Linking Is the Latch: reading the integer on a linked filament, and why a bare filament sets no floor

Closure lets a loop count; linking lets it lock. Two rings threaded through each other carry an integer linking number that no smooth motion can change. This paper builds an instrument that reads that integer directly from a curve. The reader is the discrete Gauss linking integral, which is Ampère's law read backwards. It converges onto the exact integer as the loop is refined (a Hopf link reads −1.00066 at 100 points and −1.00001 at 800), and it holds through every deformation the model makes.

Version 2 withdraws the energy-floor measurement of version 1. The energy of linked filament pairs did rise with the linking number, and every number reproduces. But the rise was the rope length the torus construction lays down. At equal rope length the starting energy falls as the linking rises (9.36 to 6.94 from linking one to six). A pair linked six times, shrunk to half size, sits at 6.81, below the unlinked reference at 8.83, with its linking unchanged. The filament energy used here, a unit current along a curve with a fixed core, sets no floor.

The floors in the literature live in a tube of conserved flux and volume (Arnold; Moffatt; Freedman and He 1991). The floor Freedman and He prove on the ordinary energy grows in proportion to the asymptotic crossing number; their "three-halves" names an energy, not a growth rate. Version 1's three-halves comparison is corrected accordingly. The next instrument carries the same reader into a tube model with thickness. The reconnection wall is named, and the helicity bookkeeping of cutting, twisting and reconnecting is credited to Moffatt and Ricca (1992).

A preprint: not peer reviewed, limited but targeted prior-art pass, all tools shipped with launchers and verbatim ledgers.

Version 2 withdraws the energy-floor measurement of version 1: the climb of the relaxed energy with the linking number was the starting rope length, and the filament energy used has no floor. It corrects the three-halves attribution to Freedman and He. It adds three checks and their tool (latch_floor_confound_check.py), a fit tool, verbatim ledgers, credit to Moffatt (1990), to Moffatt and Ricca (1992) and to Călugăreanu, and a tool and computational resource disclosure. Every version 1 run was repeated and reproduced to every printed digit. The full list is in the paper under "What changed in version 2".

Keywords

holonomy; linking number; Gauss linking; integral helicity; Arnold inequality; Freedman-He asymptotic; crossing number; ropelength; filament model; vortex knots; Calugareanu; negative result; reproducible instruments; preprint checkpoint