Cubic skein modules and Burnside groups

My thesis studies 2-tangle moves on links and their relationship to Burnside groups. Using a new invariant of links introduced in [D-P], the p th Burnside group of a link in S3, I show (in Chapters 2 and 3) that some long standing conjectures in classical Knot Theory, known as the Montesinos-Nakanishi 3-move conjecture (1981) and Nakanishi-Harikae (2,2)-move conjecture (1992), can be settled in the negative. In addition, the thesis provides, in Chapter 3, negative answers to Kawauchi's question (1985) that is related to the Nakanishi 4-move conjecture and, J. H. Przytycki's, question (2001) about unknotting property of rational pq -moves. In Chapter 3, I also discuss another family of moves on links, called D4k2s+1 - and D2k2s -moves, and prove that the pth Burnside group of a link (for p ≥ 1) is preserved by those moves. Finally, in Chapter 4, I describe some limitations of the Burnside groups: In particular, I offer an explanation as to why the Burnside group invariant of a link fails to answer the oldest problem, Nakanishi's 4-move conjecture (1979).