Structure and classification of torus theta-curves

Jack S. Calcut; Samantha E. Nieman

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arXiv · 2511.09848

Structure and classification of torus theta-curves

Jack S. Calcut·Samantha E. Nieman

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Abstract

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 explicit infinite families of prime theta-curves. We compute their constituent knots and identify the structure governing these embeddings, which leads to a complete classification of torus theta-curves up to ambient isotopy and homeomorphism of the 3-sphere. In particular, Kinoshita's theta-curve does not lie on a standard torus.

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Jack S. Calcut, Samantha E. Nieman. 2025-11-13. Structure and classification of torus theta-curves. https://arxiv.org/abs/2511.09848

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