**11**

votes

**4**answers

333 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

**2**answers

73 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 ...

**3**

votes

**1**answer

116 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 ...

**2**

votes

**1**answer

80 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 ...

**3**

votes

**1**answer

49 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 ...

**5**

votes

**2**answers

154 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

**2**answers

112 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

169 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 ...

**1**

vote

**1**answer

81 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) ...

**8**

votes

**2**answers

450 views

### Is more alternating always better?

While thinking about this interesting question asked by Dylan Thurston, it occured to me that in every case that I know, the closer a knot diagram is to being alternating, the better its properties. ...

**24**

votes

**1**answer

314 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 ...

**7**

votes

**1**answer

226 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 ...

**2**

votes

**0**answers

55 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.)
...

**9**

votes

**2**answers

346 views

### $6j$-symbols for $U_q({\mathfrak{sl}}_n)$ and colored HOMFLY polynomials

Explicit expression of quantum $6j$-symbolos for $U_q({\mathfrak{sl}_2})$ have been known due to the work of Kirillov and Reshitikhin.
My Question:
How much are known about quantum $6j$-symbolos ...

**3**

votes

**1**answer

235 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 ...

**81**

votes

**10**answers

8k 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 ...

**2**

votes

**0**answers

221 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

125 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 ...

**4**

votes

**3**answers

257 views

### slice knots, what does the locally flat condition say?

I am studying slice knots, so for example they say the cone on a trefoil knot can be embedded
in D^4 but it is not locally flat at the vertex of the cone. What I do not understand that I think every ...

**1**

vote

**1**answer

53 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

127 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

201 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

136 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 ...

**7**

votes

**0**answers

154 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 ...

**3**

votes

**1**answer

260 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 ...

**6**

votes

**1**answer

98 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 ...

**9**

votes

**2**answers

258 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 $M$, ...

**11**

votes

**3**answers

909 views

### Is there a procedure for obtaining all knots in S^3?

(Just to be precise, in this question, the word "knot" means "ambient isotopy class of a (EDIT: smooth) knot in $S^3$.") A knot in $S^3$ is called prime if it is not the connected sum of two other ...

**5**

votes

**1**answer

364 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

95 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 ...

**8**

votes

**2**answers

411 views

### Number of the Reidemeister moves needed to transform one diagram into another one

A recent question Random Reidemeister moves to unknot contains a link to the paper http://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-7/S0894-0347-01-00358-7.pdf, in which J. Hass and J. ...

**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 ...

**2**

votes

**0**answers

119 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 ...

**1**

vote

**1**answer

197 views

### Does knot Floer homology detect knot genus in rational homology spheres?

My question is the following:
Does knot Floer homology detect the genus of null-homologous knot in rational homology spheres?
If the answer is yes, I would like to have a reference for the ...

**6**

votes

**2**answers

454 views

### Reference for a fact (?) on homeomorphic knot complements

Does somebody have a reference (or an argument why it should be true) for the following statement?
“Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 ...

**8**

votes

**0**answers

133 views

### What fraction of knots are ribbon knots?

Pardon the naive question.
Perhaps there are several different variants of my question, but
essentially I am seeking to know whether ribbon knots are
vanishingly rare among all knots, or a positive ...

**2**

votes

**0**answers

65 views

### Mutant pairs distinguished by [2,1]-colored HOMFLY polynomials

Morton and Cromwell showed that the famous mutant pair, Kinoshita-Terasaka and Conway knots, can be distinguished by HOMFLY polynomials colored by [2,1] Young diagram. Are there any other mutant pairs ...

**6**

votes

**0**answers

125 views

### Stable, tame A-infinity isomorphism?

Linear duality provides a correspondence between differentials on free tensor algebras and A-infinity algebras. Under this correspondence, what is the A-infinity counterpart of "stable, tame ...

**9**

votes

**3**answers

377 views

### Unknotting number of knot diagrams

Define the "diagram unknotting number" of a knot diagram $D$ as the minimal number of crossings that need to be changed in $D$ in order to get a diagram of the trivial knot (the usual unknotting ...

**3**

votes

**1**answer

269 views

### Unknotting number and crossing number

It is well known that if $c(K)=2n+1$, then $u(K)$ is less than $n+1$. It can not be sharper because of the trefoil knot. On the other hand, if $c(K)=2n$, then similarly we have $u(K)$ is less than ...

**12**

votes

**1**answer

240 views

### Can infinitely many alternating knots have the same Alexander polynomial?

There exist many constructions of infinite families of knots with the same Alexander polynomial. However, alternating knots seem very special. While there are also many result on restricting the form ...

**6**

votes

**2**answers

300 views

### Vassilliev invariants of knots and their cables

The following is perhaps a standard question, but i could not find a plain enough answer
by simply searching online.
Q: Given a knot $K$ and its $(p,q)$-cable $K_{p,q}$ what is a relation
between ...

**6**

votes

**1**answer

348 views

### Proving that the Jones polynomial is q-holonomic

The Jones polynomial is known to have many different interpretations or definitions, by now. There are connections with QFT, quantum groups, Hilbert schemes, Cherednik algebras, etc.
My question is ...

**1**

vote

**0**answers

70 views

### Ozsvath-Szabo orientation convention for Seifert fibred spaces

I am confused by the orientation convention that Ozsvath and Szabo use in On Heegaard Floer homology and Seifert Fibered Surgeries and would appreciate if someone clarifies this for me. On page 15 ...

**2**

votes

**2**answers

254 views

### Gauss Codes that produce classical knots as opposed to virtual knots

I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that
If $K$ is a virtual knot whose underlying Gauss ...

**1**

vote

**1**answer

50 views

### Question about general torus knot lengths [closed]

Can anyone send me a reference about calculating the length of a thin string wound on a p,q torus with major radius R and minor radius r in terms of p, q, R and r? This may appear trivial but I have ...

**8**

votes

**1**answer

247 views

### Does the limit in the Volume conjecture converge?

The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...

**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 ...

**6**

votes

**6**answers

319 views

### Reference on representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations).
"Knots" by Burde and Zieschang discusses some material but it is not entirely ...

**9**

votes

**2**answers

941 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 ...