Compact and long virtual knots

Manturov, V.

doi:10.1090/S0077-1554-08-00168-4

Compact and long virtual knots
HTML articles powered by AMS MathViewer

by V. O. Manturov
Translated by: G. G. Gould

Trans. Moscow Math. Soc. 2008, 1-26

DOI: https://doi.org/10.1090/S0077-1554-08-00168-4

Published electronically: November 19, 2008

PDF | Request permission

Abstract:

The theory of virtual knots is a generalization of the theory of classical knots proposed by Kauffman in 1996. In this paper solutions of certain problems of the theory of virtual knots are given. The first part of the paper is devoted to the representation of virtual knots as knots in 3-manifolds of the form $S_g\times I$ modulo stabilization. Using Haken’s theory of normal surfaces, a theorem is proved showing that the problem of recognizing virtual links is algorithmically soluble. A theorem is proved stating that (any) connected sum of non-trivial virtual knots is non-trivial. This result is a corollary of inequalities proved in this paper for the (underlying) genus of virtual knots. The last part of this paper is concerned with long virtual knots. By contrast with the classical case, the classifications of long and compact virtual knots are different. Invariants of long virtual knots are constructed with the help of which one can establish the inequivalence of long knots having equivalent closures and also that some pairs of long virtual knots do not commute.

References

A. Bartholomew and R. Fenn, Quaternionic invariants of virtual knots and links, www.maths.sussex.ac.uk (2003).
J. Scott Carter, Seiichi Kamada, and Masahico Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11 (2002), no. 3, 311–322. Knots 2000 Korea, Vol. 1 (Yongpyong). MR 1905687, DOI 10.1142/S0218216502001639
Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Graduate Texts in Mathematics, No. 57, Springer-Verlag, New York-Heidelberg, 1977. Reprint of the 1963 original. MR 0445489
H. A. Dye and Louis H. Kauffman, Minimal surface representations of virtual knots and links, Algebr. Geom. Topol. 5 (2005), 509–535. MR 2153118, DOI 10.2140/agt.2005.5.509
Roger Fenn, Louis H. Kauffman, and Vassily O. Manturov, Virtual knot theory—unsolved problems, Fund. Math. 188 (2005), 293–323. MR 2191949, DOI 10.4064/fm188-0-13
C. F. Gauss, Zur Mathematischen Theorie der electrodynamischen Wirkungen, Werke Köningl. Gesell. Wiss. Göttingen 5 (1877).
Mikhail Goussarov, Michael Polyak, and Oleg Viro, Finite-type invariants of classical and virtual knots, Topology 39 (2000), no. 5, 1045–1068. MR 1763963, DOI 10.1016/S0040-9383(99)00054-3
Wolfgang Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245–375 (German). MR 141106, DOI 10.1007/BF02559591
Geoffrey Hemion, The classification of knots and $3$-dimensional spaces, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1211184
Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744
David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37–65. MR 638121, DOI 10.1016/0022-4049(82)90077-9
Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. MR 766964, DOI 10.1090/S0273-0979-1985-15304-2
Teruhisa Kadokami, Detecting non-triviality of virtual links, J. Knot Theory Ramifications 12 (2003), no. 6, 781–803. MR 2008880, DOI 10.1142/S0218216503002780
Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/0040-9383(87)90009-7
Roger Fenn, Louis H. Kauffman, and Vassily O. Manturov, Virtual knot theory—unsolved problems, Fund. Math. 188 (2005), 293–323. MR 2191949, DOI 10.4064/fm188-0-13
Louis H. Kauffman and Vassily O. Manturov, Virtual biquandles, Fund. Math. 188 (2005), 103–146. MR 2191942, DOI 10.4064/fm188-0-6
Louis H. Kauffman and David Radford, Bi-oriented quantum algebras, and a generalized Alexander polynomial for virtual links, Diagrammatic morphisms and applications (San Francisco, CA, 2000) Contemp. Math., vol. 318, Amer. Math. Soc., Providence, RI, 2003, pp. 113–140. MR 1973514, DOI 10.1090/conm/318/05548
Toshimasa Kishino and Shin Satoh, A note on non-classical virtual knots, J. Knot Theory Ramifications 13 (2004), no. 7, 845–856. MR 2101229, DOI 10.1142/S0218216504003548
Greg Kuperberg, What is a virtual link?, Algebr. Geom. Topol. 3 (2003), 587–591. MR 1997331, DOI 10.2140/agt.2003.3.587
Vassily Manturov, Knot theory, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2068425, DOI 10.1201/9780203402849
V. O. Manturov, Atoms and minimal diagrams of virtual links, Dokl. Akad. Nauk 391 (2003), no. 2, 166–168 (Russian). MR 2042190
V. O. Manturov, An invariant polynomial of two variables for virtual links, Uspekhi Mat. Nauk 57 (2002), no. 5(347), 141–142 (Russian); English transl., Russian Math. Surveys 57 (2002), no. 5, 997–998. MR 1992085, DOI 10.1070/RM2002v057n05ABEH000555
Vassily O. Manturov, Kauffman-like polynomial and curves in 2-surfaces, J. Knot Theory Ramifications 12 (2003), no. 8, 1145–1153. MR 2017986, DOI 10.1142/S0218216503002974
Vassily O. Manturov, Vassiliev invariants for virtual links, curves on surfaces and the Jones-Kauffman polynomial, J. Knot Theory Ramifications 14 (2005), no. 2, 231–242. MR 2128512, DOI 10.1142/S0218216505003804
Vassily O. Manturov, Long virtual knots and their invariants, J. Knot Theory Ramifications 13 (2004), no. 8, 1029–1039. MR 2108647, DOI 10.1142/S0218216504003639
V. O. Manturov, On long virtual knots, Dokl. Akad. Nauk 401 (2005), no. 5, 595–598 (Russian). MR 2160133
V. O. Manturov, Virtual links are algorithmically recognisable, (2004) arXiv:math. GT/0408255 v1.
S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(161) (1982), no. 1, 78–88, 160 (Russian). MR 672410
Sergei Matveev, Algorithmic topology and classification of 3-manifolds, Algorithms and Computation in Mathematics, vol. 9, Springer-Verlag, Berlin, 2003. MR 1997069, DOI 10.1007/978-3-662-05102-3
Edwin E. Moise, Affine structures in $3$-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96–114. MR 48805, DOI 10.2307/1969769
William Menasco and Morwen Thistlethwaite, The classification of alternating links, Ann. of Math. (2) 138 (1993), no. 1, 113–171. MR 1230928, DOI 10.2307/2946636
Kunio Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187–194. MR 895570, DOI 10.1016/0040-9383(87)90058-9
Tomotada Ohtsuki, Quantum invariants, Series on Knots and Everything, vol. 29, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. A study of knots, 3-manifolds, and their sets. MR 1881401
Abigail Thompson, Thin position and the recognition problem for $S^3$, Math. Res. Lett. 1 (1994), no. 5, 613–630. MR 1295555, DOI 10.4310/MRL.1994.v1.n5.a9
Louis H. Kauffman, Virtual knot theory, European J. Combin. 20 (1999), no. 7, 663–690. MR 1721925, DOI 10.1006/eujc.1999.0314
A. Bartholomew and R. Fenn, Quaternionic invariants of virtual knots and links,www.maths.sussex.ac.uk (2003).
Louis H. Kauffman, Detecting virtual knots, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), no. suppl., 241–282. Dedicated to the memory of Professor M. Pezzana (Italian). MR 1881100
L. H. Kauffman, Diagrammatic knot theory (in preparation).
L. H. Kauffman, Minimal surface representation of virtual knots and links (in preparation).