User:Leo algknt

Home Work 1

Question 1.

A. Prove that the set of all 3-colourings of a knot diagram is a vector space over ${\mathbb {F} }_{3}$. Hence $\lambda (K)$ is always a power of 3

Attempt: Let $D$ be a knot diagram with $n$ crossings. There are $n$ arcs. Let $a_{1},a_{1},\ldots ,a_{n}\in \mathbb {Z} /3\mathbb {Z}$ represent the arcs. Now let $a,b,c\in \mathbb {Z} /3\mathbb {Z}$ , with

$a\wedge b=\left\{{\begin{array}{cc}a,&a=b\\c,&a\not =b\end{array}}\right.$

At each crossing we have a linear equation x

Let

B. Prove that $\lambda (K)$ is computable in polynomial time in the number of crossings of K.