Khovanov homology, wedges of spheres and complexity

Our main result has topological, combinatorial and computational flavor. It is motivated by a fundamental conjecture stating that computing Khovanov homology of a closed braid of fixed number of strands has polynomial time complexity. We show that the independence simplicial complex I ( w ) associat...

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Bibliographic Details
Published in:Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2024-07, Vol.118 (3), Article 102
Main Authors: Przytycki, Jozef H., Silvero, Marithania
Format: Article
Language:English
Subjects:
Applications of Mathematics
Braiding
Combinatorial analysis
Complexity
Homology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Original Paper
Polynomials
Spheres
Strands
Theoretical
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Summary:Our main result has topological, combinatorial and computational flavor. It is motivated by a fundamental conjecture stating that computing Khovanov homology of a closed braid of fixed number of strands has polynomial time complexity. We show that the independence simplicial complex I ( w ) associated to the 4-braid diagram w (and therefore its Khovanov spectrum at extreme quantum degree) is contractible or homotopy equivalent to either a sphere, or a wedge of two spheres (possibly of different dimensions), or a wedge of three spheres (at least two of them of the same dimension), or a wedge of four spheres (at least three of them of the same dimension). On the algorithmic side we prove that finding the homotopy type of I ( w ) can be done in polynomial time with respect to the number of crossings in w . In particular, we prove the wedge of spheres conjecture for circle graphs obtained from 4-braid diagrams. We also introduce the concept of Khovanov adequate diagram and discuss criteria for a link to have a Khovanov adequate braid diagram with at most 4 strands.
ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-024-01594-z