Thick knots and ropelength-filtered knot spaces

Abstract

Ropelength is usually studied as a minimization problem for individual knot types. Here we instead use it to filter spaces of knot realizations. For a knot type $K$ and a budget $Λ$, let $Y_Λ(K)$ be the space of unit-thickness $C^{1,1}$ configurations of length at most $Λ$, modulo rigid motions and constant-speed reparametrization. We give a self-contained introduction to these filtered spaces and clarify their relation to physical knot theory and spaces of knots. We prove compact capture for smooth knot families and consequent homotopical and homological exhaustion of the ordinary knot space by finite ropelength levels. In the normalized Euclidean model, each $Y_Λ(K)$ is compact, connected merge levels are attained, and the resulting merge levels define an ultrametric on the components of the ideal stratum. A key elementary result is that $Y_L(K)$ strongly deformation retracts onto its exact-length shell. Thus path connectivity in $Y_L(K)$, unlike mere connectedness, is exactly the fixed-length physical isotopy problem. This leads naturally to distinct connected and path merge scales and gives a precise framework for Gordian phenomena. For higher homotopy we introduce ropelength widths, show that isometry-orbit classes have zero excess width, and relate stable classes to Hatcher's models of knot spaces. We also include mirror symmetry, a certified BFACF mirror-merge computation as a discrete benchmark, worked examples, an annotated literature guide, and graded open problems.