The \rm SL(2,ℂ)-character variety of 8₁₈

2025/08/17 by Chen, Haimiao
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2508.12203

Abstract

The knot 8₁₈ is the first non-arborescent hyperbolic knot. In 2020, Paoluzzi and Porti found its \rm SL(2,ℂ)-character variety with the aid of a computer, but many details were omitted. In this paper, we determine the character variety by a software-free procedure, which is easy to follow and enlightening. Along the way, we develop an efficient method for working with simultaneous conjugacy classes of four elements of \rm SL(2,ℂ).

Citations

• Turk's head knots and links: a survey
• Parabolic representations and generalized Riley polynomials
• Examples of character varieties in characteristic p and ramification

Related

• The \rm SL(2,ℂ)-character variety of an arborescent knot
• The SL(3,C)-character variety of the figure eight knot
• Character varieties for SL(3,C): the figure eight knot
• High-dimensional components of \rm SL(2,ℂ)-character varieties of prime knots
• The SL(2,C)-character varieties of torus knots