TY - JOUR

T1 - A Homological Casson Type Invariant of Knotoids

AU - Tarkaev, Vladimir

N1 - Tarkaev, V. A Homological Casson Type Invariant of Knotoids. Results Math 76, 142 (2021). https://doi.org/10.1007/s00025-021-01445-y

PY - 2021/8/1

Y1 - 2021/8/1

N2 - We consider an analogue of the well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. The value of the extension is a formal sum of subgroups of the first homology group H1(Σ) where Σ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in S2 into a knotoid diagram in the annulus by removing small disks around its endpoints. As an application of the invariants we prove two theorems: a sharp lower bound of the crossing number of a knotoid (the estimate differs from its prototype for classical knots proved by M. Polyak and O. Viro in 2001) and a sufficient condition for a knotoid in S2 to be a proper knotoid (or pure knotoid with respect to Turaev’s terminology). Finally we give a table containing values of our invariants computed for all spherical prime proper knotoids having diagrams with at most 5 crossings.

AB - We consider an analogue of the well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. The value of the extension is a formal sum of subgroups of the first homology group H1(Σ) where Σ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in S2 into a knotoid diagram in the annulus by removing small disks around its endpoints. As an application of the invariants we prove two theorems: a sharp lower bound of the crossing number of a knotoid (the estimate differs from its prototype for classical knots proved by M. Polyak and O. Viro in 2001) and a sufficient condition for a knotoid in S2 to be a proper knotoid (or pure knotoid with respect to Turaev’s terminology). Finally we give a table containing values of our invariants computed for all spherical prime proper knotoids having diagrams with at most 5 crossings.

KW - Casson knot invariant

KW - crossing number

KW - first homology group

KW - invariants of knotoids

KW - Knotoid

KW - proper knotoid

UR - http://www.scopus.com/inward/record.url?scp=85110878274&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/de3ab40e-c8d1-3bde-a43e-b20b3388a55d/

U2 - 10.1007/s00025-021-01445-y

DO - 10.1007/s00025-021-01445-y

M3 - Article

AN - SCOPUS:85110878274

VL - 76

JO - Results in Mathematics

JF - Results in Mathematics

SN - 1422-6383

IS - 3

M1 - 142

ER -