Computer Assisted Projective Rigidity

TL;DR

This paper uses computer-assisted methods to prove infinitesimal projective rigidity for certain hyperbolic Dehn fillings of the figure-eight knot complement, extending understanding of its deformation space.

Contribution

It provides the first computer-assisted proof of rigidity for specific surgeries and explicit representations in PSO(3,1), complementing prior theoretical results.

Findings

Approximately two thousand surgeries are infinitesimally projectively rigid.

Deformations near the complete structure induce non-zero maps on cohomology.

Explicit rational representations of the knot complement are constructed.

Abstract

In this paper we provide a computer assisted proof that about two thousand surgeries far away from the ideal point in the hyperbolic Dehn filling space of the figure-eight knot complement are infinitesimally projectively rigid. We also prove that for projective deformations of the figure-eight knot complement sufficiently close to the complete hyperbolic structure, the induced map on the first cohomology of the longitude of the boundary torus is non-zero. This paper provides a complementary piece to the results of Heusener and Porti who showed that for each k in Z, there is a sufficiently large Nk for which every k/n-Dehn filling on the figure-eight knot complement for n larger than Nk is infinitesimally projectively rigid. In the process of the proof, we provide explicit representations of the figure-eight knot complement in PSO(3,1) which are rational in the real and imaginary parts of the shapes of the ideal tetrahedra used to glue the knot complement together.