Swept-Area Pseudometrics on Ropelength-Filtered Knot Spaces

Abstract: We introduce swept-area pseudometrics on ropelength-filtered spaces of knot representatives. For a knot type (K) and a ropelength level (Λ), admissible isotopies are required to pass through curves of thickness at least one and length at most (Λ). The swept area is the parametrized area traced by the moving curve, and its infimum over admissible isotopies defines an extended pseudometric on each admissible component. We also define the admissible fundamental group of a based admissible component and equip it with a swept-area length function. The construction is separated from the rigidity questions it raises. The zero-distance quotient is always a metric space, while non-degeneracy before quotienting is treated separately. We prove non-degeneracy on uniformly non-collinear finite-dimensional polygonal strata. We also prove calibration lower bounds from projected signed area, including a rotation-invariant supremum over oriented planes, and use them to obtain exact distance formulas for concentric round unknots and homothetic planar ellipses. We further prove rigidity of the ideal unknot. The framework is related to static scale-free invariants such as density and compression radius, and to filtered-topological structures such as ideal strata and merge scales. We define swept-area weighted lifted Reidemeister graphs and prove that, for diagrammatically generic isotopies, the associated diagrammatic distance is bounded above by the geometric swept-area distance. We also record monotonicity in the ropelength parameter and formulate problems toward full non-degeneracy and approximation theory.

• The paper introduces an extended pseudometric based on the minimum area swept by thickness-preserving, ropelength-bounded isotopies, with a genuine metric obtained after quotienting zero-distance configurations.
• The paper uses projected-area calibrations to derive sharp formulas such as d(C₁,C_R)=π(R²−1) for concentric round unknots and proves that the ideal unknot at ropelength 2π is uniquely round.
• The paper establishes non-degeneracy on uniformly non-collinear fixed-edge polygonal strata and a one-sided comparison with weighted Reidemeister graphs, while leaving compactness, minimizer existence, and full smooth-category rigidity open.

Overview

This paper introduces a quantitative layer on top of ropelength-filtered knot spaces. For a knot type $K$ and a ropelength level $\Lambda$, the author considers admissible isotopies through curves of thickness at least one and length at most $\Lambda$, and measures each isotopy by its swept area — the parametrized area of the trace surface. The infimum of swept areas over admissible isotopies defines an extended pseudometric $d_\Lambda^K$ on each admissible component of $Y_\Lambda(K)$, and a parallel construction assigns a swept-area length function to an "admissible fundamental group" of based admissible loops. The paper is explicit about separating formal constructions from proved estimates, conditional principles, and conjectures (2605.05557).

The framework rests on three layers: static geometry (invariants such as density $\rho_D = Len/D$ and compression radius $CRad_D = D/Thi$), filtered topology (the sublevel spaces $Y_\Lambda(K)$, their components, ideal strata, and merge scales), and dynamic geometry (path costs). The factorization $Rop(\gamma) = \rho_D(\gamma)\, CRad_D(\gamma)$ is recorded as the only static identity used.

The swept-area pseudometric

For a Lipschitz isotopy $\Gamma: S^1 \times [0,1] \to \mathbb{R}^3$ with embedded $\Lambda$0 time slices satisfying $\Lambda$1 and $\Lambda$2, the swept area is

$\Lambda$3

well-defined via Federer's area formula; trace self-intersections are counted with multiplicity, which is exactly what makes calibration arguments work. The cost $\Lambda$4 is $\Lambda$5 across distinct admissible components. Non-negativity, time-reversal symmetry, additivity under concatenation, and Euclidean invariance yield an extended pseudometric on each component. Two structural facts are recorded: the zero-distance quotient is always a genuine metric space (proved formally), while non-degeneracy before quotienting — "area-rigidity" — is treated as a separate problem. The author states a compactness-and-rigidity principle that would imply area-rigidity in the full $\Lambda$6 setting but does not prove it there. Monotonicity $\Lambda$7 for $\Lambda$8 follows immediately from the larger admissible class.

Admissible fundamental groups

Restricting to closed admissible isotopies based at a fixed representative in the unquotiented space yields a group $\Lambda$9 under concatenation, with time reversal as inverse; the proof handles associativity and the inverse law via explicit reparametrizations that preserve admissibility. Equipping this group with $\Lambda$0 gives a swept-area length group satisfying $\Lambda$1, symmetry under inversion, and subadditivity. The paper notes this object is distinct from classical knot concordance: it depends on $\Lambda$2 and the thickness constraint, and is attached to a component rather than a knot type. Stable lengths, spectra, and filling functions are explicitly deferred.

