...

**5**

votes

**3**answers

454 views

### How to show whether a given knot and its mirror image are the same or not?

The title says it all:
How can I show that a knot $K$ is distinct from its mirror image?
May be I have to try different knot invariants. Not sure, I am new in this area.

**2**

votes

**1**answer

198 views

### A basic question of Khovanov-Rozansky Homology

thank you for spending time on the following question.
In [1] Khovanov and Rozansky categorifized $sl_n$ version of HOMFLY Polynomial, in page 11, they mention that what they defined in [1] is ...

**15**

votes

**1**answer

480 views

### How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, … algebras?

There is a theorem (I believe by Ocneanu) that the Markov trace on the tower of Temperley-Lieb algebras is (essentially) unique.
What about just traces on separate algebras? That is, take one of ...

**3**

votes

**0**answers

77 views

### Equivalence (or not) of two Artin/Fox wild arcs

The repeating patterns in the wikipedia articles on wild arcs and wild knots seem to me to be not continuously deformable to each other. Is this true? For clarity, here is my diagram of the repeating ...

**5**

votes

**1**answer

102 views

### Does $H_*(A^-_0(K))=\mathbb{F}[U]$ imply that $K$ is an L-space knot?

Let $K$ be a knot in the three-sphere. Let $A_s^-(K)$ be the Alexander filtrations of the knot Floer complex $CFK^{\infty}$. Would $A_0^-(K)$ has homology $\mathbb{F}[U]$ imply that K is an L-space ...

**7**

votes

**0**answers

97 views

### Kinematics of rolling knots

It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes.
(An ...

**9**

votes

**0**answers

162 views

### Doubly slice knots and an embedding of 4-manifold

It is well-known that the existence of topologically slice knot which is not smoothly slice implies the existence of exotic $\mathbb{R}^4$. For example, see the answer of K. Davis.
To prove the above ...

**2**

votes

**0**answers

81 views

### Are Markov traces matrix traces?

When starting this question I was very hesitant - literature on the subject is vast and I thought most likely the answer is already there somewhere.
Then when the list "Questions that may already ...

**18**

votes

**0**answers

480 views

### Human Knot game [duplicate]

In the popular game "Human Knot", a group of people stands in a circle and each person grabs another person's hands at random (one with the left hand and one with the right hand). The goal is to ...

**10**

votes

**0**answers

117 views

### Kernel of “Hat to Plus” in Heegaard Floer Homology

Given a 3-manifold $M$, there is a map of Heegaard Floer groups $$f:\widehat{HF}(M) \to HF^+(M)$$
induced by the inclusion $$\widehat{CF}(M) \to CF^+(M)$$ of the respective chain complexes.
Given a ...