**7**

votes

**2**answers

547 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

334 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

365 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

813 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

420 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

260 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

995 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

382 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

297 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

312 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

165 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

541 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

339 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

766 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 ...

**1**

vote

**1**answer

782 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

283 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

162 views

### Polynomial knots

I am an undergraduate studying polynomial knots. Given a polynomial knot, it is common to generate a tube around the space curve for visualization purposes. I define the set T[c(t)] as the set of all ...

**0**

votes

**0**answers

223 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

417 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

382 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

738 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

952 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

405 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

540 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

890 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

554 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

347 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

527 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

305 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

216 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

220 views

**65**

votes

**10**answers

6k 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 ...

**18**

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?)

**12**

votes

**5**answers

880 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 ...

**13**

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 ...

**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 ...

**3**

votes

**1**answer

322 views

### Homology boundary links

According the the original definition by Smythe, a homology boundary link $L\subseteq S^3$ with $m$ components is a link which sarisfies one of the following equivalent conditions:
(1): The ...

**4**

votes

**1**answer

333 views

### Cubic skein relations

Hi,
please note that this question deals with undirected knots/links!
The most generic cubic skein relation for a knot polynome would be
where w^3 is one positive writhe unit. The form is fairly ...

**3**

votes

**1**answer

494 views

### Boundary links and ribbon links.

This question is about the relation between the notions of boundary link and ribbon link.
For the definition of ribbon link see: ribbon links - counterexamples.
An n-component link ...

**3**

votes

**0**answers

278 views

### Computing Quantum Dimensions

Hi,
in "Jaeger’s Higman-Sims state model and the B2 spider" by Greg Kuperberg
(arxiv:math9601221v1, 1996) there are some quantum dimensions listed in the
"Discussion" part. Evidently quantum groups ...

**5**

votes

**1**answer

476 views

### When is a connected sum of torus knots a slice knot?

This question is about the beaviour of 4-genus of knots with respect to connected sum.
Let us indicate with $T(k)$ a Torus knot of type $(2,k)$, $k$ is an odd integer.
Fix an orientation for every ...

**2**

votes

**1**answer

605 views

### isotopy doesn't make sense (Milnor)

hello,
I am having a hard time following this isotopy put forth by Milnor in On the Total Curvature of Knots
For each $c$ and $p$ in
$\mathbb{R}^{n-1}$ such that $\|c-p\|
> < r$, there is ...

**1**

vote

**1**answer

181 views

### crookedness of convex curves (milnor)

hello,
I'm currently reading On the Total Curvature of Knots and am trying to understand one of the lemmas in it (3.3)
A closed polygon $P$ in $H^2$ is convex if and only if for every $b$ either ...

**13**

votes

**9**answers

6k 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. ...

**17**

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 ...

**13**

votes

**2**answers

832 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 ...