Homotopy properties of knots in prime manifolds

TL;DR

This paper introduces new homotopy-theoretic invariants for knots in prime 3-manifolds, extending Milnor's invariants, and explores their properties related to concordance and characteristicness.

Contribution

It defines and analyzes homotopy invariants of knots in prime 3-manifolds that generalize Milnor's invariants and are invariant under concordance and certain maps.

Findings

Invariants are preserved under concordance.

Invariants extend Milnor's invariants to broader classes of knots.

No requirement for knots to be rationally null-homologous or framed.

Abstract

We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there is a degree-one map f:M --> M throwing K onto J and mapping M-K to M-J. These invariants are invariants of concordance and of J-characteristicness when f induces the identity on the fundamental group of M, and may be viewed as extensions of Milnor's invariants. We do not require the knots considered here to be rationally null-homologous or framed.