**87**

votes

**10**answers

11k 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 ...

**48**

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 ...

**41**

votes

**3**answers

3k views

### Kirby calculus and local moves

Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and ...

**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 $\mathbb Z$ ...

**38**

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 ...

**34**

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 ...

**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 ...

**30**

votes

**0**answers

1k 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.
...

**27**

votes

**3**answers

2k views

### What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov, Kronheimer and Mrowka)

Kronheimer and Mrowka showed that the Khovanov homology detects the unknot.
Bar-Natan showed a program to compute the Khovanov homology fast: there was no rigorous complexity analysis of the ...

**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 ...

**27**

votes

**8**answers

2k views

### probabilistic knot theory

Take a smooth closed curve in the plane. At each self-intersection, randomly choose one of the two pieces and lift it up just out of the plane. (Perturb the curve so there are no triple ...

**27**

votes

**1**answer

2k views

### Does this knot invariant distinguish trefoil chiralities?

Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$.
As a corollary of something else I was playing around with, I recently ...

**26**

votes

**1**answer

1k views

### Intuitive explanation for the use of matrix factorizations in knot theory

Hello!
I read through parts of Khovanov/Rozansky's paper on the categorification of the HOMFLY polynomial using Matrix Factorizations. Technically, I can follow (though it seems to me that quite a ...

**24**

votes

**9**answers

12k 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. ...

**24**

votes

**1**answer

436 views

### The Jones polynomial at specific values of $t$.

I've been calculating some Jones polynomials lately and I was just curious if there was a physical meaning to evaluating the Jones polynomial at a particular value of $t$.
For example, if I take the ...

**24**

votes

**0**answers

241 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 ...

**23**

votes

**8**answers

4k views

### Resources for graphical languages / Penrose notation / Feynman diagrams / birdtracks?

There is an idea I've recently gotten interested in that doesn't seem to have a good agreed-upon name ("diagrammatic algebra?"). It centers around the use of two-dimensional diagrams of dots, ...

**22**

votes

**3**answers

2k views

### How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms. [closed]

With geometrization, Rubinstein's 3-sphere recognition algorithm and the Manning algorithm, 3-manifold theory has reached a certain maturity where many questions are "readily" answerable about ...

**21**

votes

**1**answer

416 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 ...

**20**

votes

**3**answers

2k views

### Why is it so hard to implement Haken's Algorithm for knot theory?

Why is it so hard to implement Haken's Algorithm for recognizing whether a knot is unknotted? (Is there a computer implementation of this algorithm?)

**20**

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)) ...

**20**

votes

**4**answers

2k views

### Utility of virtual knot theory?

Virtual knot theory is an interesting generalization of knot theory in which ``virtual" crossings are allowed. See Kauffman's Virtual Knot Theory for an introduction. Greg Kuperberg gave a nice ...

**20**

votes

**3**answers

2k views

### What would the slice-ribbon conjecture imply?

What would the slice-ribbon conjecture imply for 4-dimensional topology?
I've heard people speak of the slice-ribbon conjecture as an approach to the 4-dimensional smooth Poincare conjecture, and to ...

**20**

votes

**2**answers

1k views

### Why is the Alexander polynomial a quantum invariant?

When we think of quantum invariants, we usually think of the Jones polynomial or of the coloured HOMFLYPT. But (arguably) the simplest example of a quantum invariant of a knot or link is its Alexander ...

**19**

votes

**7**answers

3k 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) ...

**19**

votes

**4**answers

1k 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 ...

**19**

votes

**5**answers

3k 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 ...

**19**

votes

**1**answer

2k views

### Propagation of an error in the LMO invariant? (Revision: I don't think LMO is wrong!)

Edit: I think LMO is correct. Massuyeau has a nice explanation here.
Edit: Renaud Gauthier has retracted the claim of an error in the foundations of the LMO construction, and has withdrawn both ...

**19**

votes

**0**answers

346 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 ...

**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 ...

**18**

votes

**1**answer

2k views

### What is a path in K-theory space?

In a comment on Tom Goodwillie's question about relating the Alexander polynomial and the Iwasawa polynomial, Minhyong Kim makes the cryptic but tantalizing statement:
In brief, the current view is ...

**18**

votes

**2**answers

627 views

### Lie algebra automorphisms and detecting knot orientation by Vassiliev invariants

Recall that there are knots in $\mathbf{R}^3$ that are not invertible, i.e. not isotopic to themselves with the orientation reversed. However, it is not easy to tell whether or not a given knot is ...

**18**

votes

**0**answers

531 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 ...

**18**

votes

**0**answers

481 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 ...

**17**

votes

**2**answers

1k views

### Is the 4x5 chessboard complex a link complement?

The 2x3 and 3x4 chessboard complexes (form a square grid of vertices and make a simplex for any set of vertices no two of which are in the same row or column) are a 6-cycle and a triangulated torus ...

**17**

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" ...

**17**

votes

**3**answers

456 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 ...

**17**

votes

**2**answers

995 views

### Does the super Temperley-Lieb algebra have a Z-form?

Background Let V denote the standard (2-dimensional) module for the Lie algebra sl2(C), or equivalently for the universal envelope U = U(sl2(C)). The Temperley-Lieb algebra TLd is the algebra of ...

**17**

votes

**1**answer

1k views

### How are the Conway polynomial and the Alexander polynomial different?

Background story:
I have just come out from a talk by Misha Polyak on generalizations of an arrow-diagram formula for coefficients of the Conway polynomial by Chmutov, Khoury, and Rossi. In it, he ...

**16**

votes

**2**answers

411 views

### Are there piecewise-linear unknots that are not metrically unknottable?

A stick knot is a just a piecewise linear knot. We could define "stick isotopy" as isotopy that preserves the length of each linear piece.
Are there stick knots which are topologically trival, but ...

**16**

votes

**3**answers

607 views

### Looking for “large knot” examples

This question is about knots and links in the 3-sphere. I want to find an example of a "large" knot or link with some special properties. I'm looking for some fairly specific examples, but I'm also ...

**16**

votes

**4**answers

907 views

### Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?

This question was originally posted on math.SE by myself nearly a year ago. I've been thinking again about the problem after it recently received a little attention, but little progress was made in ...

**16**

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 ...

**16**

votes

**3**answers

653 views

### Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...

**15**

votes

**5**answers

1k views

### Can surfaces be interestingly knotted in five-dimensional space?

It's possible this question is trivial, in which case it will be answered quickly. In any case, I realized that it's a basic question the answer to which I should know but do not.
Everybody loves ...

**15**

votes

**4**answers

2k views

### Who thought that the Alexander polynomial was the only knot invariant of its kind?

I apologize that this is vague, but I'm trying to understand a little bit of the historical context in which the zoo of quantum invariants emerged.
For some reason, I have in my head the folklore:
...

**15**

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 ...

**15**

votes

**4**answers

962 views

### HOMFLY and homology; also superalgebras

My understanding is that an analogy along the following lines is (roughly) true:
"The Alexander polynomial is to knot Floer homology is to gl(1|1)
as the Jones polynomial is to Khovanov homology is ...

**15**

votes

**5**answers

3k 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 ...

**15**

votes

**3**answers

726 views

### What is the state of the art for algorithmic knot simplification?

Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, ...