**20**

votes

**0**answers

422 views

### Applications of arithmetic topology to number theory

There is a well-known analogy between 3-manifolds and number fields, with knots corresponding to prime ideals. Are there any results in number theory that have been proven using topology through this ...

**5**

votes

**0**answers

75 views

### Distribution of Random Knots from Braids

Let $R_{2n,l}$ be a random braid word of length $l$, where each letter is chosen uniformly from the braid generators of $B_{2n}$, $\{\sigma_1,\ldots,\sigma_{2n-1},\sigma_1^{-1},\ldots,\sigma_{2n-1}^{-...

**4**

votes

**0**answers

76 views

### Expressing knot polynomials as casimirs

I always wondered why one writhe unit (read: colored with the irrep of a quantum Lie algebra and evaluated as Reshetikhin-Turaev invariant thereof) is essentially the quadratic casimir of that irrep. (...

**5**

votes

**0**answers

119 views

### Essential surfaces in knot complements

Given any knot $K \subset \mathbb{S}^3$, one can find a closed oriented embedded surface $S$ such that $K \subset S \subset \mathbb{S}^3$. Moreover, pick such an $S$ that has minimal genus.
One can ...

**1**

vote

**1**answer

207 views

### Addition of two homology classes is zero in construction of Poincare Sphere

I ask here the question since it hasn't been answered in
Math Stack Exchange.
I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...

**3**

votes

**1**answer

93 views

### Morse function on slicing disk complement determines ribbon?

It is well-known that given a ribbon knot and the corresponding slicing disk in the 4-ball, the distance function (maybe squared) to the origin defines a Morse function in the complement of the ...

**4**

votes

**0**answers

96 views

### Connection between cardiac equations and untangling knots?

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots:
Maucher, Fabian, and Paul Sutcliffe....

**4**

votes

**2**answers

160 views

### Looking for examples of large hyperbolic two-generator knots or 3-manifolds

We say a knot $K$ in $S^3$ is small if its exterior contains no closed properly embedded incompressible surfaces and we say $K$ is large otherwise.
Does anyone know of an example of a large ...

**2**

votes

**1**answer

89 views

### A three-rope tangle that is equal to its mirror tangle

In the same way that a figure-eight knot is equal (after a suitable rotation) to its mirror knot, I am looking for simple tangles made of three ropes that are equal to their mirror tangle.
The ...

**3**

votes

**4**answers

169 views

### Knot invariants with skein relation of order 3 or 4

Many of the well-known knot/link invariants $I$, such as the Jones, Alexander, HOMFLY, Conway polynomials, satisfy a quadratic skein relation, i.e. an equation of the form $\alpha I(L_+)+\beta I(L_-)+\...

**8**

votes

**1**answer

173 views

### Are knots determined by their complements within a homotopy class?

Suppose $M$ is a closed 3-manifold and $K_1,K_2$ are two homotopic knots in $M$. That is, they are two embeddings $f_1,f_2\colon S^1 \to M$ such that there exists a homotopy $h\colon S^1 \times [0,1] ...

**7**

votes

**0**answers

82 views

### Compatibility of spherical and hyperbolic geometry for fibred knots

Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other ...

**4**

votes

**1**answer

221 views

### Understanding “Decategorified” symplectic Khovanov homology

In http://arxiv.org/abs/math/0405089 Seidel and Smith constructed a link invariant using Lagrangian Floer theory that was conjectured to be equivalent to Khovanov homology. The equivalence was ...

**4**

votes

**1**answer

184 views

### Fibered knots vs Heegaard genus

We call a knot $K$ in a 3-manifold $M$ fibered if $M\backslash K$ fibers over $S^1$ with fibers $\Sigma$ and such that $K$ is ambient isotopic to the boundary of the compactified fiber $\overline{\...

**4**

votes

**1**answer

163 views

### How many quadratic fields occur as trace fields of hyperbolic knot complements?

I am interested in when the trace field of a knot complement has the form $F(\sqrt{-d})$ for $F\subset\mathbb{R}$ and $d\in F^+$ (squarefree). Does this occur for infinitely many choices of pairs $(F,...

