Lipschitz geometry and combinatorics of circular snakes
André Costa, Davi Medeiros, Emanoel Souza
TL;DR
The paper addresses outer Lipschitz classification of circular snake germs by extending the Abn/snake framework to circular links and constructing a canonical pancake decomposition of the Valette link into segments and nodal zones. It introduces circular snake names as a combinatorial invariant and proves a Realization Theorem that associates any name to a geometric circular snake, followed by a weak outer bi-Lipschitz classification that reduces to node/zone correspondences and cluster structures. The work also clarifies when removing segments or nodal-zone pieces preserves snake structure and analyzes the binary circular snake case, including conjectured counts and ubiquitous reductions. Overall, the results provide a quantitative, combinatorial grip on the outer Lipschitz geometry of circular snakes with potential implications for understanding and classifying circle-link surface germs.
Abstract
This paper explores the Lipschitz geometric and combinatorial properties of germs of real semialgebraic surfaces (or, more generally, definable in a polynomially bounded o-minimal structure) with circular link (homeomorphic to the circle $\mathbb{S}^1$). We define and investigate the outer Lipschitz geometry of the so-called circular snakes, showing what results in the paper "Lipschitz geometry and combinatorics of abnormal surface germs" (by Andrei Gabrielov and Emanoel Souza) valid to snakes still holds for the circular case. We prove the existence of a canonical decomposition for the Valette link of a circular snake into finitely many segments and nodal zones and establish some necessary and sufficient criteria to determine when it is possible to obtain a snake from a circular snake by "removing" either one of its segments or a Hölder triangle whose Valette link is contained in one of its nodal zones. We construct a combinatorial object associated with a circular snake and prove a realization theorem for this combinatorial object. We also present a weakly outer Lipschitz classification for circular snakes. Finally, we show some results about the combinatorics of binary circular snakes, which is wildly distinct from the corresponding case shown in the work of Gabrielov and Souza.