Special Issue: Dedicated to 15th Anniversary of the Seminar “Knots and Representation Theory”; Guest Editors: V. O. Manturov, D. P. Ilyutko and I. M. NikonovNo Access

A parity map of framed chord diagrams

Denis Petrovich Ilyutko

Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

E-mail Address: ilyutko@yandex.ru

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and

Vassily Olegovich Manturov

Department of Fundamental Sciences, Bauman Moscow State Technical University, Moscow, Russia

Laboratory of Quantum Topology, Chelyabinsk State University, Chelyabinsk, Russia

E-mail Address: vomanturov@yandex.ru

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https://doi.org/10.1142/S0218216515410060Cited by:3 (Source: Crossref)

Abstract

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo $4$ T-relations admit a Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define a multiplication is not known yet. In this paper, we first define a new module $ℳ_{2}$ which is generated by chord diagrams on two circles and factored by $4$ T-relations. Then we construct a “parity” map from the module of framed chord diagrams into $ℳ_{2}$ and a weight system on $ℳ_{2}$ . Using the map and weight system we show that a connected sum for framed chord diagrams is not a well-defined operation. In the end of the paper we touch linear diagrams, the circle replaced by a directed line.

Keywords:

AMSC: 16T05, 57M15, 57M25, 57M27

References

1. D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423–472. Crossref, Web of Science, Google Scholar
2. D. Bar-Natan and S. Garoufalidis, On the Melvin–Morton–Rozansky conjecture, Invent. Math. 125 (1996) 103–133. Crossref, Web of Science, Google Scholar
3. S. V. Chmutov, S. V. Duzhin and S. K. Lando, Vassiliev knot invariants I, II, III, Adv. Sov. Math. 21 (1994) 117–147. Google Scholar
4. S. Chmutov, S. Duzhin and J. Mostovoy, Introduction to Vassiliev Knot Invariants (Cambridge University Press, 2012). Crossref, Google Scholar
5. M. Cohn and A. Lempel, Cycle decomposition by disjoint transpositions, J. Combin. Theory Ser. A 13 (1972) 83–89. Crossref, Google Scholar
6. V. V. Goryunov, Finite order invariants of framed knots in a solid torus and in Arnold’s $(J^{+})$ -theory of plane curves, Geometry and Physics, eds. J. E. Andersen, J. Dupont, H. Pedersen and A. Swann, Lecture Notes in Pure and Applied Mathematics, Vol. 184 (Marcel Dekker, New York, 1997), pp. 549–556. Google Scholar
7. M. Goussarov, M. Polyak and O. Viro, Finite type invariants of classical and virtual knots, Topology 39 (2000) 1045–1068. Crossref, Web of Science, Google Scholar
8. D. P. Ilyutko, Framed 4-valent graphs: Euler tours, Gauss circuits and rotating circuits, Sb. Math. 202(9) (2011) 1303–1326; Mat. Sb. 202(9) (2011) 53–76 (in Russian). Google Scholar
9. D. P. Ilyutko, An equivalence between the set of graph-knots and the set of homotopy classes of looped graphs, J. Knot Theory Ramifications 21(1) (2012) Article ID: 1250001, 30pp. Link, Web of Science, Google Scholar
10. D. P. Ilyutko, V. O. Manturov and I. M. Nikonov, Parity in knot theory and graph-links, J. Math. Sci. 193(6) (2011) 809–965. Crossref, Google Scholar
11. D. P. Ilyutko, V. O. Manturov and I. M. Nikonov, Virtual knot invariants arising from parities, Banach Center Publ. 100 (2013) 99–130. Crossref, Google Scholar
12. M. Karev, The space of framed chord diagrams as a Hopf module, J. Knot Theory Ramifications 24 (2015) 1550014 (17 pages), doi:10.1142/S0218216515500145. Link, Web of Science, Google Scholar
13. S. K. Lando, $J$ -invariants of plane curves and framed chord diagrams, Funktsional. Anal. i Prilozhen. 40(1) (2006) 1–13. Crossref, Web of Science, Google Scholar
14. B. Mellor, A few weight systems arising from intersection graphs, Michigan Math. J. 51(3) (2003) 509–536. Crossref, Web of Science, Google Scholar
15. V. O. Manturov, A proof of Vassiliev’s conjecture on the planarity of singular links, Izv. Math. 69(5) (2005) 1025–1033, Izv. Ross. Akad. Nauk Ser. Mat. 69(5) (2005) 169–178 (in Russian). Google Scholar
16. V. O. Manturov, Embeddings of 4-valent framed graphs into 2-surfaces, Dokl. Math. 79(1) (2009) 56–58; Dokl. Akad. Nauk 424(3) (2009) 308–310 (in Russian). Google Scholar
17. V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214. Google Scholar
18. V. O. Manturov, On free knots and links, preprint (2009), arXiv:math.GT/0902.0127. Google Scholar
19. V. O. Manturov, Free knots are not invertible, preprint (2009), arXiv:math.GT/0909.2230v2. Google Scholar
20. V. O. Manturov, Parity and cobordisms of free knots, preprint (2010), arXiv:math.GT/1001.2827. Google Scholar
21. V. O. Manturov, Parity in knot theory, Sb. Math. 201(5) (2010) 693–733; Mat. Sb. 201(5) (2010) 65–110 (in Russian). Google Scholar
22. V. O. Manturov, Embeddings of four-valent framed graphs into 2-surfaces, in The Mathematics of Knots, Contributions in the Mathematical and Computational Science, Vol. 1 (Springer, 2010), pp. 169–197. Google Scholar
23. V. O. Manturov, A functorial map from knots in thickened surfaces to classical knots and generalisations of parity, preprint (2010), arXiv:math.GT/1011.4640v4. Google Scholar
24. V. O. Manturov, Parity and cobordism of free knots, Mat. Sb. 203(2) (2012) 45–76 (in Russian). Crossref, Web of Science, Google Scholar
25. V. O. Manturov, Free knots and parity, in Introductory Lectures on Knot Theory: Selected Lectures Presented at the Advanced School and Conference on Knot Theory and its Applications to Physics and Biology, Series on Knots and Everything, Vol. 46 (World Scientific, 2012), pp. 321–345. Link, Google Scholar
26. V. O. Manturov, Parity and projection from virtual knots to classical knots, J. Knot Theory Ramifications 22 (2013) Article ID: 1350044, 20pp. Link, Web of Science, Google Scholar
27. G. Moran, Chords in a circle and linear algebra over $G F (2)$ , J. Combin. Theory Ser. A 37 (1984) 239–247. Crossref, Web of Science, Google Scholar
28. I. Nikonov, A new proof of Vassiliev’s conjecture, J. Knot Theory Ramifications 23 (2014) Article ID: 1460005, 28pp. Link, Web of Science, Google Scholar
29. E. Soboleva, Vassiliev knot invariants coming from Lie algebras and 4-invariants, J. Knot Theory Ramifications 10(1) (2001) 161–169. Link, Web of Science, Google Scholar
30. L. Traldi, Binary nullity, Euler circuits and interlace polynomials, preprint (2009), arXiv:math.CO/0903.4405. Google Scholar