**0**

votes

**0**answers

135 views

### Tying knots in $\mathbb{C}$

As far as I know, there is no deep significance to this question, but I've been playing around with it for a bit and it seems interesting:
Fix a complex number $c$, and consider the map $J_c: \mathbb{...

**5**

votes

**0**answers

76 views

### How long does it take for the action of the braid monoids on Laver tables to become trivial?

Let $A_{n}$ denote the classical Laver table of cardinality $2^{n}$.
Let $B_{n}^{*}$ denote the positive (including the identity) braid monoid on $n$ elements generated by $\sigma_{1},...,\sigma_{n}$....

**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 algorithm,...

**3**

votes

**0**answers

61 views

### Linking circles inside an immersed surface

(Migrated from Math Stack Exchange)
A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the ...

**3**

votes

**1**answer

92 views

### Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?

Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by $...

**4**

votes

**2**answers

632 views

### What is the mathematical significance of the IHES logo?

The logo of the IHES
http://www.ihes.fr/jsp/site/Portal.jsp
(upper left) is lovely, but what exactly does represent mathematically?
(There's a slightly larger version at
http://www.ihes.fr/~...

**6**

votes

**1**answer

364 views

### Is a “knot knot” or “double knot” a thing in knot theory?

I apologize in advance for my rudimentary knowledge of knot theory, but I've been trying to find out about the significance (if any) of taking a knot (particularly a torus knot), cutting it, and ...

**13**

votes

**3**answers

687 views

### Classification of knots by geometrization theorem

I read this interview with Ian Agol, where he says:
"...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots."
My question is:
How does ...

**5**

votes

**1**answer

122 views

### Ascending surfaces in the 4-ball

Let the standard symplectic structure on $B^4$ (viewed in $\mathbb{R}^4$ or $\mathbb{C}^2$) be given by $\omega=(1/2) d \eta$, for
\begin{align*}
\eta &:=
x_1 \, dy_1 - y_1 \, dx_1 + x_2 \, ...

**1**

vote

**1**answer

187 views

### On Knot Equivalence problem statement

How is the knot equivalence problem represented?
By this I mean I am looking for an analogy that compares with graph equivalence. For graph equivalence, we have two graphs $G_1$ and $G_2$ with ...

**11**

votes

**4**answers

757 views

### Distance between two knots

Are there some well-studied functions defining natural distance measures between two knots? One can imagine a function that counts, say, the minimum number of
moves, each of which passes one strand of ...

**3**

votes

**0**answers

71 views

### Knot invariants which arise as the solution of differential equations along the knot

Do there exist knot invariants which can be computed by solving some kind of differential equation along the knot?

**8**

votes

**1**answer

205 views

### What does the representation category of the knot group know?

Let $K, K'$ be knots in $S^3$, and $T, T'$ the boundaries of their tubular neighborhoods.
Recall that by theorems of Waldhausen, and Gordon and Luecke, one knows the following: an isomorphism $[\...

**1**

vote

**0**answers

94 views

### Unnormalized Kauffman homology of the unknot

Is the unnormalized Kauffman homology of the unknot known? The Poincare polynomial of HOMFLY homology of the unknot is known as
$$\frac{1+at}{1-q}.$$
Is the Poincare polynomials of Kauffman homology ...

**1**

vote

**1**answer

96 views

### Cancellation of 2-component links

Consider a two-component tame link in 3-space, consisting of an arc from $(-1,1,0)$ to $(1,1,0)$ and an arc from $(-1,-1,0)$ to $(1,-1,0)$, confined to the slab $-1 \leq x \leq 1$. Call such a link ...

**6**

votes

**1**answer

205 views

### Can one twist fibred knots and still get fibred knots?

Suppose we have a fibred knot $K$ with a fiber surface $F$ and let $c$ be an unknot disjoint from $F$ (but not homotopically trivial in the complement of $F$). Is it possible that every twist along $c$...

