Moduli-space structure of knots with intersections

TL;DR

The paper shows that knots with intersections possess moduli spaces of d-knot classes under diffeomorphisms, in contrast to ordinary knots. By analyzing the local matching problem between two intersecting strands via Taylor expansions and a gauge structure, it derives a finite-dimensional description for the moduli at a nondegenerate intersection of valence $N$, yielding a lower bound on the moduli dimension $d(N)$ that depends on $N$. It demonstrates that for $N\le 4$ there are no moduli, while for $N=5$ there are at least two continuous moduli, and it provides explicit invariants $\lambda_1,\lambda_2$ illustrating the two-parameter moduli. The results generalize to a space ${\cal K}_d$ that decomposes into fibers over the countable ${\cal K}_c$, with moduli factors associated to each intersection, implying a richly structured, finite-dimensional moduli space for intersecting knots with diffeomorphism equivalence.

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.