1
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1answer
82 views

Knot theory: Braids

Show by using a picture, that the two braids $\sigma_{i} \sigma_{i+1} \sigma_{i}$ and $\sigma_{i+1} \sigma_{i} \sigma_{i+1}$ are equivalent. This is 5.26 in knot book by Colin Adams. Need some ...
4
votes
1answer
42 views

What does the tensor product in the definition of Combinatorial Floer knot homology look like?

I am working on a project that involves summarizing the article A combinatorial description of knot Floer homology (http://arxiv.org/abs/math/0607691) and doing some example computations with the ...
2
votes
0answers
59 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 ...
1
vote
0answers
114 views

Conditions for a projection of a Knot to be a Knot diagram.

friends. I'm working on a problem, the broad scope of which is to show that given a map $f:S^1\rightarrow \mathbb{R}^3$ be a smooth embedding, and a projection map $\pi_v:S^2\rightarrow P_v$, where ...
2
votes
0answers
32 views

Seifert surface and crossing number

i am sitting here with the problem of Seifert Surfaces. I know from a theorem that every knot does have a Seifert surface. We can also make a so called disc-and-band surface $F$ by gluing $v$ discs ...
1
vote
1answer
114 views

A question about orthogonal projections of Smooth Embeddings of the circle.

Question: Let $f$ be a smooth embedding of $S^1\rightarrow \mathbb{R}^3$. Given an element $v\in S^2$ we have the orthogonal projection $\pi_v:\mathbb{R}^3\rightarrow P_v$ to the plane $P_v$ = the ...
2
votes
0answers
27 views

$3\tau(K_1$#$K_2)$=$\tau(K_1)\tau(K_2)$

Suppose we have two knots $K_1$ and $K_2$. Then look to the connected sum of $K_1$ and $K_2$ denoted by $K_1$#$K_2$ (defined for knots). Suppose $\tau$ is the number of $3$-colourings (definition for ...
1
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1answer
112 views

Crossing number and Torus links

We define the crossing number of a knot $K$ to be the minimal number of crossings in any diagram of $K$. Surely we can easy prove that there do not exist knots with crossing number $1$ and $2$ ...