**4**

votes

**2**answers

96 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

65 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 ...

**17**

votes

**0**answers

147 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

169 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

51 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

101 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

160 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 ...

**0**

votes

**0**answers

81 views

### Codimension one embeddings

For smooth knots in $S^3$ "isotopy" and "ambient isotopy" are equivalent (although this is not true in the topological category). I guess that therefore also for tori in $S^3$ "isotopy" and "ambient ...

**6**

votes

**1**answer

306 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

68 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

417 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 ...

**5**

votes

**3**answers

398 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.

**4**

votes

**2**answers

215 views

### hyperbolic structure on Figure–8 knot complement

I was trying to understand the proof of the fact that there is a hyperbolic structure on Figure–8 knot complement initially from Thurston's notes and then from some online notes; but unfortunately I ...

**7**

votes

**0**answers

82 views

### Kinematics of rolling knots

It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes.
(An ...

**5**

votes

**1**answer

86 views

### Does $H_*(A^-_0(K))=\mathbb{F}[U]$ imply that $K$ is an L-space knot?

Let $K$ be a knot in the three-sphere. Let $A_s^-(K)$ be the Alexander filtrations of the knot Floer complex $CFK^{\infty}$. Would $A_0^-(K)$ has homology $\mathbb{F}[U]$ imply that K is an L-space ...

**9**

votes

**0**answers

129 views

### Doubly slice knots and an embedding of 4-manifold

It is well-known that the existence of topologically slice knot which is not smoothly slice implies the existence of exotic $\mathbb{R}^4$. For example, see the answer of K. Davis.
To prove the above ...

**2**

votes

**0**answers

77 views

### Are Markov traces matrix traces?

When starting this question I was very hesitant - literature on the subject is vast and I thought most likely the answer is already there somewhere.
Then when the list "Questions that may already ...

**18**

votes

**0**answers

448 views

### Human Knot game

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 ...

**9**

votes

**0**answers

94 views

### Kernel of “Hat to Plus” in Heegaard Floer Homology

Given a 3-manifold $M$, there is a map of Heegaard Floer groups $$f:\widehat{HF}(M) \to HF^+(M)$$
induced by the inclusion $$\widehat{CF}(M) \to CF^+(M)$$ of the respective chain complexes.
Given a ...

**4**

votes

**1**answer

132 views

### Can every large point set be connected to a given knot?

Let $K$ be a given knot, and
$P$ a set of points in $\mathbb{R}^3$ in general position,
general position in the sense that no three points are collinear
and no four coplanar.
Define the point-set ...

**2**

votes

**1**answer

179 views

### A basic question of Khovanov-Rozansky Homology

thank you for spending time on the following question.
In [1] Khovanov and Rozansky categorifized $sl_n$ version of HOMFLY Polynomial, in page 11, they mention that what they defined in [1] is ...

**1**

vote

**0**answers

146 views

### Surgery to unlink $S^p$ and $S^q$ in $S^d$

We know that $S^p$ and $S^q$ can be linked in $S^d$ if $p+q<d$. Let us consider the simplest case where both $S^p$ and $S^q$ are un-knotted spheres.
I am looking for a surgery to unlink $S^p$ and ...

**9**

votes

**2**answers

187 views

### Descriptive Complexity of Knot Equivalence

I was reading a little about knots (in a popular math book that wasn't very good) and the book put forth several knot invariants like the Alexander and Jones polynomials. But these are not complete ...

**6**

votes

**1**answer

108 views

### Tunnel number of Pretzel knots

I would like to know the tunnel number of $n$-pretzel knots. I have searched and found nothing for any $n>3$. When $n=2$, $t(K)=1$ or $2$ depending on the number of twists, which is proved in a ...

**3**

votes

**1**answer

132 views

### Hopf link from analytic geometry

I am a condensed matter physicist, and want to understand the Hopf link from analytic point of view. My question is as follows.
We have two sets of equations, and each set of equations describes a ...

**2**

votes

**2**answers

164 views

### How can i change 8_19 to (3,4)-torus knot K(3,4)?

In the knot table, it is well-known that 8_19 is (3,4)-torus knot. But, it is not clear to me. How can i change 8_19 to (3,4)-torus knot K(3,4)? Moreover, it is well-known that braids of two ...

**2**

votes

**0**answers

55 views

### Classes of knots that have known Bridge spectra

Bridge spectra is a knot invariant first defined by Doll, who established some basic properties. Tomova has shown that high distance knots have bridge spectra $(n,n-1,\ldots,2,1,0)$. Zupan has ...

**4**

votes

**3**answers

117 views

### Polynomial invariants for unoriented links

I have seen that usually one finds polynomial invariants for oriented links (for example the Jones polynomial, the Hompfly polynomial). Does anyone know what polynomial invariants exist for ...

**4**

votes

**1**answer

133 views

### arc length of a knot in the solid torus

As motivation, consider the knot in the solid torus in the first (left) picture below.
Put a metric on the torus -- for concreteness, let's assume it's induced by the standard euclidean metric on ...

**12**

votes

**4**answers

417 views

### Obtain any 3-manifold from repeating surgeries on knots in $S^3$

In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) ...

