Ropelength-Filtered Swept-Area Geometry

Abstract

This paper studies the swept-area cost of isotopies between thick knot representatives when the isotopy is required to stay inside a ropelength window: every intermediate curve has thickness at least one and length at most Λ. Without these constraints the swept area is the classical homotopy-area length on spaces of curves, and the induced distance on unparametrized curves is bounded below by the flat norm; see Yezzi--Mennucci and Michor--Mumford. We record the corresponding non-degeneracy on the ropelength-filtered moduli space, taken modulo orientation-preserving reparametrizations and Euclidean isometries, as a consequence of this classical lower bound. The ropelength window changes the theory in two ways. Distances are infinite between classes that are not yet connected at the level Λ, and all costs depend monotonically on the budget. We organize this dependence through budget--cost profiles and swept-area merge costs of admissible components. We prove their monotonicity and a transport estimate under whole-path simulations, and relate them to the merge scales of ropelength-filtered knot spaces. On the diagrammatic side, we construct a network with exact spatial endpoints whose path cost equals the infimal cost over diagrammatically generic isotopies, and show that the graph obtained by collapsing projection fibres gives only a lower bound, which can lose positive cost inside a fibre. We also give projected-area calibrations, exact formulas for concentric round and homothetic elliptical unknots, in which the window is not active, a labelled polygonal estimate, and a based loop-length structure on admissible fundamental groups. Existence of minimizing isotopies under the thickness and length constraints is left open.

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BibTeX RIS

BibTeX

@misc{ozawa2026ropelengthfiltered,
  title = {{Ropelength-Filtered Swept-Area Geometry}},
  author = {Makoto Ozawa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.05557},
  eprint = {2605.05557},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.05557},
}

RIS

TY  - UNPB
TI  - Ropelength-Filtered Swept-Area Geometry
AU  - Makoto Ozawa
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2605.05557
UR  - https://arxiv.org/abs/2605.05557
AB  - This paper studies the swept-area cost of isotopies between thick knot representatives when the isotopy is required to stay inside a ropelength window: every intermediate curve has thickness at least one and length at most Λ. Without these constraints the swept area is the classical homotopy-area length on spaces of curves, and the induced distance on unparametrized curves is bounded below by the flat norm; see Yezzi--Mennucci and Michor--Mumford. We record the corresponding non-degeneracy on the ropelength-filtered moduli space, taken modulo orientation-preserving reparametrizations and Euclidean isometries, as a consequence of this classical lower bound. The ropelength window changes the theory in two ways. Distances are infinite between classes that are not yet connected at the level Λ, and all costs depend monotonically on the budget. We organize this dependence through budget--cost profiles and swept-area merge costs of admissible components. We prove their monotonicity and a transport estimate under whole-path simulations, and relate them to the merge scales of ropelength-filtered knot spaces. On the diagrammatic side, we construct a network with exact spatial endpoints whose path cost equals the infimal cost over diagrammatically generic isotopies, and show that the graph obtained by collapsing projection fibres gives only a lower bound, which can lose positive cost inside a fibre. We also give projected-area calibrations, exact formulas for concentric round and homothetic elliptical unknots, in which the window is not active, a labelled polygonal estimate, and a based loop-length structure on admissible fundamental groups. Existence of minimizing isotopies under the thickness and length constraints is left open.
ER  -

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