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.