Distinguishing link-homotopy classes by pre-peripheral structures

James R. Hughes, Elizabethtown College


An open problem in link-homotopy of links in S is classification using peripheral invariants, analogous to that of Waldhausen for links up to ambient isotopy. An approach to such a classification was outlined by Levine, but shown not to be feasible by the author. Here, we develop an approach to finding classification counterexamples. The approach requires non-injectivity of a group homomorphism that is completely determined by minimal-weight commutator numbers (equivalent to the first non-vanishing μ̄ invariants of Milnor). For non-injectivity, the minimal-weight commutator numbers must all be non-zero, and satisfy a certain system of polynomials, which we compute for 4- and 5-component links. 3