Virtual Knot Cobordism and Bounding the Slice Genus

ABSTRACT In this paper, we compute the slice genus for many low-crossing virtual knots. For instance, we show that 1295 out of 92,800 virtual knots with six or fewer crossings are slice, and that all but 248 of the rest are not slice. Key to these results are computations of Turaev’s graded genus, which we show extends to give an invariant of virtual knot concordance. The graded genus is remarkably effective as a slice obstruction, and we develop an algorithm that applies virtual unknotting operations to determine the slice genus of many virtual knots with six or fewer crossings.

[1]  H. Boden,et al.  Signature and concordance of virtual knots , 2017, 1708.08090.

[2]  W. Rushworth Computations of the slice genus of virtual knots , 2017, Topology and its Applications.

[3]  A. Henrich,et al.  Knots, Links, Spatial Graphs, and Algebraic Invariants , 2017 .

[4]  M. Nagel,et al.  Concordance group of virtual knots , 2016, 1606.06404.

[5]  M. Chrisman Band-Passes and Long Virtual Knot Concordance , 2016, 1603.00267.

[6]  Andrew J. Nicas,et al.  Virtual knot groups and almost classical knots , 2015, 1506.01726.

[7]  L. Kauffman,et al.  Khovanov Homology, Lee Homology and a Rasmussen Invariant for Virtual Knots , 2014, 1409.5088.

[8]  Andrew J. Nicas,et al.  Alexander invariants for virtual knots , 2014, 1409.1459.

[9]  L. Kauffman Virtual Knot Cobordism , 2014, 1409.0324.

[10]  Zhiyun Cheng,et al.  A polynomial invariant of virtual links , 2013, 1301.1755.

[11]  Sang Youl Lee,et al.  Signature, Nullity And Determinant Of Checkerboard Colorable Virtual Links , 2010 .

[12]  A. Henrich A Sequence of Degree One Vassiliev Invariants for Virtual Knots , 2008, 0803.0754.

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

[14]  V. Manturov Khovanov homology for virtual knots with arbitrary coefficients , 2007 .

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

[16]  Susan G. Williams,et al.  POLYNOMIAL INVARIANTS OF VIRTUAL LINKS , 2003 .

[17]  M. Saito,et al.  STABLE EQUIVALENCE OF KNOTS ON SURFACES AND VIRTUAL KNOT COBORDISMS , 2000, math/0008118.

[18]  S. Kamada,et al.  ABSTRACT LINK DIAGRAMS AND VIRTUAL KNOTS , 2000 .

[19]  J. Sawollek On Alexander-Conway Polynomials for Virtual Knots and Links , 1999, math/9912173.

[20]  L. Kauffman Virtual Knot Theory , 1998, Eur. J. Comb..

[21]  Grant Cairns,et al.  THE PLANARITY PROBLEM FOR SIGNED GAUSS WORDS , 1993 .

[22]  J. Carter CLOSED CURVES THAT NEVER EXTEND TO PROPER MAPS OF DISKS , 1991 .

[23]  Ulrich Amsel,et al.  Surfaces In 4 Space , 2016 .

[24]  Sam Nelson,et al.  The Combinatorial Revolution in Knot Theory , 2011 .

[25]  V. Turaev Virtual strings@@@Cordes virtuelles , 2004 .

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

[27]  Peter C. Jurs,et al.  Mathematica , 2019, J. Chem. Inf. Comput. Sci..