**2**

votes

**1**answer

56 views

### Calculating the Upper Bound on the Sphere Radius of Knotted Channel Surfaces

This question is motivated by trying to determine the upper bound on the thickness of a rope of fixed length (w.l.o.g. $2\pi$), with which a knot of given topology can be realized under the further ...

**4**

votes

**1**answer

185 views

### Physical strength of a link [closed]

Assuming that we construct a link/chain using a collection of knots.
Is there a way to measure the physical strength of this chain?

**6**

votes

**2**answers

272 views

### Fibered example of topologically slice knots

Is there any known example of fibered knot which is topologically slice but not (expected to be) smoothly slice?

**21**

votes

**1**answer

422 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 ...

**8**

votes

**1**answer

219 views

### Questions on poincare homology spheres and branched covers

I have two questions:
Question 1. Suppose that $K$ is a knot in $S^3$. Let $\Sigma(K)$ be the double branched cover of $S^3$ branched along $K$. If $\Sigma(K)=\#_{i=1}^n\Sigma(2,3,5)$, then $K=\#_{i=...

**18**

votes

**0**answers

592 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 $...

**12**

votes

**0**answers

387 views

### Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?

Let $K$ be a knot in the 3-sphere $S^3$.
Here we denote by $s(K)$ Rasmussen's s-invariant for $K$,
and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$
by attaching a $2$-...

**11**

votes

**1**answer

554 views

### A fun game related to knot theory

I recently learned the following rather fun game: a group of people is standing up roughly in circle, facing each other. Then participants randomly join hands, in such a way that nobody holds its own ...

**6**

votes

**3**answers

235 views

### Surgery along an arc connecting the components of a $2$-component link gives the unknot

Math Overflow seems to have a dearth of low dimensional topology, but this seems like an interesting question. Let $L$ be a $2$-component link in $S^3$. Suppose that there is a framed arc joining the ...

**1**

vote

**1**answer

96 views

### Essential surfaces in the Exterior of Montesinos knots

Hatcher and Oertel computed the boundary slopes of essential surfaces of Montesinos knots in this paper. But they do not consider surfaces that do not intersect the boundary of the exterior. An ...

**24**

votes

**0**answers

248 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 (...

**7**

votes

**2**answers

289 views

### What is the easiest way to compute Ozsváth-Szabó tau invariant of a knot?

Suppose that we have a knot $K$ with 40 crossings which is not a cable knot or an alternating knot.
Then, what is the easiest way to compute Ozsváth-Szabó's invariant $\tau(K)$?
Are there any ...

**1**

vote

**0**answers

77 views

### Distiguishing mutant knots

Can an invariant from a quantum Lie algebra ever distinguish mutant knots?
(Maybe if it is "chiral"...whatever that means :-)
(Note that Kauffman abstract tensors/skein equations CAN distinguish ...

**5**

votes

**0**answers

116 views

### Framed singular knots

I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let $D$ be ...

**8**

votes

**0**answers

213 views

### Union of knots and its Alexander polynomial

In 1957 Kinoshita and Terasaka in their article "On unions of knots" generalized the operation of connected sum; it's called $union$.
I have several questions :
1.) If the Alexander polynomial of a ...

**7**

votes

**1**answer

542 views

### Introductory article of knot Heegaard Floer Homology

I am looking for some article that gives an introduction to Heegaard Floer homology of knot.
I heard that it is very useful to determine the unknotting number of a knot, but I couldn't find any ...

**3**

votes

**0**answers

99 views

### Equivalence (or not) of two Artin/Fox wild arcs

The repeating patterns in the wikipedia articles on wild arcs and wild knots seem to me to be not continuously deformable to each other. Is this true? For clarity, here is my diagram of the repeating ...

**15**

votes

**1**answer

499 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 them,...

**5**

votes

**3**answers

496 views

### How to show whether a given knot and its mirror image are the same or not?

The title says it all:
How can I show that a knot $K$ is distinct from its mirror image?
May be I have to try different knot invariants. Not sure, I am new in this area.