Chord index for knots in thickened surfaces

In this note, we construct a chord index homomorphism from a subgroup of H1(Σ, Z) to the group of chord indices of a knot K in Σ × I. Some knot invariants derived from this homomorphism are discussed.

[1]  Y. Im,et al.  THE PARITY WRITHE POLYNOMIALS FOR VIRTUAL KNOTS AND FLAT VIRTUAL KNOTS , 2013 .

[2]  L. Kauffman.  An Affine Index Polynomial Invariant of Virtual Knots , 2012, 1211.1601.

[3]  Minimal surface representations of virtual knots and links , 2004, math/0401035.

[4]  Zhiyun Cheng.  The Chord Index, its Definitions, Applications, and Generalizations , 2016, Canadian Journal of Mathematics.

[5]  V. Vassiliev,et al.  Two constructions of weight systems for invariants of knots in non-trivial 3-manifolds. , 2008 .

[6]  Zhiyun Cheng.  A TRANSCENDENTAL FUNCTION INVARIANT OF VIRTUAL KNOTS , 2015, 1511.08459.

[7]  Louis H. Kauffman.  Virtual Knot Theory , 1999, Eur. J. Comb..

[8]  Hans U. Boden,et al.  Virtual Knot Cobordism and Bounding the Slice Genus , 2019, Exp. Math..

[9]  V. Vassiliev,et al.  Fiedler type combinatorial formulas for generalized Fiedler type invariants of knots in M2×R1 , 2009 .

[10]  Louis H. Kauffman,et al.  A self-linking invariant of virtual knots , 2004, math/0405049.

[11]  V. Turaev.  Cobordism of knots on surfaces , 2007, math/0703055.

[12]  S. Kamada,et al.  VIRTUAL STRINGS , 2004 .

[13]  L. Kauffman,et al.  A Linking Number Definition of the Affine Index Polynomial and Applications , 2012, 1211.1747.

[14]  Susan G. Williams,et al.  Invariants of links in thickened surfaces , 2013, 1304.4655.

[15]  T. Fiedler.  A SMALL STATE SUM FOR KNOTS , 1993 .

[16]  Mark D. Meyerson,et al.  Representing homology classes of closed orientable surfaces , 1976 .

[17]  H. Dye.  Vassiliev Invariants from Parity Mappings , 2012, 1203.2939.

[18]  Zhiyun Cheng.  A polynomial invariant of virtual knots , 2012, 1202.3850.

[19]  S. Satoh,et al.  The writhes of a virtual knot , 2014 .

[20]  Zhiyun Cheng,et al.  Some remarks on the chord index , 2018, 1811.09061.