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Narrowing the gap between combinatorial and hyperbolic knot invariants via deep learning. (English) Zbl 1497.57004

The Jones polynomial \(J_K\) of a knot \(K\) is a well-known invariant. Evaluating \(J_K\) at a root of unity yields a scalar invariant. Another scalar invariant is obtained by computing the Mahler measure of \(J_K\). Because \(J_K\) can be computed directly from a diagram of \(K\), these are referred to as combinatorial invariants. The Khovanov polynomial, which generalizes the Jones polynomial, is another combinatorial invariant. In contrast, if \(K\) is assumed be a hyperbolic knot, so that its complement in the three-sphere is a hyperbolic 3-manifold, then there are geometric invariants. Among these are the Chern-Simons invariant and the hyperbolic volume. Others are obtained by considering the maximal cusp of a knot, namely the maximal cusp volume, the longitude and meridian lengths, and the meridian translations. In recent years, artificial neural networks have been used to investigate relations between combinatorial and geometric invariants. In particular, a simple empirical formula that approximates the hyperbolic volume from the Jones polynomial evaluated at \(e^{3\pi i/4}\) has been discovered. Using this as motivation, the author of the article uses a combination of linear regression analysis and artificial neural networks on a database of all hyperbolic knots up to 12 crossings to compare the ability of a combinatorial invariant to predict a geometric invariant. Numerical experiments were performed on alternating and non-alternating knots both separately and combined, and the results are presented in tabular form.

MSC:

57K10 Knot theory
68T07 Artificial neural networks and deep learning
57K32 Hyperbolic 3-manifolds
57K31 Invariants of 3-manifolds (including skein modules, character varieties)
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