Faisal Leo Quraishi | Searcharxiv

arXiv2026

Classical invariants of spiral knots

Torus knots are an important family of knots about which much is understood; invariants of torus knots often exhibit nice formulas, making them convenient and fundamental building blocks for examples in knot theory. Spiral knots, defined and first studied by Brothers et al., are a braid-theoretic generalization of torus knots, but comparatively not much is known about this broader family of knots. We give a general recursive formula for the Alexander polynomials of spiral knots, and from this we derive several properties of spiral knots, including a simple genus formula. Additionally, we investigate the consequences these results have on classification questions.

math.GT↗

arXiv2026

Characterizing Hyperbolicity in Graphs

Gromov's delta-hyperbolicity, the classical measure of how tree-like a metric space is, works well on spaces with unbounded diameter but behaves poorly on finite graphs, where it depends primarily on diameter rather than geometry. We introduce a function relating Gromov's delta of a quadruple to its diameter and use it to define a normalized invariant that characterizes the hyperbolicity of any finite graph, taking values between zero (trees) and one (large lattice graphs). We derive a closed-form formula for this function on the hyperbolic plane. Using this formula, we give an alternate proof of the best constant of Gromov's delta-hyperbolicity for the hyperbolic plane. We also give the first theoretical proof of the optimal constant for the scaled Gromov four-point condition under diameter scaling, previously known only from numerical computations.

math.MG↗