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

Thumbnail Image
Open Access Location
Journal Title
Journal ISSN
Volume Title
Massey University
The Author
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.