What question did this study set out to answer?

The aim is to determine constraints on the knot types of two knots undergoing Hamiltonian isotopy that result in clean intersections.

May 9, 2026

On knot types of clean Lagrangian intersections in T^*mathbbR³

Key Points

The aim is to determine constraints on the knot types of two knots undergoing Hamiltonian isotopy that result in clean intersections.
Analyzed the isotopy of the conormal bundle of knots in T*R^3
Utilized Chekanov-Eliashberg DGAs to develop a map between unit conormal bundles
Investigated specific cases such as the unknot and torus knots.
If K_0 is the unknot, then K is also proven to be the unknot.
Results indicate that certain knot types, like torus knots and connected sums, yield specific constraints.
Established a relation between the augmentation varieties of K_0 and K.

Abstract

Let K₀ and K be knots in R³. Suppose that by a compactly supported Hamiltonian isotopy on T^*R³, the conormal bundle of K₀ is isotopic to a Lagrangian submanifold which intersects the zero section cleanly along K. In this paper, we prove some constraints on the pair of knot types of K₀ and K. One example is that if K₀ is the unknot, then K is also the unknot. We also consider some cases where K₀ and K have specific knot types, such as torus knots and connected sums of trefoil knots. The key step is finding a DGA map between the Chekanov-Eliashberg DGAs of the unit conormal bundles of knots. The main results are deduced from a relation between the augmentation varieties of K₀ and K determined by these DGAs.

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Cite This Study

Yukihiro Okamoto (Thu,) studied this question.

synapsesocial.com/papers/69fececcb9154b0b82875fde https://doi.org/https://doi.org/10.4310/jsg.260424001608

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