**1**

vote

**0**answers

155 views

### Measuring the complexity of a knot by minimum number of simplices to tile the complement

This is essentially a duplicate of: Lower bound on number of tetrahedra needed to triangulate a knot complement
Suppose a knot $K\subseteq\mathbb S^3$ is such that the complement $\mathbb ...

**4**

votes

**1**answer

208 views

### Growth of knots possible with rope of length L

What is the asymptotics (in L) for the number of topologically different knots possible using a perfectly flexible, non-selfintersecting rope of length L and radius 1? (With ends glued together after ...

**11**

votes

**5**answers

1k views

### Why “Categorify”? Relating to link/knot homologies…

Hey Everyone!
So I am new blood in the topic of Khovanov Homology and related topics. According to my basic reading the idea is to get the Jones polynomial as the Euler Characteristic of a certain ...

**9**

votes

**1**answer

986 views

### Self-tightening knot

Is there a way, for some finite L>1, to tie two pieces of rope together, such that any finite force is not enough to pull them apart?
The type of rope I have in mind is something like cylindrical ...

**1**

vote

**1**answer

520 views

### Knots that are not knots [closed]

1) Classic Knotting problem: Classify embeddings of circle into 3D Euclidean space up to isotopy. http://en.wikipedia.org/wiki/Knot_theory
2) General topological knotting problem: Classify embeddings ...

**6**

votes

**0**answers

460 views

### Does anyone know this sequence of polynomials?

A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$.
...

**3**

votes

**0**answers

329 views

### Connected Sum Decomposition of a Knot

Given a composite knot, is it possible to decompose it in prime knots by an algorithm that runs in polynomial time?

**3**

votes

**2**answers

357 views

### Is there a known method for finding the minimum bridge index of a knot?

It is easy to establish an upper bound $n$ for the bridge index of a knot by producing a diagram with the knot in $n$-bridge position.
Is there a known method to produce a reasonable lower bound ...

**10**

votes

**1**answer

849 views

### Is there a periodic table for knots?

When I see knot tables, I have two feeling: ah, it's beautiful, and... painful.
I don't see how knots are ordered in the knot table, the way to go from one knot of a certain crossing number to ...

**3**

votes

**1**answer

215 views

### a special type of 2 component link complement

It is well know that a 2 component link complement may doesn't detect the link type.
My question is whether the following type of 2 component links detect their link types?
Such a link is composed ...

**2**

votes

**0**answers

234 views

### Knots that turn around an axis [closed]

Take a thick cord (the alim cord of your laptop or your mouse cord for example) and wrap it around your hand (or finger) turning always in the same direction but possibly knotting it. Then try to ...

**4**

votes

**1**answer

608 views

### Does there exist infinitely many prime knots?

I'm not a topologist and I just saw the definition of prime knot a while ago.
Today I'm somewhat supprised to realize that I don't even know if there are infinitely many prime knots? If this ...

**3**

votes

**0**answers

198 views

### Knot polynomials of non-crystallographic Coxeter groups?

I learnt that the Coxeter groups have a few members more than the
classic simple Lie groups: $H_3, H_4$ and $I_2(p)$. Is there a Reshetikhin-Turaev
invariant for those, too? If not, where does the ...

**2**

votes

**1**answer

268 views

### First cohomology of the space of long knots in R^4

Let's consider the space of long knots in $\mathbb R^n, n>3$. I know that there are many results (Vassiliev, Turchin, Sinha, Kontsevich) about different expressions of cohomology of this space. I ...

**10**

votes

**1**answer

474 views

### Traces on Hecke algebras and the Jones polynomial

In his famous paper "Hecke algebra representations of braid groups and link polynomials," (Annals 1987), Jones uses a compatible family of traces $tr_z$ on the Iwahori-Hecke algebras $H(q,n)$ of type ...

**12**

votes

**1**answer

724 views

### Fox differential calculus and the Alexander invariant of a link

I am teaching a course in knot theory, and I would like to describe the presentation of the Alexander module of a link obtained via Fox differential calculus. In doing this, I should prove the ...

**19**

votes

**7**answers

2k views

### Is there a “knot theory” for graphs?

I think knot theory has been studied for quite a while (like a century or so), so I'm just wondering whether there is a "knot theory" for graphs, i.e. the study of (topological properties of) ...

