2
votes
2answers
61 views

A comment on a proof of equivalent knots

The following theorem and its proof is from A First Course in Algebraic Topology By Czes Kosniowski pp. 219-220 ...
4
votes
0answers
169 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, ...
2
votes
1answer
90 views

Dehn and Wirtinger Presentations of Knot Groups and their connection

I'm currently working through N.D. Gilbert and T. Porter's Knots and Surfaces. In it the idea of a Wirtinger presentation and a Dehn presentation for a group associated with a given knot is ...
3
votes
0answers
76 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
2
votes
0answers
101 views

Zero exponent sum w.r.t group words in knot group's presentation

I am reading, "Plane Curves Associated to Character Varieties of 3-Manifolds" by Cooper, Culler, Gillet, Long, and Shalen and on page 28 ( http://www.math.uic.edu/~culler/papers/PlaneCurves/curves.pdf ...
14
votes
0answers
144 views

Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
3
votes
0answers
45 views

Alexanderpolynomial of connected sum via Fox calculus and Wirtinger presentation

Hello :) i have just reading the question "How to compute the Alexander polynomial of general torus knot" and i was suprised how strong it works if someone have a difficult question. I am also very ...
2
votes
0answers
58 views

Alexanderpolynomial of torus knot

i want to compute the Alexanderpolynomial of the torus knot $T_{p,q}$ with $p$ and $q$ coprime. I should work with the groups presentation $G(T_{p,q})=<x,y:x^p=y^q>$ of $T_{p,q}$. I have to use ...
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 $S^{3}$?
0
votes
1answer
104 views

Pure braid group, stabilizer

From group theory we know that a homomorphism $\phi: G \to \operatorname{Sym}(S)$, where S is a set, then $\operatorname{Sym}(S) \cong \Sigma_n $. Its kernel is given as $\bigcap_{s \in S}G_s$, which ...
10
votes
1answer
312 views

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

This question has been cross posted on MathOverflow with some very interesting answers and discussion. I'm currently writing a project on the braid groups and their analogues on closed surfaces. ...
4
votes
1answer
231 views

Computing knot/link groups

The knot group of a knot $K$ is the fundamental group of $\mathbb R^3 \smallsetminus K$; that is, the set of possibly self-crossing closed paths (starting and ending at any single point in space) ...