Link-homotopy Classes Distinguished by Pre-peripheral Structures
by
James R. Hughes
Elizabethtown College

An open problem in link-homotopy of links in the three-sphere is classification using some version of peripheral structures on reduced groups of links. We seek cases where a particularly straightforward version, called pre-peripheral structures, achieves classification, and also conditions under which counterexamples may occur. By formulating pre-peripheral structures in terms of string links (or pure braids), we can capture the relationship between link-homotopy equivalence and equivalence of pre-peripheral structures in a commutative diagram. The classification question then reduces to whether a certain group homomorphism is injective. For appropriately chosen collections of links, the groups are free abelian, so the question further reduces whether a certain matrix (whose entries are polynomials in generalized linking numbers) has non-trivial nullspace. To find counterexamples, then, we must find integer solutions to systems of polynomial equations derived from these matrices.

Date received: January 31, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaa-50.