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 [math]\displaystyle{ {\mathbb F}_3 }[/math]. Hence [math]\displaystyle{ \lambda(K) }[/math] is always a power of 3

Attempt: Let [math]\displaystyle{ D }[/math] be a knot diagram with [math]\displaystyle{ n }[/math] crossings. There are [math]\displaystyle{ n }[/math] arcs. Let [math]\displaystyle{ a_1, a_1, \ldots, a_n \in \mathbb{Z}/3\mathbb{Z} }[/math] represent the arcs. Now let [math]\displaystyle{ a,b,c \in \mathbb{Z}/3\mathbb{Z} }[/math] , with

[math]\displaystyle{ a\wedge b = \left\{ \begin{array}{cc} a, & a = b\\ c, & a\not= b \end{array} \right. }[/math]

At each crossing we have a linear equation x

Let

B. Prove that [math]\displaystyle{ \lambda(K) }[/math] is computable in polynomial time in the number of crossings of K.