Geodesic
Parity in knot theory
Sbornik. Mathematics, Tome 201 (2010) no. 5, pp. 693-733 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this work we study knot theories with a parity property for crossings: every crossing is declared to be even or odd according to a certain preassigned rule. If this rule satisfies a set of simple axioms related to the Reidemeister moves, then certain simple invariants solving the minimality problem can be defined, and invariant maps on the set of knots can be constructed. The most important example of a knot theory with parity is the theory of virtual knots. Using the parity property arising from Gauss diagrams we show that even a gross simplification of the theory of virtual knots, namely, the theory of free knots, admits simple and highly nontrivial invariants. This gives a solution to a problem of Turaev, who conjectured that all free knots are trivial. In this work we show that free knots are generally not invertible, and provide invariants which detect the invertibility of free knots. The passage to ordinary virtual knots allows us to strengthen known invariants (such as the Kauffman bracket) using parity considerations. We also discuss other examples of knot theories with parity. Bibliography: 27 items.
Keywords: knot, link, graph, atom, virtual knot, parity, Kauffman bracket, minimality. 
@article{SM_2010_201_5_a4,
     author = {V. O. Manturov},
     title = {Parity in knot theory},
     journal = {Sbornik. Mathematics},
     pages = {693--733},
     year = {2010},
     volume = {201},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2010_201_5_a4/}
}
TY  - JOUR
AU  - V. O. Manturov
TI  - Parity in knot theory
JO  - Sbornik. Mathematics
PY  - 2010
SP  - 693
EP  - 733
VL  - 201
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/SM_2010_201_5_a4/
LA  - en
ID  - SM_2010_201_5_a4
ER  - 
%0 Journal Article
%A V. O. Manturov
%T Parity in knot theory
%J Sbornik. Mathematics
%D 2010
%P 693-733
%V 201
%N 5
%U http://geodesic.mathdoc.fr/item/SM_2010_201_5_a4/
%G en
%F SM_2010_201_5_a4
V. O. Manturov. Parity in knot theory. Sbornik. Mathematics, Tome 201 (2010) no. 5, pp. 693-733. http://geodesic.mathdoc.fr/item/SM_2010_201_5_a4/

[1] L. H. Kauffman, “Virtual knot theory”, European J. Combin., 20:7 (1999), 663–691 | DOI | MR | Zbl

[2] L. H. Kauffman, “A self-linking invariant of virtual knots”, Fund. Math., 184 (2004), 135–158 | DOI | MR | Zbl

[3] M. Goussarov, M. Polyak, O. Viro, “Finite-type invariants of classical and virtual knots”, Topology, 39:5 (2000), 1045–1068 | DOI | MR | Zbl

[4] D. P. Ilyutko, V. O. Manturov, “Introduction to graph-link theory”, J. Knot Theory Ramifications, 18:6 (2009), 791–823 | DOI | MR | Zbl

[5] D. P. Il'yutko, V. O. Manturov, “Graph-links”, Dokl. Math., 80:2 (2009), 739–742 | DOI | Zbl

[6] L. Traldi, L. Zulli, “A bracket polynomial for graphs, I”, J. Knot Theory Ramifications, 18:12 (2009), 1681–1709, arXiv: 0808.3392 | DOI | MR | Zbl

[7] V. Turaev, “Topology of words”, Proc. Lond. Math. Soc. (3), 95:2 (2007), 360–412 | DOI | MR | Zbl

[8] V. O. Manturov, On free knots, arXiv: 0901.2214

[9] V. O. Manturov, On free knots and links, arXiv: 0902.0127

[10] V. O. Manturov, Free knots are not invertible, arXiv: 0909.2230

[11] A. Gibson, Homotopy invariants of Gauss words, arXiv: 0902.0062

[12] D. P. Il'yutko, V. O. Manturov, “Cobordisms of free knots”, Dokl. Math., 80:3 (2009), 844–846 | DOI | Zbl

[13] V. O. Manturov, Parity and cobordisms of free knots, arXiv: 1001.2827

[14] V. O. Manturov, Free knots and parity, arXiv: math.gt/0912.5348

[15] W. M. Goldman, “Invariant functions on Lie groups and Hamiltonian flows of surface group representations”, Invent. Math., 85:2 (1986), 263–302 | DOI | MR | Zbl

[16] V. G. Turaev, “Skein quantization of Poisson algebras of loops on surfaces”, Ann. Sci. École Norm. Sup. (4), 24:6 (1991), 635–704 | MR | Zbl

[17] A. T. Fomenko, “The theory of invariants of multidimensional integrable Hamiltonian systems (with arbitrary many degrees of freedom). Molecular table of all integrable systems with two degrees of freedom”, Topological classification of integrable systems, Adv. Soviet Math., 6, Amer. Math. Soc., Providence, RI, 1991, 1–35 | MR | Zbl

[18] A. T. Fomenko, “Morse theory of integrable Hamiltonian systems”, Soviet Math. Dokl., 33:2 (1986), 502–506 | MR | Zbl

[19] A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658 | DOI | MR | Zbl | Zbl

[20] A. T. Fomenko, H. Zieschang, “On the topology of the three-dimensional manifolds arising in Hamiltonian mechanics”, Soviet Math. Dokl., 35:2 (1987), 529–534 | MR | Zbl

[21] S. V. Matveev, A. T. Fomenko, “Constant energy surfaces of Hamiltonian systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds”, Russian Math. Surveys, 43:1 (1988), 3–24 | DOI | MR | Zbl

[22] V. O. Manturov, Knot theory, Chapman Hall, Boca Raton, FL, 2004 | MR | Zbl

[23] V. O. Manturov, “A proof of Vassiliev's conjecture on the planarity of singular links”, Izv. Math., 69:5 (2005), 1025–1033 | DOI | MR | Zbl

[24] V. O. Manturov, “Embeddings of 4-valent framed graphs into 2-surfaces”, Dokl. Math., 79:1 (2009), 56–58 | DOI | MR

[25] H. A. Dye, L. H. Kauffman, “Virtual crossing number and the arrow polynomial”, J. Knot Theory Ramifications, 18:10 (2009), 1335–1357 | DOI | MR | Zbl

[26] Y. Miyazawa, “Magnetic graphs and an invariant for virtual links”, J. Knot Theory Ramifications, 15:10 (2006), 1319–1334 | DOI | MR | Zbl

[27] M. O. Bourgoin, “Twisted link theory”, Algebr. Geom. Topol., 8:3 (2008), 1249–1279, arXiv: math/0608233 | DOI | MR | Zbl