Benjamin Hamblin | Searcharxiv

arXiv2026

Thermodynamic Invariants of Coupled Channels: A Many-Channel Tolman-Ehrenfest Effect

When multiple thermodynamic channels are coupled, single-channel equilibrium conditions fail. Extending the Tolman--Ehrenfest effect to the entropy manifold, we derive the unique $n$-channel invariant $\zeta_i T_i = C$, where $\zeta_i$ is the holonomy of the Ruppeiner connection. For the granular volume--stress ensemble, Rowe's dilatancy ratio and energy restriction emerge as geometric consequences of the off-diagonal curvature $g_{V\sigma}$, and the 60-year puzzle of state-dependent $K_\mu$ is resolved: the correction $\zeta_V$ reaches $O(1)$ near jamming. The prediction $\zeta_V\chi=\mathrm{const}$ across a shear band is experimentally testable.

math-ph↗

arXiv2025

All Tied Up: A First Look At Modern Yo-Yo and Knot Theory

Modern Yo-Yo play has developed into a sophisticated international subculture, featuring elite competition and intricate tricks. Despite this, no systematic knot-theoretic treatment has yet been applied to the numerous string configurations realised in contemporary Yo-Yo play. This paper takes initial steps in addressing this gap, recalling fundamental results from knot theory and embeddings to develop a methodology for classifying string arrangements, known as `mounts', in both beginner and advanced single Yo-Yo play. We classify a range of mounts according to the knots they form under appropriate post-processing procedures and identify Yo-Yo maneuvers that correspond to Reidemeister moves. Furthermore, we analyse the impact of certain mounts on the writhe of their diagrammatic projections and introduce operations to facilitate discussion of composite mounts. This work seeks to initiate a dialogue between Yo-Yo practitioners and knot theorists, fostering further advancements in high-level Yo-Yo play and enabling novel physical realisations of knots, links, braids, surgeries, and other topological transformations.

math.GT↗