Dehn surgery and hyperbolic knot complements without hidden symmetries

TL;DR

This paper provides evidence supporting the conjecture that only three hyperbolic knot complements admit hidden symmetries, using obstructions to fillings and volume bounds, and proves the uniqueness of the figure-eight knot in this context.

Contribution

It introduces new obstructions to infinite fillings producing hidden symmetries and proves the figure-eight knot is the only small-volume hyperbolic knot with hidden symmetries.

Findings

Finitely many fillings of two-bridge links can produce knot complements with hidden symmetries.

The figure-eight knot complement is the unique small-volume hyperbolic knot with hidden symmetries.

Two independent proofs confirm the uniqueness of the figure-eight knot in admitting hidden symmetries.

Abstract

Neumann and Reid conjecture that there are exactly three knot complements which admit hidden symmetries. This paper establishes several results that provide evidence for the conjecture. Our main technical tools provide obstructions to having infinitely many fillings of a cusped manifold produce knot complements admitting hidden symmetries. Applying these tools, we show for any two-bridge link complement, at most finitely many fillings of one cusp can be covered by knot complements admitting hidden symmetries. We also show that the figure-eight knot complement is the unique knot complement with volume less than $6v_0 \approx 6.0896496$ that admits hidden symmetries. We then conclude with two independent proofs that among hyperbolic knot complements only the figure-eight knot complement can admit hidden symmetries and cover a filling of the two-bridge link complement $\mathbb{S}^3\setminus 6^2_2$. Each of these proofs shows that the technical tools established earlier can be made effective.