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We show that Khovanov link homology is trivial in a range of gradings and utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and the chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper, we define the girth of a link, discuss relations to other knot invariants, and describe possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to [Formula: see text], then the girth of the link is equal to [Formula: see text]
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Learning knot invariants across dimensions
We use deep neural networks to machine learn correlations betweenknot invariants in various dimensions. The three-dimensional invariantof interest is the Jones polynomial J(q) J ( q ) ,and the four-dimensional invariants are the Khovanov polynomial \text{Kh}(q,t) Kh ( q , t ) ,smooth slice genus g g ,and Rasmussen’s s s -invariant.We find that a two-layer feed-forward neural network can predict s s from \text{Kh}(q,-q^{-4}) Kh ( q , − q − 4 ) with greater than 99% 99 % accuracy. A theoretical explanation for this performance exists in knottheory via the now disproven knight move conjecture, which is obeyed byall knots in our dataset. More surprisingly, we find similar performancefor the prediction of s s from \text{Kh}(q,-q^{-2}) Kh ( q , − q − 2 ) ,which suggests a novel relationship between the Khovanov and Leehomology theories of a knot. The network predicts g g from \text{Kh}(q,t) Kh ( q , t ) with similarly high accuracy, and we discuss the extent to which themachine is learning s s as opposed to g g ,since there is a general inequality |s| ≤2g | s | ≤ 2 g .The Jones polynomial, as a three-dimensional invariant, is not obviouslyrelated to s s or g g ,but the network achieves greater than 95% 95 % accuracy in predicting either from J(q) J ( q ) .Moreover, similar accuracy can be achieved by evaluating J(q) J ( q ) at roots of unity. This suggests a relationship with SU(2) S U ( 2 ) Chern—Simons theory, and we review the gauge theory construction ofKhovanov homology which may be relevant for explaining the network’sperformance.
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- Award ID(s):
- 2213295
- PAR ID:
- 10466138
- Date Published:
- Journal Name:
- SciPost Physics
- Volume:
- 14
- Issue:
- 2
- ISSN:
- 2542-4653
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.21468/SciPostPhys.14.2.021
