On Khovanov Homology and Related Invariants

Caprau, Carmen; González, Nicolle; Lee, Christine Ruey Shan; Lowrance, Adam M.; Sazdanovic, Radmila; Zhang, Melissa

doi:10.48550/arxiv.2002.05247

Cited by 2 publications

(1 citation statement)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, Note that Khovanov width is related to the Turaev genus g T (K): 1 2 |σ(K) − s(K) ≤ g T (K) [12]. There are other invariants that can be derived from Khovanov homology such as its width or the page on which various spectral sequences collapse that can also be related to the unknotting number and other numerical invariants [3,4,5,10]. Khovanov width w KH (K) is equal to the largest horizontal distance between two lines of slope two that contain non-trivial Khovanov homology groups [29].…”

Section: Knot Theory: Brief Introductionmentioning

confidence: 99%

Mapper-type algorithms for complex data and relations

Dlotko¹,

Gurnari²,

Sazdanovic³

2021

Preprint

View full text Add to dashboard Cite

This work brings methods from topological data analysis to knot theory and develops new data analysis tools inspired by this application. We explore a vast collection of knot invariants and relations between then using Mapper and Ball Mapper algorithms. In particular, we develop versions of the Ball Mapper algorithm that incorporate symmetries and other relations within the data, and provide ways to compare data arising from different descriptors, such as knot invariants. Additionally, we extend the Mapper construction to the case where the range of the lens function is high dimensional rather than a 1-dimensional space, that also provides ways of visualizing functions between high-dimensional spaces. We illustrate the use of these techniques on knot theory data and draw attention to potential implications of our findings in knot theory.

show abstract