**4**

votes

**0**answers

243 views

### Reshetikhin-Turaev and links with a distinguished component

Hi,
This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question.
Let $T$ be the category whose objects are ...

**1**

vote

**0**answers

196 views

### A knot complexity measure

Construct a knot/link by fusing two n-tangles together.
(A tangle matrix shows how
this might look for tangles with 6 legs. But lets use 4 legs for a start
as this is far simpler.)
Now, any rational ...

**5**

votes

**1**answer

674 views

### Kontsevich Integral without associators?

Recall the fact that the representations of a quantum group form a braided tensor category, and this corresponds to the fact that $U_q(\mathfrak g)$ is a quasi-triangular Hopf algebra. The braiding ...

**1**

vote

**0**answers

342 views

### A Category of Knot Diagrams

A brief explanation of my motivation before I ask my question. I am trying to understand Skein relations, the Jones polynomial, and their relations to Khovanov homology. To me, the natural setting to ...

**1**

vote

**0**answers

116 views

### Spectral decomposition of R matrix -> Wenzl projectors?

Just curious: if you take a R matrix from knot theory and apply
a spectral decomposition (see. e.g. my following post
Matrix decomposition the other way)
you'll get projectors: ...

**8**

votes

**2**answers

409 views

### Number of the Reidemeister moves needed to transform one diagram into another one

A recent question Random Reidemeister moves to unknot contains a link to the paper http://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-7/S0894-0347-01-00358-7.pdf, in which J. Hass and J. ...

**0**

votes

**0**answers

146 views

### Around the Montesino-Nakanishi 3-move

I have a few questions around the 3-move. I know it's NOT an unknotting
move (but who needs knots with 20+ crossings anyway :-) by the recent proof
of Przytycki.
1. In another paper about the third ...

**6**

votes

**2**answers

666 views

### Random Reidemeister moves to unknot

Suppose one has a link diagram of the unknot, and applies random Reidemeister moves
until the unknot is reached.
Surely it requires an exponential number of moves, exponential in, say, the crossing ...

**1**

vote

**1**answer

506 views

### Recovering the Alexander Polynomial from Ocneanu's HOMFLYPT

Let $H_q(n)$ denote the Hecke algebra associated to the symmetric group $S(n)$: this is the $\mathbb{Z}[q^{\pm 1/2}]$ algebra generated by $T_1, \ldots, T_{n-1}$ satisfying the braid relations along ...

**1**

vote

**2**answers

352 views

### Untangling a graph

Assume you have a 4-valent graph (i.e., a knot universe, i.e. a collection
of self-intersecting curves). Your allowed moves are the equivalents
of Reidemeister 1, 2, 3, just with 4-nodes instead of ...

**2**

votes

**2**answers

398 views

### Is it possible to reliably generate a particular approximation of an ideal knot via a simulated annealing approach?

Say I take a cord, tie a loose knot in three-dimensional space, and pull tightly on the ends to generate an approximation of an ideal knot. If the cord has a fixed knot topology and a random initial ...

**4**

votes

**1**answer

776 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 ...

**1**

vote

**1**answer

326 views

### Knot polynomials: Skein>Matrix>Group?

OK, the heading was a bit tersely formulated...
If you have a quantum group and an irrep, you theoretically know the
R matrix (mathematicians are a notoriously idle lot, they give the
general formula ...

**6**

votes

**2**answers

647 views

### Examples of Non-algebraic Fibered Knots?

I am currently reading a monograph by Jose Seade, " On the topology of isolated singularities in analytic spaces".
I have following questions but before asking questions I recall the definition of ...

**4**

votes

**1**answer

389 views

### How does one relate the monodromy of the KZ equations with the WRT representation of the braid group?

The KZ equations on the configuration space of $n$ distinct points in $\mathbb C$ give rise to a representation of $B_n$ on $V^{\otimes n}$, where $V$ is any given representation of $SL(2)$ (we'll ...

**7**

votes

**1**answer

567 views

### What are the statistics of prime knots in 3d Random walk?

This question on physics stackexchange http://physics.stackexchange.com/questions/12973/the-entropic-cost-of-tying-knots-in-polymers has a formulation which is perhaps more appropriate for this forum.
...