Calibrations and exact computations

The main proved bridge between static and dynamic quantities is a calibration argument via projected signed area. Since $\Lambda$3 has comass one, Stokes' theorem gives $\Lambda$4 for every admissible isotopy, hence

$\Lambda$5

with the supremum over oriented planes. This yields positive distance whenever projected signed areas differ.

Two exact computations follow. For concentric round unknots $\Lambda$6 with $\Lambda$7:

$\Lambda$8

with the radial expansion giving the upper bound and the $\Lambda$9-calibration the matching lower bound. The analogous formula $d_\Lambda^K$0 holds for homothetic ellipses under the explicit admissibility hypothesis $d_\Lambda^K$1 (so thickness stays at least one along the homothety); the paper notes this hypothesis is sufficient rather than necessary. Finally, rigidity of the ideal unknot is proved: combining Fenchel's theorem with the curvature bound from thickness forces any element of $d_\Lambda^K$2 to be the unit round circle, so $d_\Lambda^K$3 is a single point. A crude displacement estimate $d_\Lambda^K$4 provides a general upper bound.

Merge costs and polygonal non-degeneracy

For multi-component ideal strata, the merge scale $d_\Lambda^K$5 records when two components become connectable; the swept-area merge cost $d_\Lambda^K$6 then quantifies how much area the cheapest connection requires, and is monotone nonincreasing in $d_\Lambda^K$7. The relaxed quantity $d_\Lambda^K$8 need not agree with the immediate right-hand limit at the merge scale — a distinction the paper makes deliberately.

In the polygonal category, with fixed labelled $d_\Lambda^K$9-edge polygons and Rawdon's thickness, the key result is non-degeneracy on uniformly non-collinear strata (edge lengths $Y_\Lambda(K)$0, exterior angles in $Y_\Lambda(K)$1): the infinitesimal swept-area functional dominates a Euclidean norm on the tangent space of a compact gauge slice, with constant $Y_\Lambda(K)$2, so the restricted pseudometric is a genuine metric. This is a fixed-$Y_\Lambda(K)$3 statement; variable-$Y_\Lambda(K)$4 models reintroduce zero-area degeneracies and are better treated via the metric quotient. Convergence of $Y_\Lambda(K)$5 to $Y_\Lambda(K)$6 as $Y_\Lambda(K)$7 is left open.

Weighted Reidemeister graphs

The diagrammatic counterpart assigns to each lifted one-move transition in a ropelength-filtered lifted Reidemeister graph the weight $Y_\Lambda(K)$8 over admissible realizations. The main theorem here is one-sided: for diagrammatically generic admissible isotopies,

$Y_\Lambda(K)$9

because only Reidemeister-time intervals contribute edges while generic intervals carry geometric cost but no diagrammatic change. The graph therefore can underestimate geometric distance; equality would require restricting the isotopy class and is open. An R1 example shows the projected-area bound $\rho_D = Len/D$0 for creating a round planar loop is sharp when the radial contraction is admissible. Dependence of the weighted graph on the projection direction $\rho_D = Len/D$1 is explicitly not claimed to vanish.

Limitations and open problems

The paper is candid about what remains unproved. Non-degeneracy of $\rho_D = Len/D$2 before the zero-distance quotient is conjectured for compact $\rho_D = Len/D$3 classes but established only in the polygonal stratum. Existence of minimizing admissible isotopies is open. Area-Lipschitz control for diameter, enclosing radius, mean-chord size, density, or compression radius — which would convert static variation into distance lower bounds — is not claimed; the diameter-variation conjecture proposes $\rho_D = Len/D$4 with $\rho_D = Len/D$5 as a first candidate. Total curvature variation cannot be controlled by swept area without extra hypotheses, since swept area controls the trace but not tangent variation. Behavior under connected sum, satellite operations, and cabling, and the use of weighted Reidemeister graphs for finite recognition, are posed as questions.

Conclusion

The paper delivers a coherent metric framework for ropelength-filtered deformation theory: a formally clean extended pseudometric with a canonical metric quotient, a group-theoretic loop-cost invariant, sharp calibrated computations for round and elliptical unknots, rigidity of the ideal unknot, finite-dimensional polygonal non-degeneracy, and a diagrammatic weighted-graph shadow with a proved one-sided comparison. Its value lies as much in the careful separation of proved estimates from conditional principles and conjectures as in the results themselves; the analytic program it identifies — compactness, zero-area rigidity, and approximation — constitutes the remaining gap between the pseudometric and a full metric theory of thick knot deformations.