2023/05/17 by Yukihiro Okamoto, Okamoto, Yukihiro
Mathematics · #math.SG #math.GT
paper · pdf · doi:10.48550/arxiv.2305.09912
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.