arXiv subjects

Orion Zymaris

Publications and source records attributed to Orion Zymaris.

4 recordsLinked to original sources

arXiv2026

An algebraic approach to circle packing

We show that for certain triangulations of surfaces, circle packings realising the triangulation can be found by solving a system of polynomial equations. We also present a similar system of equations for unbranched circle packings. The variables in these equations are associated to corners of triangles in the complex, with equations for interior vertices, edges, faces, and generators of first homology. The vertex equations are generalisations of the Descartes circle theorem, of higher degree but more symmetric than those previously found by the authors. We also provide some connections between the spinorial approach of previous work of the authors, and classical Euclidean geometry.

math.GT↗

arXiv2026

Triangulating Spun 2-Knot Complements

A $2$-knot is an embedding of a $2$-sphere into the $4$-sphere. Similar to the case of embedding circles into the $3$-sphere, this allows the $2$-sphere to be knotted. In this short paper, we present an algorithm to generate triangulations of the exteriors of $2$-knots obtained by spinning $1$-knots. We give an implementation of the algorithm in \emph{Regina} and present triangulations of exteriors obtained from all $1$-knots with up to eight crossings.

math.GT↗

arXiv2026

The Lipschitz Spinor-Higher Horosphere Correspondence

In a paper of Mathews, an isomorphism is constructed between two-component complex spinors and horospheres in H^3 carrying `spin decorations'. A recent arXiv preprint of Mathews and Varsha arXiv:2412.06572 extends this result to the case of `quaternionic spinors' and spin decorated horospheres in H^4. The following work generalises these results to an equivariant correspondence between two-component `Lipschitz spinors' with entries drawn from the Lipschitz group of a Clifford algebra, null multiflags in generalised Minkowski space, and higher-dimensional horospheres that carry an extension of the Mathews spin decoration. This correspondence allows spinors to be applied to horospheres in any dimension of hyperbolic space.

math.GT↗

arXiv2023

Spinors and Descartes' Theorem

Descartes' circle theorem relates the curvatures of four mutually externally tangent circles, three "petal" circles around the exterior of a central circle, forming a "$3$-flower" configuration. We generalise this theorem to the case of an "$n$-flower", consisting of $n$ tangent circles around the exterior of a central circle, and give an explicit equation satisfied by their curvatures. The proof uses a spinorial description of horospheres in hyperbolic geometry.

math.GT↗