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 ${\mathbb {Z} }_{3}$. Hence $\lambda (K)$ is always a power of 3

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

$a\wedge b=\left\{{\begin{array}{cc}a,&a=b\\c,&a\not =b\end{array}}\right.$ so that $a\wedge b+a+b\cong 0\mod 3$

With the above definition, we get a linear equation $a_{i_{1}}+a_{i_{2}}+a_{i_{3}}$ for each each of the $n$ crossings, where $i_{1},i_{2},i_{3}\in {1,2,\ldots ,n}$. Thus we get a system of $n$ linear equation.

Let

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