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On filamentations and virtual knots. (English) Zbl 1031.57008

Virtual knots may be regarded as equivalence classes of Gauss diagrams under appropriate versions of the Reidemeister moves. The authors begin by introducing a new type of Gauss diagram, called an oriented chord diagram, which corresponds to a flat virtual link. The notion of a filamentation on an oriented chord diagram is defined. The authors then show that existence of a filamentation is an invariant of knot type. In particular, since every classical knot admits a filamentation, any Gauss diagram which does not admit a filamentation represents a non-classical (and hence non-trivial) virtual knot. An infinite family of non-trivial virtual knots is then defined, and these are shown to have trivial Jones polynomials and trivial quandles (and hence trivial fundamental group), but are distinguished by the Alexander biquandle.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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[1] Bar-Natan, D., On the Vassiliev knot invariants, Topology, 34, 2, 423-472 (1995) · Zbl 0898.57001
[2] Carter, J. S., Classifying immersed curves, Proc. Amer. Math. Soc., 111, 1, 281-287 (1991) · Zbl 0742.57008
[3] Carter, J. S., Closed curves that never extend to proper maps of disks, Proc. Amer. Math. Soc., 113, 3, 879-888 (1991) · Zbl 0733.57012
[4] Carter, J. S., Extending immersed curves to proper immersions of surface, Topology Appl., 40, 287-306 (1991) · Zbl 0753.57019
[5] J.S. Carter, Private communication, 1999; J.S. Carter, Private communication, 1999
[6] Carter, J. S.; Kamada, S.; Saito, M., Stable equivalence of knots on surfaces and virtual knot cobordisms, Knots 2000 Korea, Vol. 1. Knots 2000 Korea, Vol. 1, J. Knot Theory Ramifications, 11, 3, 311-322 (2002) · Zbl 1004.57007
[7] Fenn, R., The Braid-Permutation group, Topology, 36, 1, 123-135 (1997) · Zbl 0861.57010
[8] R. Fenn, M. Jordan, L.H. Kauffman, Biracks, biquandles and virtual knots, Preprint, 2002; R. Fenn, M. Jordan, L.H. Kauffman, Biracks, biquandles and virtual knots, Preprint, 2002 · Zbl 1063.57006
[9] Fenn, R.; Rimànyi, R.; Rourke, C., Some remarks on the braid-permutation group, Topics in Knot Theory, 57-68 (1993) · Zbl 0830.57005
[10] Goussarov, M.; Polyak, M.; Viro, O., Finite type invariants of classical and virtual knots, Topology, 39, 1045-1068 (2000) · Zbl 1006.57005
[11] D. Hrencecin, On filamentations and virtual knot invariants, Ph.D. Thesis, Univ. of Illinois at Chicago, October 2001; D. Hrencecin, On filamentations and virtual knot invariants, Ph.D. Thesis, Univ. of Illinois at Chicago, October 2001
[12] Jaeger, F.; Kauffman, L. H.; Saleur, H., The Conway polynomial in \(R^3\) and in thickened surfaces: A new determinant formulation, J. Combin. Theory, 61, 237-259 (1994) · Zbl 0802.57003
[13] Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra, 23, 1, 37-65 (1982) · Zbl 0474.57003
[14] Kauffman, L. H., On Knots, Ann. Math. Stud., 115 (1987), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0749.57002
[15] Kauffman, L. H., Virtual knot theory, European J. Combin., 20, 663-690 (1999) · Zbl 0938.57006
[16] Kauffman, L. H., Detecting virtual knots, Atti Sem. Mat. Fis. Univ. Modena, 49, 241-282 (2001) · Zbl 1072.57004
[17] Kauffman, L. H.; Radford, D. E., Bi-oriented quantum algebras and a generalized Alexander polynomial for virtual links, (Radford, D. E.; Souza, F. J.; Yetter, D. N., Diagrammatic Morphisms and Applications. Diagrammatic Morphisms and Applications, Contemp. Math., 318 (2003), American Mathematical Society: American Mathematical Society Providence, RI), 113-140 · Zbl 1031.57009
[18] Kawauchi, A., A Survey of Knot Theory (1996), Birkhäuser: Birkhäuser Basel
[19] T. Kishino, S. Satoh, A note on non-classical virtual knots, J. Knot Theory Ramifications (2002), submitted for publication; T. Kishino, S. Satoh, A note on non-classical virtual knots, J. Knot Theory Ramifications (2002), submitted for publication · Zbl 1094.57012
[20] Polyak, M., On the algebra of Arrow diagrams, Lectures Math. Phys., 51, 275-291 (2000) · Zbl 0976.57012
[21] Polyak, M.; Viro, O., Gauss diagram formulas for Vassiliev invariants, Internat. Math. Res. Notices, 11, 445-454 (1994) · Zbl 0851.57010
[22] Sawollek, J., On Alexander-Conway polynomials for virtual knots and links
[23] Silver, D. S.; Williams, S. G., Alexander groups and virtual links, J. Knot Theory Ramifications, 10, 151-160 (2001) · Zbl 0997.57019
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