# zbMATH — the first resource for mathematics

Minimal surface representations of virtual knots and links. (English) Zbl 1083.57007
The authors study virtual links considered as links in a surface cross an interval $$\Sigma \times I$$ (where the genus of $$\Sigma$$ is greater than or equal to the virtual genus of the virtual link), using the surface bracket polynomial. They obtain results about virtual links which differ from classical links by only one virtual crossing or one virtualization (i.e., replacing a classical crossing with the opposite crossing sandwiched between two virtual crossings). In particular, some sufficient conditions are found for such links to be non-classical. A Kishino knot and some related virtual knots are analyzed using the surface bracket polynomial, and the paper concludes with a non-classicality result for a class of virtual links which differ from classical links by two virtualizations.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010)
Full Text:
##### References:
 [1] G E Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer (1997) · Zbl 0934.55001 [2] A Bartholomew, R Fenn, Quaternionic invariants of virtual knots and links, J. Knot Theory Ramifications 17 (2008) 231 · Zbl 1149.57301 [3] J S Carter, S Kamada, M Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11 (2002) 311 · Zbl 1004.57007 [4] H A Dye, Virtual knots undetected by 1- and 2-strand bracket polynomials, Topology Appl. 153 (2005) 141 · Zbl 1082.57003 [5] R Fenn, C Rourke, B Sanderson, The rack space, Trans. Amer. Math. Soc. 359 (2007) 701 · Zbl 1123.55006 [6] M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer (1994) [7] N Kamada, S Kamada, Abstract link diagrams and virtual knots, J. Knot Theory Ramifications 9 (2000) 93 · Zbl 0997.57018 [8] L H Kauffman, Detecting virtual knots, Atti Sem. Mat. Fis. Univ. Modena 49 (2001) 241 · Zbl 1072.57004 [9] L H Kauffman, Virtual knot theory, European J. Combin. 20 (1999) 663 · Zbl 0938.57006 [10] L H Kauffman, S L Lins, Temperley-Lieb recoupling theory and invariants of 3-manifolds, Annals of Mathematics Studies 134, Princeton University Press (1994) · Zbl 0821.57003 [11] L H Kauffman, V O Manturov, Virtual biquandles, Fund. Math. 188 (2005) 103 · Zbl 1088.57006 [12] L K Kaufman, V O Manturov, Virtual knots and links, Tr. Mat. Inst. Steklova 252 (2006) 114 · Zbl 1351.57012 [13] T Kishino, S Satoh, A note on non-classical virtual knots, J. Knot Theory Ramifications 13 (2004) 845 · Zbl 1094.57012 [14] T Kadokami, Detecting non-triviality of virtual links, J. Knot Theory Ramifications 12 (2003) 781 · Zbl 1049.57006 [15] G Kuperberg, What is a virtual link?, Algebr. Geom. Topol. 3 (2003) 587 · Zbl 1031.57010 [16] G Kuperberg, private conversation [17] V O Manturov, Multi-variable polynomial invariants for virtual links, J. Knot Theory Ramifications 12 (2003) 1131 · Zbl 1061.57013 [18] S Matveev, Algorithmic topology and classification of 3-manifolds, Algorithms and Computation in Mathematics 9, Springer (2003) · Zbl 1048.57001 [19] S Eliahou, L H Kauffman, M B Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology 42 (2003) 155 · Zbl 1013.57005 [20] V V Prasolov, A B Sossinsky, Knots, links, braids and 3-manifolds, Translations of Mathematical Monographs 154, American Mathematical Society (1997) · Zbl 0864.57002 [21] D S Silver, S G Williams, On a class of virtual knots with unit Jones polynomial, J. Knot Theory Ramifications 13 (2004) 367 · Zbl 1077.57007 [22] V F R Jones, The annular structure of subfactors, Monogr. Enseign. Math. 38, Enseignement Math. (2001) 401 · Zbl 1019.46036 [23] V F R Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. $$(2)$$ 126 (1987) 335 · Zbl 0631.57005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.