**6**

votes

**1**answer

247 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

489 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

219 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 ...

**26**

votes

**0**answers

928 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

445 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

530 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

182 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

621 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

1k 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

235 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

459 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

445 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

322 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

550 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

339 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

372 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

831 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

230 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

426 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

263 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

997 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

383 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

302 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

128 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

320 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 ...

**0**

votes

**0**answers

166 views

### On Birman-Wenzlyfying the B2 spider

Prelude: First of all, let "S matrix" denote "an abstract 4D tensor satisfying
the usual isotypy rules (with no arrows!)". I'm busy trying to classify all possible
S matrices (paper pending) - it's ...

**5**

votes

**1**answer

572 views

### Link of singularities

For an isolated plane curve singularity, given by homogeneous equation $F=0 \subset \mathbb{C}^2$, one consider the curve $(F=0) \cap S^3 \subset S^3$, and we call it the link of singularity. some ...

**1**

vote

**1**answer

342 views

### Braid*Temperley-Lieb=?

I would be very astonished if this algebra isn't named.
You simply have the braid AND the Temperley-Lieb generator in
the algebra. Rules are the usual Reidemeister equivalents
plus the kink and ...

**7**

votes

**2**answers

803 views

### Is there a table of (fibred knot) monodromies?

Background/motivation
I'm working on contact topology (in dimension three): a fundamental theorem of Giroux gives us a bijection between contact structures (up to isotopy) and open books (up to ...

**0**

votes

**1**answer

812 views

### Is the HOMFLY Polynomial the best knot invariant? [closed]

Is the HOMFLY Polynomial the best polynomial invariant that can be calculation from skein relation?

**0**

votes

**0**answers

291 views

### Properties of the Jones polynomial

Let $V(L)$ be the Jones polynomial of the oriented
link $L$. For $\alpha \in B_n$, we write $V(\alpha)$ for
$V(\hat{\alpha})$, where $\hat{\alpha}$ is the closed braid
associated to $\alpha$. The ...

**0**

votes

**0**answers

227 views

### “Lorentz-invariant” Morse theory possible?

Hi,
I hope filing this under "Morse theory" is correct.
For the "abstract tensor" approach to knot theory you need:
a) a 4D tensor to emulate particle interaction,
b) a 2D tensor to emulate pair ...

**7**

votes

**2**answers

419 views

### Do the results of (1/n)-surgery determine the link?…

Knowing the result of knot surgery is often not enough to determine the knot. Indeed, there are 3-manifolds admitting an infinite number of descriptions as surgery on a (1-component) knot in $S^3$. ...

**9**

votes

**2**answers

392 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 ...

**4**

votes

**3**answers

759 views

### Whitehead doubles of any knots

I was curious about the fact that the Whitehead doubles of all knots have Alexander polynomial equal to 1, which is the same as a unknot. How to prove this?

**26**

votes

**3**answers

979 views

### Random knot on six vertices

This question is inspired by Joseph O'Rourke's beautiful question on random knots. Choose an random ordered 6-tuple of points on the unit sphere in $\mathbf{R}^3$, and form a knot by connecting ...

**8**

votes

**2**answers

415 views

### Is every virtual knot group an HNN extension?

A basic fact in knot theory is that a knot group $\pi(K)$ is an HNN extension of $\pi(F)$, the fundamental group of a Seifert surface complement. A nice discussion of this may be found in Chapter 11 ...

**1**

vote

**1**answer

579 views

### Markov Trace and Markov Property

Hey guys,
I'm a computer science student attempting to understand a quantum algorithm that uses braid theory - something I'm completely unfamiliar. I've getting through the algorithm but I can't ...

**18**

votes

**4**answers

905 views

### Complexity of random knot with vertices on sphere

Connect $n$ random points on a sphere in a cycle of
segments between succesive points:
I would like to know the growth rate, with respect to $n$, of the crossing ...

**-1**

votes

**1**answer

558 views

### monodromy defects and Chern-Simons

In the context of string theory I recently read "The formulation of Chern-Simons theory in terms of monodromy defects can be carried through all the dualities of the present paper, leading to ...

**5**

votes

**2**answers

356 views

### What is known about links with a countably-infinite number of tame components?

I want to be clear about my phrasing, but I'm not a topologist, so when I say "knot" or "link" I mean the equivalence class under ambient isotopy of an embedding of the circle into $\mathbb{R}^3$.
...

**6**

votes

**1**answer

529 views

### Length of shortest possible knot

Consider a line L in R^3 in the shape of a trefoil knot. Consider the surface S that is the union of all unit circles that have centers on this line and whose tangent vectors are all perpendicular to ...

**9**

votes

**1**answer

307 views

### Ramified cover of 3-ball

I it true that any connected oriented 3-manifold with nonempty boundary can be obtained as a ramified cover of 3-ball with a ramification at a link?
link = a 1-dimensional submanifold with possibly ...

**1**

vote

**1**answer

218 views

### Simplified Jones trace invariant for links

Jones (1985) defines a simplified trace invariant for knots by $W_K(t)=\frac{1-V_K(t)}{(1-t^3)(1-t)}$. Then, e.g., the Arf invariant for $K$ is $Arf(K)=W_K(i)$. Does this work for oriented links as ...

**0**

votes

**1**answer

221 views

**65**

votes

**10**answers

7k 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 ...