Finite type link homotopy invariants of k-trivial links
Journal of Knot Theory and its Ramifications
In a recent paper, Xiao-Song Lin gave an example of a finite type invariant of links up to link homotopy that is not simply a polynomial in the pairwise linking numbers. Here we present a reformulation of the problem of finding such polynomials using the primary geometric obstruction homomorphism, previously used to study realizability of link group automorphisms by link homotopies. Using this reformulation, we generalize Lin's results to k-trivial links (links that become homotopically trivial when any k components are deleted). Our approach also gives a method for finding torsion finite type link homotopy invariants within "linking classes," generalizing an idea explored earlier in  and , and yielding torsion invariants within linking classes that are different from Milnor's invariants in their original indeterminacy.
Hughes, James R., "Finite type link homotopy invariants of k-trivial links" (2003). Faculty Publications. 1461.