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Abstract We prove a formula for the involutive concordance invariants of the cabled knots in terms of those of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer homology.
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Illuminating new and known relations between knot invariants
Abstract We automate the process of machine learning correlations between knot invariants. For nearly 200 000 distinct sets of input knot invariants together with an output invariant, we attempt to learn the output invariant by training a neural network on the input invariants. Correlation between invariants is measured by the accuracy of the neural network prediction, and bipartite or tripartite correlations are sequentially filtered from the input invariant sets so that experiments with larger input sets are checking for true multipartite correlation. We rediscover several known relationships between polynomial, homological, and hyperbolic knot invariants, while also finding novel correlations which are not explained by known results in knot theory. These unexplained correlations strengthen previous observations concerning links between Khovanov and knot Floer homology. Our results also point to a new connection between quantum algebraic and hyperbolic invariants, similar to the generalized volume conjecture.
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- Award ID(s):
- 2213295
- PAR ID:
- 10559531
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Machine Learning: Science and Technology
- Volume:
- 5
- Issue:
- 4
- ISSN:
- 2632-2153
- Format(s):
- Medium: X Size: Article No. 045061
- Size(s):
- Article No. 045061
- Sponsoring Org:
- National Science Foundation
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