What question did this study set out to answer?

The research investigates how topological complexity influences the physical compactness of knotted biopolymers and resolves inconsistencies in previous findings.

April 16, 2026Open Access

The Topo-Physical Sign Change in Polymer Knots: Resolving Heuristic Artifacts via Direct Integer Invariants and Ropelength Scaling

Key Points

The research investigates how topological complexity influences the physical compactness of knotted biopolymers and resolves inconsistencies in previous findings.
Utilized a deterministic C-language hash to replace the unstable Python hash function.
Redesigned the scalar descriptor with six integer-valued topological invariants.
Applied statistical analysis under the Mansfield–Douglas theoretical scaling limit with bootstrapping techniques.
Elimination of the sign change confirmed the original descriptor's instability.
Robust recovery of the compactness sign change under high statistical confidence (p = 0.000032).
Results were consistent across three independent logistic map seeds.

Abstract

ABSTRACT A fundamental question in polymer physics is whether topological complexity alone can predict the physical compaction of a knotted biopolymer. In 2025, a scalar descriptor based on a chaotic projection of knot invariants was introduced, and a counterintuitive phenomenon was reported: within the non-trivial knot regime, topological complexity correlates positively with polymer compactness—the opposite of the global trend. However, this result relied on a heuristic hash function whose stochastic behavior across different operating systems rendered it non-reproducible. This paper resolves the resulting inconsistency in three stages. First, we demonstrate that replacing the Python hash with a deterministic C-language hash eliminates the observed sign change entirely, confirming that the original descriptor was numerically unstable. Second, we redesign the descriptor using six direct integer-valued topological invariants: the Alexander determinant, absolute knot signature, Alexander polynomial span, crossing number, bridge number, and braid index. Third, under the Mansfield–Douglas theoretical scaling limit (n = 48 non-trivial knots, B = 10, 000 bootstrap cycles), the sign change is recovered with high statistical confidence: compactness (p = 0. 000032, CI₉₅+0. 299, +0. 790, NCZ = YES) and mobility (p = 0. 000244, NCZ = YES). The result is robust across three independent logistic map seeds. These findings indicate that the unknot must be treated as a thermodynamically decoupled state in both biopolymer and synthetic polymer models. Beyond resolving a fundamental debate in polymer physics, the deterministic structure of this framework provides an inexpensive predictive tool for the rational design of topological plastics and the computational screening of DNA-unknotting therapeutics, effectively bypassing molecular dynamics simulations. ==============================================================COMPUTATIONAL BENCHMARK============================================================== CVS Engine (this work) Hardware: Consumer mobile device (Snapdragon 8 Gen 2) Language: C++ (standard, hardware-agnostic) Dataset: n=49 knot types Runtime: 300, 000 License cost: 10, 000 - 1, 000, 000/year Statistical result: rNT = +0. 562 (p = 0. 000032, NCZ=YES) Achieved on consumer hardware in under 1 second. "The same physical prediction. Free for science. Licensed for industry. " ==============================================================Source code: available on academic requestContact: lctrnc1@gmail. com (institutional email required) ==============================================================

The Topo-Physical Sign Change in Polymer Knots: Resolving Heuristic Artifacts via Direct Integer Invariants and Ropelength Scaling

Key Points

Abstract

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