Home Work 1
Question 1.
A. Prove that the set of all 3-colourings of a knot diagram is a vector space over
. Hence
is always a power of 3
Attempt: Let
be a knot diagram with
crossings. There are
arcs. Let
represent the arcs. Now let
. Define
by
so that
With the above definition, we get a linear equation
for each each of the
crossings, where
. Thus we get a system of
linear equation.
Let
B. Prove that
is computable in polynomial time in the number of crossings of K.