Divisibility of the Conway polynomial of links
Lampazzi, Amy M. Noel
Cochran, Tim D.
Doctor of Philosophy
The Conway polynomial ∇K = c0 + c1z + c2z2...of a link K is an invariant of links. In this paper we extend a theorem of J. Levine  regarding divisibility of the Conway polynomial by monomials of the form zi. Three different definitions of finite type invariants of links are presented, including a definition of surgery finite type. When the coefficients ci are considered as finite type invariants, the type of these invariants is closely related to the degree of the monomials which can be factored out of ∇K. With this in mind, we prove an extension of a conjecture by T. Cochran and P. Melvin  concerning the divisibility of an alternating sum sum S<L ∇K(Sigma S) of Conway polynomials of an algebraically split link K in various surgered spheres. This result was also proved for a more general case in which the link K is not necessarily algebraically split. Finally, corollaries relate these theorems to the type of the coefficients ci, considered as finite type invariants.