Moduli-space structure of knots with intersections

TL;DR

This paper explores the moduli-space structure of knots with intersections, revealing their uncountable nature and characterizing their dimensions, which depend on intersection valence, with implications for physical applications.

Contribution

It introduces the concept of moduli spaces for knots with intersections, providing a characterization and dimension bounds, advancing understanding beyond classical knot theory.

Findings

Knots with intersections form uncountable moduli spaces.

Derived a lower bound on the dimension of these moduli spaces.

Conjecture that the lower bound is sharp for non-degenerate components.

Abstract

It is well known that knots are countable in ordinary knot theory. Recently, knots {\it with intersections} have raised a certain interest, and have been found to have physical applications. We point out that such knots --equivalence classes of loops in $R^3$ under diffeomorphisms-- are not countable; rather, they exhibit a moduli-space structure. We characterize these spaces of moduli and study their dimension. We derive a lower bound (which we conjecture being actually attained) on the dimension of the (non-degenerate components) of the moduli spaces, as a function of the valence of the intersection.