On the isotopies of tangles in periodic 3-manifolds using finite covers

Yuka Kotorii; Sonia Mahmoudi; Elisabetta A. Matsumoto; Ken'ichi Yoshida

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arXiv · 2505.20940

On the isotopies of tangles in periodic 3-manifolds using finite covers

Yuka Kotorii·Sonia Mahmoudi·Elisabetta A. Matsumoto·Ken'ichi Yoshida

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Abstract

A periodic tangle is a one-dimensional submanifold in $\mathbb{R}^3$ that has translational symmetry in one, two or three transverse directions. A periodic tangle can be seen as the universal cover of a link in the solid torus, the thickened torus, or the three-torus, respectively. Our goal is to study equivalence relations of such periodic tangles. Since all finite covers of a link lift to the same periodic tangle, it is necessary to prove that isotopies between different finite covers are preserved. In this paper, we show that if two links have isotopic lifts in a common finite cover, then they are isotopic. To do so, we employ techniques from 3-manifold topology to study the complements of such links.

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Yuka Kotorii, Sonia Mahmoudi, Elisabetta A. Matsumoto, Ken'ichi Yoshida. 2025-05-27. On the isotopies of tangles in periodic 3-manifolds using finite covers. https://arxiv.org/abs/2505.20940

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