Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Strong Geometry : Knots

Baptiste Gros, Jorge L. Ramirez Alfonsin

TL;DR

The paper defines strong geometry \(\mathsf{SGeom}_{\mathsf{Lin}}(X)=(M_{\mathsf{Lin}}(X),M_{\mathsf{Wedge}}(X))\) by pairing the usual chirotope with a wedge chirotope derived from hyperplanes spanned by a point configuration. Central to the method is the witness chirotope, which links wedge and linear chirotopes and yields a dimension-reduction relation: in rank 3 the wedge chirotope expresses the linear chirotope in terms of the wedge data, while in higher rank the wedge information determines the linear chirotope up to a sign depending on parity. This framework enables a positive answer to Las Vergnas’ question on reconstructing polygonal knots from chirotopes and shows that linear spatial graphs are likewise determined by their strong geometries. The work connects Gauss-diagram based knot invariants (knotoids, graphoids) to strong geometry, proving that knots and spatial graphs are determined by their associated strong geometries, and outlines rich directions for universality, k-equivalence, and adjoints. Overall, the approach provides a combinatorial, matroid-theoretic route to reconstructing classical geometric objects from chirotopes and their adjoint structures, with potential broad impact on knot theory and spatial graph analysis.

Abstract

In this paper, we introduce the notion of strong geometry, a structure composed by both the chirotope of a set of points X in the d-dimensional space and the wedge chirotope which is the specific adjoint chirotope induced by the hyperplanes spanned by X. We present various properties relating these two chirotopes, for instance, by introducing the witness chirotope, we are able to give a formula expressing the wedge chirotope in terms of the usual chirotope. With this on hand, we answer positively a strong geometry version of a question due to M. Las Vergnas about reconstructing polygonal knots via chirotopes. Moreover, we also show that linear spatial graphs are determined by their corresponding strong geometries.

Strong Geometry : Knots

TL;DR

The paper defines strong geometry \(\mathsf{SGeom}_{\mathsf{Lin}}(X)=(M_{\mathsf{Lin}}(X),M_{\mathsf{Wedge}}(X))\) by pairing the usual chirotope with a wedge chirotope derived from hyperplanes spanned by a point configuration. Central to the method is the witness chirotope, which links wedge and linear chirotopes and yields a dimension-reduction relation: in rank 3 the wedge chirotope expresses the linear chirotope in terms of the wedge data, while in higher rank the wedge information determines the linear chirotope up to a sign depending on parity. This framework enables a positive answer to Las Vergnas’ question on reconstructing polygonal knots from chirotopes and shows that linear spatial graphs are likewise determined by their strong geometries. The work connects Gauss-diagram based knot invariants (knotoids, graphoids) to strong geometry, proving that knots and spatial graphs are determined by their associated strong geometries, and outlines rich directions for universality, k-equivalence, and adjoints. Overall, the approach provides a combinatorial, matroid-theoretic route to reconstructing classical geometric objects from chirotopes and their adjoint structures, with potential broad impact on knot theory and spatial graph analysis.

Abstract

In this paper, we introduce the notion of strong geometry, a structure composed by both the chirotope of a set of points X in the d-dimensional space and the wedge chirotope which is the specific adjoint chirotope induced by the hyperplanes spanned by X. We present various properties relating these two chirotopes, for instance, by introducing the witness chirotope, we are able to give a formula expressing the wedge chirotope in terms of the usual chirotope. With this on hand, we answer positively a strong geometry version of a question due to M. Las Vergnas about reconstructing polygonal knots via chirotopes. Moreover, we also show that linear spatial graphs are determined by their corresponding strong geometries.

Paper Structure

This paper contains 14 sections, 13 theorems, 36 equations, 24 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Oriented matroid preliminaries
  3. Wedge matroids
  4. Planar case
  5. General setting
  6. Geometric knots
  7. Gauss code
  8. Knotoids
  9. Main result
  10. Spatial graphs
  11. Further research directions
  12. Universality
  13. $k$-equivalence
  14. Adjoints

Key Result

Theorem 1

Let $X=(x_0, \dots, x_{n-1})$ and $X'=(x'_0, \dots, x'_{n-1})$ be two $n$-tuples of points in $\mathbb{R}^3$ in general position. Let $\mathsf{SGeom}_{\mathsf{Aff}}(X)$ and $\mathsf{SGeom}_{\mathsf{Aff}}(X')$ be the strong geometries associated to $X$ and $X'$ respectively. If $\mathsf{SGeom}_{\math

Figures (24)

  • Figure 1: (Left) A regular hexagon where point $p$ (in gray) is the intersection of lines joining opposite vertices. (Right) The same regular hexagon in which a vertex (in white) has been slightly moved to the right. The lines joining opposite vertices form a triangle (in gray).
  • Figure 2: Unit vectors associated to points in the plane.
  • Figure 4: Some of the non-isotopic triplets of oriented lines.
  • Figure 5: Positive and negative triplets of equators.
  • Figure 6:
  • ...and 19 more figures

Theorems & Definitions (24)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Lemma 1
  • Proposition 1
  • proof
  • Corollary 2
  • proof
  • Lemma 2
  • proof
  • ...and 14 more