Transverse string topology and the cord algebra

TL;DR

This paper introduces a coalgebra structure for open strings transverse to framed codimension 2 submanifolds, which recovers a specialized form of the Ng cord algebra, a significant knot invariant.

Contribution

It defines a new coalgebra structure that generalizes the Ng cord algebra for knots, providing a novel algebraic perspective in string topology.

Findings

Recovers a specialization of the Ng cord algebra for knots in R^3.

Establishes a coalgebra structure for open strings transverse to framed codimension 2 submanifolds.

Shows the structure captures non-trivial knot invariants.

Abstract

We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold. When the submanifold is a knot in R^3, we show this structure recovers a specialization of the Ng cord algebra, a non-trivial knot invariant which is not determined by a number of other knot invariants.