1
vote
2answers
115 views

Knot group epimorphism from a prime knot

The knot group of a knot $K$ is the fundamental group of the knot complement $\pi_{1} (S^{3} \backslash K )$ (sometimes denoted $\pi_{1} (K)$ ). Let $f: \pi_{1} (K) \rightarrow \pi_{1} (L) $ be a ...
1
vote
1answer
193 views

The knot group of a prime knot

The fundamental group of a knot $K$ (otherwise known as the knot group) is the fundamental group of the knot complement $S^{3} \backslash K $ in $S^{3} $. In "Virtual Knots: The State of the Art" ...
12
votes
2answers
312 views

Semidirect product decomposition of the Borromean rings group

Let $X=S^3\setminus B$ be the link complement of the Borromean rings. Then $G=\pi_1(X)$ has a presentation of the form $$ G = \langle \; a,b,c \mid [a,[b^{-1},c]],\; [b,[c^{-1},a]], \; ...
2
votes
2answers
409 views

How to compute the Alexander polynomial of general torus knot

Hello, i am very interested in knot theory, especially in knot groups and knot polynomials. Therefore i am reading the book of Crowell and Fox (Introduction to knot theory). I want to compute some ...
1
vote
1answer
354 views

Dehn presentation of a knot group

The knot group is the fundamental group of the knot complement in $S^{3} $. The Dehn presentation of the knot group is a particular group presentation obtained by looking at the regions and crossings ...
0
votes
0answers
85 views

Difference between $MCG(ext(K))$ and $MCG(S^{3} - K)$

Can someone explain the difference between the mapping class groups of the knot complement and knot exterior?
14
votes
4answers
788 views

Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?

This question was originally posted on math.SE by myself nearly a year ago. I've been thinking again about the problem after it recently received a little attention, but little progress was made in ...
1
vote
1answer
196 views

Reference for a proof of the Dehn presentation

I would like a reference for a proof that the Dehn presentation is a presentation of the fundamental group of the knot complement in $\mathbb{S}^{3} $.
11
votes
1answer
603 views

Fox differential calculus and the Alexander invariant of a link

I am teaching a course in knot theory, and I would like to describe the presentation of the Alexander module of a link obtained via Fox differential calculus. In doing this, I should prove the ...
14
votes
3answers
1k views

Is anything known about this braid group quotient?

Let $B_n$ be the braid group on $n$ strands. As is well known, if $\sigma_i$ is the operation of crossing the string in position $i$ over the string in position $i+1$, then the elements ...
5
votes
4answers
884 views

Higher-dimensional braid group?

Let $\Delta$ be 2-disk. Let $C(\Delta;n)$ be a configuration space. i.e.) $C(\Delta;n)= \lbrace (z_1,\ldots,z_n)\in \Delta\times\ldots\Delta | z_i\neq z_j ~\textrm{if}~ i\neq j \rbrace $ Then, it ...
10
votes
6answers
980 views

Computing the structure of the group completion of an abelian monoid, how hard can it be?

Cherry Kearton, Bayer-Fluckiger and others have results that say the monoid of isotopy classes of smooth oriented embeddings of $S^n$ in $S^{n+2}$ is not a free commutative monoid provided $n \geq 3$. ...