**8**

votes

**2**answers

645 views

### Torus knots in Euclidean space — a symmetry argument

Consider a $(p,q)$ torus knot $K$ in 3-dimensional Euclidean space $\mathbb R^3$ where $p,q \geq 2$ and $\operatorname{GCD}(p,q)=1$.
Let $\operatorname{Isom}(\mathbb R^3,K)$ be the isometries of ...

**6**

votes

**1**answer

261 views

### What is the complex structure on the boundary torus of a hyperbolic knot complement?

Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary ...

**6**

votes

**0**answers

256 views

### Is the category of tangles that includes, X, Y, and Lambda a free Frobenius braided category?

Consider the category whose objects are non-negative integers that are represented as dots along a line, and whose morphisms are generated by $X$---positive crossing, $\bar{X}$ --- negative crossing, ...

**6**

votes

**1**answer

518 views

### Is the complete functorial structure for Khovanov--Lee homology known?

I'm interested in Lee's modification of Khovanov homology, which I'll denote $\operatorname{Kh}_{\operatorname{Lee}}^\ast$. Below $L$ is a link in $\mathbb R^3$.
The groups ...

**1**

vote

**0**answers

222 views

### subset embedding gives trefoil knot [closed]

Let $X$ be a topological space and $E_n(X)$ the space of finite sets of cardinality $\leq n$.
It is a theorem of Bott that $E_3(S^1)=S^3$. What is the idea to show that
the embedding ...

**27**

votes

**0**answers

983 views

### What is the three-dimensional hyperbolic volume of a four-manifold?

Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds.
...

**0**

votes

**0**answers

459 views

### Knot Numerology

EDIT: Ok, I condense it to only that what is needed.
Assume it's possible to use the method described here
Matrix decomposition the other way
to decompose a $S$ matrix from knot theory. Then each ...

**5**

votes

**1**answer

555 views

### Trace identities and the Kauffman Bracket skein module

Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the ...

**2**

votes

**1**answer

188 views

### Invariants of unframed, oriented links from Reshetikhin Turaev construction

Hello!
I have a few questions on Reshetikhin Turaev invariants.
By RT any ribbon category ${\mathcal C}$ yields an invariant of oriented, framed links labelled with objects of ${\mathcal C}$.
Is ...

**5**

votes

**2**answers

634 views

### Are there any very hard unlinks?

This question is closely related to a question of Gowers: Are there any very hard unknots? .
I'm thinking about how to create interesting knots from small numbers of local moves on unlinks. The ...

**15**

votes

**2**answers

2k views

### How many definitions are there of the Jones polynomial?

Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...

**3**

votes

**0**answers

242 views

### A tangle matrix

The image should already say everything. But in case it is mute... :-)
List tangles with 2n legs downward as column header, and with 2n legs
upward as row header. (n=3 here.) In the crossing point ...

**2**

votes

**2**answers

494 views

### is twisted torus knot T(7,2;4,1) prime?

what kind of twisted torus knot is prime?
even more , is twisted torus knot T(7,2;4,1) prime?

**3**

votes

**1**answer

303 views

### A question about Dehn filling in the unknot.

Hello,
If you have a link of 2 components $U$ and $V$ in $S^3$. $U$ is the unknot, and you make $1/q$ dehn filling in $U$, you can visualize the resulting knot $V'$ from $V$, by twisting it $q$ ...

**2**

votes

**1**answer

452 views

### Now that I got a mutant-discriminating invariant…

...what can I do with the darn thing?
Background: I read that still no Vassiliev Invariant with mutant-discriminating
power is known (correct me if this is outdated). Now, my research lead to a
whole ...

**0**

votes

**1**answer

333 views

### Some Vassiliev Invariant Questions

The V.I. definition goes doublepoint=overpass-underpass (or was it the other
way around? If it's 50:50, I score 0 always :-). Would it lead anywhere
to define doublepoint=overpass+underpass? (Even if ...

**7**

votes

**2**answers

557 views

### Does every knot contain all four vertices of an isosceles trapezoid?

I ask this question with some trepidation, because it may be trivial and/or of entirely recreational interest.
Erika Pannwitz proved in 1933 that every non-trivial knot contains a quadrisecant (four ...