Geodesic
Invariants of links in thickened surfaces
Carter, J Scott1 ; Silver, Daniel S1 ; Williams, Susan G2
1Department of Mathematics and Statistics, University of South Alabama, ILB 325, 411 University Boulevard North, Mobile, AL 36688, USA
2Department of Mathematics and Statistics, University of South Alabama, ILB325, 411 University Boulevard North, Mobile, AL 36688, USA
Algebraic and Geometric Topology, Tome 14 (2014) no. 3, pp. 1377-1394
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A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects the nontriviality of a virtual link and determines its virtual genus.

DOI : 10.2140/agt.2014.14.1377
Classification : 57M25, 20E22
Keywords: knot, link, operator group, virtual link, virtual genus

Carter, J Scott  1   ; Silver, Daniel S  1   ; Williams, Susan G  2

1 Department of Mathematics and Statistics, University of South Alabama, ILB 325, 411 University Boulevard North, Mobile, AL 36688, USA
2 Department of Mathematics and Statistics, University of South Alabama, ILB325, 411 University Boulevard North, Mobile, AL 36688, USA 
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Carter, J Scott; Silver, Daniel S; Williams, Susan G. Invariants of links in thickened surfaces. Algebraic and Geometric Topology, Tome 14 (2014) no. 3, pp. 1377-1394. doi: 10.2140/agt.2014.14.1377

[1] N Bourbaki, Algebra I, Chapters $1$–$3$, Springer (1989)

[2] J S Carter, S Kamada, M Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11 (2002) 311

[3] D J Collins, H Zieschang, Combinatorial group theory and fundamental groups, from: "Algebra, VII", Encyclopaedia Math. Sci. 58, Springer (1993) 1, 233

[4] H A Dye, L H Kauffman, Minimal surface representations of virtual knots and links, Algebr. Geom. Topol. 5 (2005) 509

[5] A Hatcher, Notes on basic $3$–manifold topology

[6] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[7] N Kamada, On alternating simple formal presentation (1993)

[8] N Kamada, The crossing number of alternating link diagrams on a surface, from: "KNOTS '96 (Tokyo)" (editor S Suzuki), World Sci. Publ. (1997) 377

[9] N Kamada, S Kamada, Abstract link diagrams and virtual knots, J. Knot Theory Ramifications 9 (2000) 93

[10] L H Kauffman, private communication

[11] L H Kauffman, Virtual knot theory, European J. Combin. 20 (1999) 663

[12] G Kuperberg, What is a virtual link?, Algebr. Geom. Topol. 3 (2003) 587

[13] A G Kurosh, The theory of groups, Chelsea (1960)

[14] R C Lyndon, P E Schupp, Combinatorial group theory, Springer (1977)

[15] H R Morton, S Grishanov, Doubly periodic textile structures, J. Knot Theory Ramifications 18 (2009) 1597

[16] D J S Robinson, A course in the theory of groups, Graduate Texts in Mathematics 80, Springer (1982)

[17] P Scott, A new proof of the annulus and torus theorems, Amer. J. Math. 102 (1980) 241

[18] D S Silver, S G Williams, Virtual genus of satellite links, J. Knot Theory Ramifications 22 (2013) 4

[19] F Waldhausen, Gruppen mit Zentrum und $3$–dimensionale Mannigfaltigkeiten, Topology 6 (1967) 505

[20] F Waldhausen, On irreducible $3$–manifolds which are sufficiently large, Ann. of Math. 87 (1968) 56

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