**81**

votes

**10**answers

9k 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 ...

**13**

votes

**1**answer

2k views

### What is the knot associated to a prime?

I can't help but ask this question, having found out about arithmetic topology here on MO. There is a concise description of the MKR dictionary central to this philosophy here. This dictionary is used ...

**19**

votes

**3**answers

2k views

### Complete knot invariant?

I've seen a couple papers (that I now can't find) that say that in his paper "On irreducible 3-manifolds which are sufficiently large" Waldhausen proved that the data $\pi_1(\partial (S^3\setminus K)) ...

**19**

votes

**5**answers

2k views

### Proof of the Reidemeister theorem

While preparing for my introduction to topology course, I've realized that I don't know where to find a detailed proof of the Reidemeister theorem (two link diagrams give isotopic links, iff they can ...

**3**

votes

**1**answer

239 views

### Min Bend Orthogonal Knots

I am seeking literature on 3D orthogonal drawings of knots,
especially minimum bend drawings.
An orthogonal drawing employs segments parallel to the axes of
a Cartesian coordinate system.
A bend is a ...

**41**

votes

**2**answers

3k views

### Prime numbers as knots: Alexander polynomial

A naive and idle number theory question from a topologist (but not a knot theorist):
I have heard it said (and this came up recently at MO) that there is a fruitful analogy between Spec $\mathbb Z$ ...

**47**

votes

**4**answers

3k views

### Tying knots with reflecting lightrays

Let a lightray bounce around inside a cube whose faces
are (internal) mirrors.
If its slopes are rational, it will eventually form a cycle.
For example, starting with a point $p_0$ in the interior
of ...

**33**

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 ...

**33**

votes

**2**answers

2k views

### Knot security (When to trust your life with a knot)

This question is related to a a question about self-tightening knots.
I am supervising a senior thesis and my student is interested in knots. My student is also a rock climber and has an ...

**14**

votes

**2**answers

2k views

### What's about “quantum modular forms”?

Do you know where one could read on "Modular Forms, K-theory and Knots"? The combination of themes sounds thrilling!
Edit: Zagier's paper on "quantum modular forms" will be published in Clay's ...

**37**

votes

**2**answers

2k views

### Can knot diagrams be monotonically simplified using under moves?

It is well known that knot diagrams cannot be monotonically simplified using Reidemeister moves. For instance, the Goeritz unknot cannot be directly simplified. On the other hand, there is a stronger ...

**27**

votes

**3**answers

1k 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 ...

**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" ...

**19**

votes

**0**answers

292 views

### Topological description of inverting a knot

The first figure shows an offset overhand knot. To tie it, you simply bring the two cords together and make an overhand knot in them. It's more secure than it looks, and several climbers have been ...

**14**

votes

**5**answers

2k views

### can you fool SnapPea?

A while back I thought I had some simple knots that "fooled" SnapPea. But I no longer remember those examples, if I ever had them to begin with.
What I'm looking for is a non-hyperbolizable knot ...

**8**

votes

**2**answers

1k views

### Slice knots and exotic $\mathbb R^4$

In the http://arxiv.org/abs/math/0606464v1 I read
"If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in
possession of a knot which is ...

**8**

votes

**0**answers

338 views

### What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds ?

The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...

**7**

votes

**1**answer

236 views

### A $k$-component link defines a map $T^{k}\rightarrow \mathrm{Conf}_{k} S^{3}$. Does the homotopy type capture Milnor's invariants?

A $k$-component link defines a map $T^{k}\rightarrow \mathrm{Conf}_{k} S^{3}$. Does the homotopy type of this map capture the Milnor invariants?
Some special cases:
$k=2$, no, it's null ...

**15**

votes

**1**answer

406 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 ...

**9**

votes

**3**answers

337 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 ...

**7**

votes

**1**answer

478 views

### Diagrammatic proof of unique prime decomposition of knots

Consider a knot to be a diagram in a plane--- i.e. a drawing of a finite connected planar graph (loops and multiple edges allowed) whose vertices are 4-valent with cyclic ordering for the incident ...

**6**

votes

**1**answer

181 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 ...

**4**

votes

**1**answer

239 views

### Equivalent to Oriented knot complement conjecture

I would like to see why the following two statements in Kirby's list of problem are equivalent:
Statement 1:
If $K_1$ and $K_2$ are knots in a closed oriented 3-manifold $M$ whose complements are ...

**4**

votes

**0**answers

184 views

### Constructing Markov traces simply

Short version: I wondering how to simply check if a proposed Markov trace, $\phi$ had the correct property using techniques similar to those from the Akutsu-Wadati 1987 paper `Exactly solvable models ...

**8**

votes

**0**answers

160 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 $I^3$, ...

**2**

votes

**1**answer

154 views

### Oriented knot complement conjecture for fibered knots

Suppose I have two inequivalent fibered knots in a homology sphere. When I say 'inequivalent', I mean that there is no orientation-preserving homeomorphism of the space that takes one to the other. ...

**2**

votes

**2**answers

613 views

### How to compute the Alexander polynomial of general torus knot

Hello,
i am very interested in knot theory, especially in knot groups and knot polynomials. Therefore i am reading the book of Crowell and Fox (Introduction to knot theory). I want to compute some ...

**2**

votes

**1**answer

328 views

### Why Tristram-Levine signature jumps at the zeros of alexander polynomial?

It seems easy but I can't prove it. Can anyone give proof or reference?

**2**

votes

**1**answer

465 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 ...