A graph theoretic proof that Wada's type seven link invariant is determined by the double branched cover : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University, Manawatū, New Zealand
The fundamental group of a link L is a group-valued link invariant that can be defined by assigning a generator to each arc of a link diagram of L, and introducing a relation between them at each crossing. Wada studied what he called shift representations to look for other crossing relations that might define group-valued link invariants. He found seven shift representations, two of which he noted do not define group-valued link invariants. One of the seven defines an infinite family Gm of invariants that includes the fundamental group as G₁, and these have since been shown to distinguish knots up to reflection for m ≥ 2. Wada showed that three of the remaining four give no new information, leaving just his type seven invariant, which we call W₇. Sakuma showed that the seventh of Wada’s shift representations is isomorphic to the free product of Z and the fundamental group of the double branched cover of L, π₁(L˜₂), that is W₇(L) ∼= π₁(L˜₂) ∗ Z. We will use graph theoretic methods to give a new proof of Sakuma’s result.