TY - JOUR

T1 - Homological invariants of links in a thickened surface

AU - Tarkaev, Vladimir

N1 - Publisher Copyright: © 2019, The Royal Academy of Sciences, Madrid.

PY - 2020/1

Y1 - 2020/1

N2 - In the paper we introduce a new invariant of oriented links in a thickened surface. The invariant is a generalization of polynomial invariants u±(t) introduced by Turaev in 2008. Our invariant Q(ℓ) is defined as follows. Consider a diagram representing an oriented link ℓ⊂ Σ× [0 , 1] , where Σ is a closed orientable surface of positive genus. A value of Q(ℓ) is the formal sum over all crossings in the diagram terms of the form sign (c) [h1(c) , h2(c)] , where sign (c) denotes the sign of a crossing c and [h1(c) , h2(c)] denotes a ordered pair of homology classes of two loops associated with the crossing. As an application we prove a low bounds for the crossing number and for the virtual genus of a link. Additionally we describe an analogous constructions in some other situations, in particular, in the case of long virtual knots and prove a low bound for the virtual genus of a virtual knot which can be represented by concatenation of long virtual knots. Finally we show that Turaev’s invariants u±(t) is weaker than Q(ℓ) and discuss the results of a computing experiment which illustrates the fact.

AB - In the paper we introduce a new invariant of oriented links in a thickened surface. The invariant is a generalization of polynomial invariants u±(t) introduced by Turaev in 2008. Our invariant Q(ℓ) is defined as follows. Consider a diagram representing an oriented link ℓ⊂ Σ× [0 , 1] , where Σ is a closed orientable surface of positive genus. A value of Q(ℓ) is the formal sum over all crossings in the diagram terms of the form sign (c) [h1(c) , h2(c)] , where sign (c) denotes the sign of a crossing c and [h1(c) , h2(c)] denotes a ordered pair of homology classes of two loops associated with the crossing. As an application we prove a low bounds for the crossing number and for the virtual genus of a link. Additionally we describe an analogous constructions in some other situations, in particular, in the case of long virtual knots and prove a low bound for the virtual genus of a virtual knot which can be represented by concatenation of long virtual knots. Finally we show that Turaev’s invariants u±(t) is weaker than Q(ℓ) and discuss the results of a computing experiment which illustrates the fact.

KW - Crossing number

KW - First homology group

KW - Knot in thickened surface

KW - Link in thickened surface

KW - Long virtual knot

KW - Minimal surface representation

KW - Virtual genus

KW - Virtual link

KW - VIRTUAL KNOTS

UR - http://www.scopus.com/inward/record.url?scp=85076434444&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/c21196de-fecb-3569-810e-754af5dddb5f/

U2 - 10.1007/s13398-019-00752-y

DO - 10.1007/s13398-019-00752-y

M3 - Article

AN - SCOPUS:85076434444

VL - 114

JO - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

JF - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

SN - 1578-7303

IS - 1

M1 - 17

ER -