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math.GT2026
Mazur's knot and the Octahedron
Jack S. Calcut, Yangyang Du
Mazur's knot exterior admits a geometric description using a single regular ideal octahedron. The resulting hyperbolic structure is closely related to the Whitehead link exterior t…
math.GT2026
Structure and classification of torus theta-curves
Jack S. Calcut, Samantha E. Nieman
We study theta-curves embedded in a standard torus in the 3-sphere. We show that each nontrivial torus knot together with an essential arc determines a prime theta-curve, yielding…
math.GT2025
Prime theta-curves on minimal genus Seifert surfaces
Jack S. Calcut, Jamie Phillips-Freedman
We prove that each prime knot union an essential arc on a minimal genus Seifert surface is a prime theta-curve.