TY - JOUR

T1 - Khovanov-Lipshitz-Sarkar homotopy type for links in thickened higher genus surfaces

AU - Kauffman, Louis H.

AU - Nikonov, Igor Mikhailovich

AU - Ogasa, Eiji

N1 - Funding Information: Kauffman's work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation). Nikonov is supported by Russian Fund of Fundamental Researches, grant 20-51-53022. Publisher Copyright: © 2021 World Scientific Publishing Company.

PY - 2021/7/1

Y1 - 2021/7/1

N2 - We discuss links in thickened surfaces. We define the Khovanov-Lipshitz-Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus > 1. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov-Lipshitz-Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus > 1 are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus > 1. It is the first meaningful Khovanov-Lipshitz-Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.

AB - We discuss links in thickened surfaces. We define the Khovanov-Lipshitz-Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus > 1. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov-Lipshitz-Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus > 1 are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus > 1. It is the first meaningful Khovanov-Lipshitz-Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.

KW - homotopical Khovanov homology

KW - Khovanov-Lipshitz-Sarkar stable homotopy type

KW - Knot

KW - link

KW - Steenrod square

KW - thickened surface

UR - http://www.scopus.com/inward/record.url?scp=85118558624&partnerID=8YFLogxK

U2 - 10.1142/S0218216521500528

DO - 10.1142/S0218216521500528

M3 - Article

AN - SCOPUS:85118558624

VL - 30

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 8

M1 - 2150052

ER -