Euclidean 4-simplices and invariants of four-dimensional manifolds: III. Moves 1 <-> 5 and related structures

Euclidean 4-simplices and invariants of four-dimensional manifolds: I. Moves 3 -> 3

We construct invariants of four-dimensional piecewise-linear manifolds, represented as simplicial complexes, with respect to rebuildings that transform a cluster of three 4-simplices having a common two-dimensional face in a different cluster of the same type and having the same boundary. Our construction is based on the usage of Euclidean geometric values.

Euclidean 4-simplices and invariants of four-dimensional manifolds: II. An algebraic complex and moves 2 <-> 4

We write out some sequences of linear maps of vector spaces with fixed bases. Each term of a sequence is a linear space of differentials of metric values ascribed to the elements of a simplicial complex - a triangulation of a manifold. If the sequence turns out to be an acyclic complex then one can construct a manifold invariant out of its torsion. We demonstrate this first for three-dimensional manifolds, and then we conduct the part, related to moves 2 <-> 4, of the corresponding work for four-dimensional manifolds.

Euclidean simplices and invariants of three-manifolds: a modification of the invariant for lens spaces

We propose a modification of the three-manifold invariant based on the use of Euclidean metric values ascribed to the elements of manifold triangulation. We thus obtain a nontrivial invariant that can, in particular, distinguish non-homeomorphic lens spaces.

Braid groups and euclidean simplices

When Daan Krammer and Stephen Bigelow independently proved that braid groups are linear, they used the Lawrence-Krammer-Bigelow representation for generic values of its variables q and t. The t variable is closely connected to the traditional Garside structure of the braid group and plays a major role in Krammer's algebraic proof. The q variable, associated with the dual Garside structure of the braid group, has received less attention. In this article we give a geometric interpretation of the q portion of the LKB representation in terms of an action of the braid group on the space of non-dege…

Synthetic Geometry in Hyperbolic Simplices

Let $τ$ be an $n$-simplex and let $g$ be a metric on $τ$ with constant curvature $κ$. The lengths that $g$ assigns to the edges of $τ$, along with the value of $κ$, uniquely determine all of the geometry of $(τ, g)$. In this paper we focus on hyperbolic simplices ($κ= -1$) and develop geometric formulas which rely only on the edge lengths of $τ$. Our main results are distance and projection formulas in hyperbolic simplices, as well as a projection formula in Euclidean simplices. We also provide analogous formulas in simplices with arbitrary constant curvature $κ$.

单纯形的阿波罗尼奥斯定理推广及相关问题

Generalization of the Apollonius theorem for simplices and related problems

本文将阿波罗尼奥斯定理推广至n维欧几里得空间的m-单纯形，基于与中线相关的结果，给出了求最小包围球面、单纯形厚度度量及寻根二分法收敛性和误差估计的应用。