3
votes
0answers
43 views

Actual Definition of the term: Hopf Band?

Sorry if this is too trivial: I need an actual working definition of the term: Hopf band. I see references to it in many searches, but never an actual precise definition. All I know so far is that ...
3
votes
2answers
50 views

More knots as crossing number increases?

Reading knot tables, it seems that as $n$ increases, more prime knots have crossing number $n$. Is this a proven fact? More precisely, If $k(n)$ is the number of knots with crossing number $n$, is ...
0
votes
0answers
19 views

Subsets of immersions that are embeddings

Let $X$ be a manifold and let $Y$ be a submanifold, possibly with boundary. I am dealing with a situation where $f:X\to \mathbb{R^3}$ is an immersion, but $f\vert_Y:Y\to \mathbb{R}^3$ is an embedding. ...
4
votes
0answers
170 views

Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.

A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) /b=(a/b)*b=a$ If we have a group $(G, \cdot, ...
0
votes
0answers
21 views

branched cover along a closed curve in the $3$-sphere

Let $c$ be a closed embedded smooth curve in the $3$-sphere $\mathbb S^3$. I was told that $\mathbb S^3$ admits a two fold branched cover $X(c)$, branched along $c$, which corresponds to the ...
4
votes
0answers
40 views

Name of a link invariant?

Below I will describe a link invariant, denoted by me as $inv(L)$. Has anyone encountered this invariant in the literature? If so, what is its name? Also, any references to papers or books that ...
4
votes
2answers
82 views

Recommended books on knot invariants

I've been reading the books "An introduction to knot theory" by Lickorish and "Knots, Links, Braids and 3-Manifolds" by Prosolov and Sossinsky, and while both seem to me as good books, sometimes I'd ...
4
votes
2answers
97 views

Burau matrix of braid

What is the definition of a Burau matrix of a braid? Where can I find a definition?
1
vote
1answer
70 views

Braid invariants resource

What are some braid invariants (analogous to the idea of knot invariants) or a resource where I can find them?
5
votes
1answer
280 views

An introduction to Khovanov homology, Heegaard-Floer homology

I am interested in knot theory and low dimensional topology. I would like to start studying Khovanov homology and Heegaard-Floer homology. I (partially) read the original paper of Khovanov and then ...
1
vote
1answer
46 views

Are there any combinatorial studies of Kirby calculus?

All of the other diagrammatic calculi I know of can be utilised with basically just combinatorial knowledge - for instance calculating knot and link polynomials. Are there similar combinatorial ...
1
vote
0answers
17 views

Skein trees resource

Is there a resource where a collection of Skein trees for the Conway polynomials of knots have been presented? Thanks
2
votes
1answer
153 views

Dehn presentation proof reference request

Can someone give me a reference for a proof that the Dehn presentation of a knot group gives us the fundamental group of the knot complement in $S^{3}$?
7
votes
4answers
262 views

Reference for an unknotting move

Consider the following move on diagrams. I dimly recall hearing or reading that a sequence of such moves is sufficient to unknot any knot but I don't recall where I saw this. The strands in the ...
10
votes
8answers
955 views

A good quick introduction to Knot Theory?

Is there a good quick introduction to knot theory? I am relatively mathematically savvy so any level is appreciated.