Conjugate Generators of Knot and Link Groups

TL;DR

This paper proves that in certain knot and link groups, conjugate generators with equal trace must be parabolic or peripheral, providing new insights into the structure of these groups.

Contribution

It establishes that conjugate generators of arithmetic two-bridge knot/link groups are parabolic, and those of the trefoil knot group are peripheral, revealing structural constraints.

Findings

Conjugate generators of arithmetic two-bridge knot/link groups are parabolic.

Conjugate generators of the trefoil knot group are peripheral.

Includes specific cases like the figure-eight knot and Whitehead link groups.

Abstract

This note shows that if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic. This includes the figure-eight knot and Whitehead link groups. Similarly, if two conjugate elements generate the trefoil knot group, then the elements are peripheral.