**2**

votes

**1**answer

265 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

456 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

692 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 ...

**17**

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

237 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

195 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

662 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

330 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

113 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: ...

**6**

votes

**1**answer

324 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

145 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

647 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

495 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

347 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

387 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

723 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

324 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

617 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

374 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

551 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

639 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

257 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

251 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

507 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

221 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

958 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

453 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

543 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

185 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

625 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

237 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

481 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

450 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

328 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

552 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 ...

**3**

votes

**0**answers

341 views

### Positivity of braid monodromy of curve singularities

I recall the notion of braid monodromy. Let $C \subset \mathbb{C}^2$ be an algebraic curve, and choose a projection $\pi: \mathbb{C}^2 \to \mathbb{C}$ such that the restriction $\pi: C \to ...

**3**

votes

**2**answers

382 views

### Construction of Kirby-diagram for slice-complement

Let $(S^3,K)=\partial (D^4,D)$ be a slice knot. (i.e. a knot in $S^3$ bounds a disk in $D^4$.)
Are there any relation between the knot diagram of $K$ and the kirby diagram for $(D^4-D,M_K)$, where ...

**5**

votes

**1**answer

865 views

### smooth proof of reidemeister theorem

Hello!
In another post Ryan Budney has mentioned a "smooth proof" of the theorem of Reidemeister...
(@Ryan Budney:) Do You know if there is a book (or paper) containing such a proof? I am interrested ...

**14**

votes

**3**answers

1k views

### Is anything known about this braid group quotient?

Let $B_n$ be the braid group on $n$ strands. As is well known, if $\sigma_i$ is the operation of crossing the string in position $i$ over the string in position $i+1$, then the elements ...

**0**

votes

**1**answer

231 views

### “Skein” equations sets that can reduce any graph

Consider for example the approach to the Kuperberg G2 or the Yamada polynomial. I don't know whether relations of graph (in contrast to knot) theory are also usually called "skein" but
I simply carry ...

**1**

vote

**2**answers

431 views

### Which knots can appear as a space-time cut/slice of an open trefoil?

An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space is a surface in space-time. Now, other observers in space-time have other time ...

**2**

votes

**1**answer

265 views

### Can a closed trefoil appear as a space-time “cut” of an open trefoil?

An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space, in space-time is a knotted hypersurface.
Different observers in space-time have ...

**2**

votes

**2**answers

998 views

### How many lanes has a freeway? (Crossing free Kuperberg G2, that is.) EDITED

For referencing, I keep the original title and post and ask
only about the simplest case (and forget the freeway with crossings
for now).
Consider a trivalent graph, e.g. the dodecahedron or cube ...

**1**

vote

**1**answer

385 views

### What is the shape of a tight open trefoil?

Take an infinitely long rope of diameter 1, ideally flexible and ideally slippery. Tie a trefoil knot into it and pull it tight. Describe the resulting rope shape analytically.
The problem is ...

**31**

votes

**9**answers

2k views

### In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?

In several textbooks on knot theory (e.g. Lickorish's, Rolfsen's) knots are considered in $\mathbb{R}^3$ or $S^3$. The reason for working in $S^3$ is sometimes given at the beginning of a text as that ...

**0**

votes

**0**answers

308 views

### A knot theory form of the carpenter's ruler question

Hey,
The carpenter's ruler problem is about polygons you can make with planar joints. You can formulate a similar question for knots when you introduce ball joints:
Given m ball joints connected by ...

**2**

votes

**0**answers

131 views

### Simple terminology question about the Dubrovnik (Kauffman) polynomial

In my S matrix classification attempts I encounter a lot of
Dubrovnik polynomials of the form D(z-1/z,z^n) and D(-z+1/z,z^n).
[Second variable is for writhe, n is an integer; for the first I don't
...

**1**

vote

**1**answer

338 views

### Handle slides homeomorphism

For handle decomposition of surface, suppose I have a twisted 1-handle(twisted only once) adjacent to an isolated pair of linked handles, is the handle slide operation enough to move the previous one ...