Concordance

In this two-part video, we will introduce you to the world of knot concordance. Part 1 discusses definitions for sliceness and concordance, the concordance group, and three classical invariants (signature, Alexander polynomial, and Alexander modules). Part 2 discusses modern approaches to studying concordance.

Milnor's invariants for knots and links in closed orientable 3-manifolds

Abstract: In his 1957 paper, John Milnor introduced a collection of invariants for links in $S^3$ detecting higher-order linking phenomena by studying lower central quotients of link groups and comparing them to those of the unlink. These invariants, now known as Milnor's $\overline{\mu}$-invariants, were later shown to be topological link concordance invariants and have since inspired decades of consequential research. Milnor's invariants have many interpretations, and there have been numerous attempts to extend them to other settings. In this two-part talk, we discuss an extension of Milnor's invariants to topological concordance invariants of knots and links in general closed orientable 3-manifolds. These invariants unify and generalize all previous versions of Milnor's invariants, including Milnor's original invariants for links in $S^3$. The first part of the talk focuses on background information, including interesting complications that arise when studying concordance outside the setting of knots in $S^3$. The second half centers around the definitions of the new invariants and some of their most important properties.

A higher-order Fox-Milnor Theorem

Abstract:In a celebrated theorem, Fox and Milnor showed the Alexander polynomial of a slice knot factors as $f(t)f(t^{-1})$. We introduce a family of Alexander modules $\mathcal{A}_n^\mathcal{P}$ that peer more deeply into the knot group than the classical Alexander module. Our main result is a new sequence of Fox-Milnor Theorems for (n+1)-solvable knots.