**3**

votes

**1**answer

59 views

### Blanchfield pairing: knot exterior versus $0$-framed surgery

The Blanchfield pairing is usually defined on the homology of the infinite cyclic cover over the knot exterior.
In his article "cobordism of satellite knots", Litherland works with the $0$-framed ...

**3**

votes

**1**answer

146 views

### Chern-Simons invariants of 2-bridge knots

2-bridge links $L(p/q)$ are described by the continuous fraction expansion $\frac{p}{q}=\left[a_1,a_2,\ldots,a_n\right]$, where the $a_i$ are the numbers of twists in the boxes below:
Looking at ...

**3**

votes

**2**answers

143 views

### Algorithm for Computing the Arf Invariant of a Knot

According to "The knot book", by Colin Adams, two knots are pass equivalent if they are related by a finite sequence of pass-moves. Moreover every knot is pass-equivalent to either the unknot or the ...

**7**

votes

**2**answers

199 views

### What are known examples of a 3-manifold $Y$ embedded into $Y'\times I$ where $Y'$ is another 3-manifold?

The question I have is the following:
Let $Y,Y'$ be two integer homology 3-spheres. Can we embed $Y'$ into $Y\times I$ such that $Y'$ separates the two boundary components apart?
Do we know any ...

**5**

votes

**2**answers

190 views

### 2-bridge knots in the Rolfsen's table

2-bridge knots (aka rational knots) $K(p,q)$ are described by a rational number $\frac{p}{q}$ or likewise its continued fraction expansion $\left[a_1,a_2,\ldots,a_k\right]$.
Has somebody worked out a ...

**3**

votes

**1**answer

121 views

### Why does the Gluck twist on a spun knot give the standard $S^4$?

Given a knotted arc $A \subset D^3$ (whose endpoints are, say, at $(\pm 1,0,0)$), the spun knot on this arc is $$\partial\left((D^3, A) \times D^2\right), = (\partial(D^3,A) \times D^2) \cup ((D^3,A) ...

**2**

votes

**0**answers

72 views

### Self-diffeomorphisms of fibered knot complements

A knot $K \subset S^3$ is fibered if the complement $S^3 \setminus K$ of (a small open neighborhood of) $K$ is a fiber bundle over $S^1$. (The fiber will be a surface with one boundary component.)
...

**2**

votes

**0**answers

246 views

### Is this approach to the combinatorics of knots well known?

I am teaching a course on knots for the first time, and this led me to play with an approach which I have not seen in the literature. I would be surprised if no one had used it before, so I am ...

**2**

votes

**2**answers

165 views

### Minimal genus of Seifert surface of torus knot

Let $(p,q)$ be a pair of coprime (positive) integers. Consider the torus knot $T_{p,q}$. What is the minimal genus of an (embedded) oriented Seifert surface for this knot?
It is not had to convince ...

**1**

vote

**1**answer

61 views

### link which is the closure of some power of a braid

Recently I have stumbled upon links which are closures of braids, of the form $\sigma = \tau^{n}$. Such links generalize torus links. Are there any papers studying such links? In particular I am ...

**2**

votes

**1**answer

157 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. ...

**7**

votes

**1**answer

219 views

### Can two fibered knots have the same exterior?

Suppose I have two distinct fibered knots in a homology sphere. Is it possible for them to have (orientation-preservingly) homeomorphic exteriors?
See Oriented knot complement conjecture for ...

**2**

votes

**1**answer

151 views

### Two-bridge knots and CW-complex

The fundamental group of any two-bridge knot K in $\mathbb{S}^3$ has a presentation with two generators and one relation.
On the other hand, it's possible to provide a CW-complex with only one 0-cell ...

**6**

votes

**1**answer

108 views

### What is the original reference for disorientations on tangle diagrams?

There are several invariants whose "natural" domain is a category of disoriented tangles, that is tangles which are piecewise-oriented, but which contain points called `disorientations' at which the ...

**8**

votes

**0**answers

175 views

### Alexander polynomial in branched covers

Suppose I am given a 3-manifold as a double branched cover over a link. Let a null-homologus knot in this space be given as a lift of an arc with endpoints on the link (it is automatically ...

**9**

votes

**2**answers

312 views

### How many knots are there with hyperbolic volume less than a given constant

Are there any known upper bounds on:
$$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$
? I expect this grows at least exponentially in $M$, ...

**5**

votes

**1**answer

377 views

### Are there spaces in which there are no fibered knots?

I am looking for orientable closed 3-manifolds in which there are no fibered knots. Although I know little about this, I think for links the answer to the question above is "no", and the result is ...

**4**

votes

**1**answer

106 views

### Is there a known Legendrian simple link?

Several knots like unknot, $4_1$, $3_1$ are known to be Legendrian simple, i.e., Thurston-Bennequin number and rotation number determine Legendrian type completely.
How about the same notion for link ...

**3**

votes

**1**answer

295 views

### Jones polynomial of tangles using Temperley-Lieb algbra?

The definition of the Jones polynomial of tangles (a la Reshetikhin and Turaev) uses the space of invariants for $U_q sl_2$ and R-matrices. It seems to me the same thing cane be done in terms of the ...

**2**

votes

**0**answers

127 views

### Jones Polynomial as an Equivariant Index

In Khovanov homology the Jones polynomial of a given link is computed as the graded Euler characteristic of a certain complex associated with the link. From other side the Atiyah-Singer theorem ...