# Questions tagged [knot-theory]

Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.

633
questions

**2**

votes

**0**answers

149 views

### Tracking down an elusive book

A few weeks ago I had a very engaging talk with a faculty member, where he told me lots of interesting things about quantum algebras, know theory and Reshetikhin-Turaev invariants (this field is not ...

**2**

votes

**1**answer

129 views

### Distinguishing Square Knot and Granny Knot using Quandles

It is known that the square knot and the granny knot are nonequivalent although they have isomorphic fundamental groups.
I want to write a work on knot theory and provide these knots as an example ...

**4**

votes

**0**answers

149 views

### Reference request: A knot is tame if and only if it has a tubular neighbourhood

Definitions:
A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal).
Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...

**5**

votes

**0**answers

193 views

### Why does the longitude correspond to Frobenius in Arithmetic Topology, and other strange phenomena

I am trying to adress Morishita's book Knots and Primes to discover a bit about Arithmetic Topology, but some analogies puzzle me. I know that the correspondence should be addressed with a grain of ...

**4**

votes

**0**answers

65 views

### Finding a presentation matrix with low dimension

Let $R=\mathbb Z[t^{\pm}]$ and $M$ a finitely generated $R$-module. With $A$ a presentation matrix, i.e we have the following exact sequence (usually I'm working with the case where $A$ is an square ...

**4**

votes

**0**answers

132 views

### Are Turaev-Viro invariants holonomic?

Consider a 3-manifold $M$ with a boundary, which is a genus $g\geq 1$ surface $\Sigma$. Fix a triangulation $T$ of $\Sigma$. Then Turaev-Viro invariants $TV_q(M)$ are functions, assigning to integer ...

**10**

votes

**1**answer

91 views

### What are the possible linking matrices of a quasi-positive link?

I was surprised recently to come across a 3-component link where the linking number of two of the components was negative. For a while I thought I had made a mistake, then I thought a little more and ...

**3**

votes

**0**answers

109 views

### The bridge index and crookedness of a knot

I am reading Dale Rolfsen's book KNOTS AND LINKS, at page 115, I can't figure out why the crookedness of a knot equals its bridge index. Please give me some hints or any references available, much ...

**5**

votes

**0**answers

147 views

### What are these 3-manifolds from surgery?

I know that surgery on the unlink with +0 slope gives $S^2 \times S^1$ (where all the links above are embedded in $S^3$). I figured (I think) that surgery on the hopf link (with +0 on both) returns $S^...

**2**

votes

**2**answers

133 views

### Does a knot and a tunnel exterior having free fundamental group imply it's an unknotting tunnel?

The title is just about it. Assume we have a nontrivial knot $K$ in $S^3$ and the exterior of $K$, $E(K)$, is $S^3 \setminus N(K)$. Here $N(K)$ is a regular neighborhood.
Let $\tau$ be a properly ...

**3**

votes

**1**answer

211 views

### Sliceness of knots

For a subring $R? \mathbb Q$, a knot $K?S^3$ is called $R$-slice if there exists an embedded disk $D$ in an $R$-homology $4$-ball $B$ such that $?(B,D) = (S^3,K)$, see [Definition 1.3, KW16]. We say $...

**3**

votes

**1**answer

167 views

### Racks with “trichotomy”

(This is a follow-up question; the original question was about shelves.)
A rack $(R, \rhd, \lhd)$ is a set $R$ with two binary operations $\rhd$ and $\lhd$ such that for all $x, y, z \in R$:
$x \rhd ...

**3**

votes

**1**answer

326 views

### Shelves with “trichotomy”

A left shelf $(S, \rhd)$ is a magma with the left self-distributive law:
$$
\forall x, y, z \in S: x \rhd (y \rhd z) = (x \rhd y) \rhd (x \rhd z).
$$
Shelves are generalization of racks and quandles ...

**0**

votes

**1**answer

66 views

### Genus of the surface traced out by a knot

I am no good at visualizing things, and to add to the misery, have only a passing acquaintance with knot theory, so at the risk of sounding silly I dare ask in loose terms: if I take a knot $K$ in $\...

**4**

votes

**2**answers

152 views

### Chirality and Anti-Chirality of links in 3 and in 5 dimensions

We know there is a chiral knot which is a knot that is not equivalent to its mirror image. It is well known in the mathematical field of knot theory:
https://en.wikipedia.org/wiki/Chiral_knot
My ...