We investigate the use of artificial neural networks (NNs) as an alternative tool to current analytical methods
for recognizing knots in a given polymer conformation. The motivation is twofold. First, it is of interest to
examine whether NNs are effective at learning the global and sequential properties that uniquely define a knot.
Second, knot classification is an important and unsolved problem in mathematical and physical sciences, and
NNs may provide insights into this problem. Motivated by these points, we generate millions of polymer
conformations for five knot types: 0, 3_1, 4_1, 5_1, and 5_2, and we design various NN models for classification.
Our best model achieves a five-class classification accuracy of above 99% on a polymer of 100 monomers. We
find that the sequential modeling ability of recurrent NNs is crucial for this result, as it outperforms feed-forward
NNs and successfully generalizes to differently sized conformations as well. We present our methods and suggest
that deep learning may be used in specific applications of knot detection where some error is permissible.
Hopefully, with further development, NNs can offer an alternative computational method for knot identification
and facilitate knot research in mathematical and physical sciences.
In this paper, we study contact surgeries along Legendrian links in the standard contact 3-sphere. On one hand, we use algebraic methods to prove the vanishing of the contact Ozsváth–Szabó invariant for contact (+1)-surgery along certain Legendrian two component links. The main tool is a link surgery formula for Heegaard Floer homology developed by Manolescu and Ozsváth.
On the other hand, we use contact-geometric argument to show the overtwistedness of the contact 3-manifolds obtained by contact (+1)-surgeries along Legendrian two-component links whose two components are linked in some special configurations.