Symplectic structure on colorings, Lagrangian tangles and Tits buildings

• Autorzy: Dymara Jan , Januszkiewicz Tadeusz , Przytycki Józef H.

Identyfikatory

DOI

10.4064/ba201230-2-3

• Języki publikacji: EN

Abstrakty

EN

We define a symplectic form ϕ on a free R-module R^2n-2 associated to 2n points on a circle. Using this form, we establish a relation between submodules of R^2n-2 induced by Fox R-colorings of an n-tangle and Lagrangians or virtual Lagrangians in the symplectic structure (R^2n-2; ϕ) depending on whether R is a field or a PID. We prove that when R = Z_p, p > 2, all Lagrangians are induced by Fox R-colorings of some n-tangles and note that for p = 2 and n > 3 this is no longer true. For any ring, every 2π/n-rotation of an n-tangle yields an isometry of the symplectic space R^2n-2. We analyze invariant Lagrangian subspaces of this rotation and we partially answer the question whether an operation of rotation (generalized mutation) defined in [A-P-R] preserves the first homology group of the double branched cover of S³ along a given link.

Słowa kluczowe

EN

knot; link; rotor; tangle; branched cover; symplectic form; Lagrangian; Fox coloring; Tits building

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2020
• Tom: Vol. 68, no. 2
• Strony: 169--194
• Opis fizyczny: Bibliogr.26 poz., rys.

Twórcy

autor

Dymara Jan

• Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, Poland

autor

Januszkiewicz Tadeusz

• Instytut Matematyczny, PAN Warszawa, Poland
• Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, Poland

autor

Przytycki Józef H.

• Department of Mathematics, The George Washington, University Washington, DC, U.S.A.
• Institute of Mathematics, University of Gdansk, Gdansk, Poland

Bibliografia

• [A-P-R] R. P. Anstee, J. H. Przytycki and D. Rolfsen, Knot polynomials and generalized mutation, Topology Appl. 32 (1989), 237-249.
• [A] J. Assion, Einige endliche Faktorgruppen der Zopfgruppen, Math. Z. 163 (1978), 291-302.
• [B-H] J. S. Birman and H. M. Hilden, On the mapping class groups of closed surfaces as covering spaces, in: Advances in the Theory of Riemann Surfaces, Princeton Univ. Press, Ann. of Math. Stud. 66, Princeton, NJ, 1972, 81-115.
• [Brown] K. S. Brown, Buildings, Springer, New York, 1989.
• [C-T-1] D. Cimasoni and V. Turaev, A Lagrangian representation of tangles, Topology 44 (2005), 747-767.
• [C-T-2] D. Cimasoni and V. Turaev, A Lagrangian representation of tangles. II, Fund. Math. 190 (2006), 11-27.
• [DIPY] M. Dąbkowski, M. Ishiwata, J. H. Przytycki and A. Yasuhara, Signature of rotors, Fund. Math. 184 (2004), 79-97.
• [DP-1] M. K. Dąbkowski and J. H. Przytycki, Burnside obstructions to the Montesinos-Nakanishi 3-move conjecture, Geom. Topol. 6 (2002), 355-360.
• [DP-2] M. K. Dąbkowski and J. H. Przytycki, Unexpected connection between Burnside groups and knot theory, Proc. Nat. Acad. Sci. USA 101 (2004), 17357-17360.
• [Ga] P. Garrett, Buildings and Classical Groups, Chapman & Hall, London, 1997.
• [Hu] S. P. Humphries, Generators for the mapping class group, in: Topology of Low-Dimensional Manifolds (Chetwood Gate, 1977), R. Fenn (ed.), Lecture Notes in Math. 722, Springer, 1979, 44-47.
• [Lic] W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math. (2) 76 (1962), 531-540.
• [O] O. T. O’Meara, Symplectic Groups, Math. Surveys 16, Amer. Math. Soc., Providence, RI, 1978.
• [Pla] A. Plans, Contribution to the study of the homology groups of the cyclic ramified coverings corresponding to a knot, Rev. Acad. Ciencias Madrid 47 (1953), 161-193 (in Spanish).
• [Pr-1] J. H. Przytycki, 3-coloring and other elementary invariants of knots, in: Knot Theory, Banach Center Publ. 42, Inst. Math., Polish Acad. Sci., Warszawa, 1998, 275-295.
• [Pr-2] J. H. Przytycki, Search for different links with the same Jones’ type polynomials: Ideas from graph theory and statistical mechanics, in: Panoramas of Mathematics, Banach Center Publ. 34, Inst. Math., Polish Acad. Sci., Warszawa, 1995, 121-148.
• [Pr-3] J. H. Przytycki, Skein module deformations of elementary moves on links, in: Invariants of Knots and 3-Manifolds (Kyoto, 2001), Geom. Topol. Monogr. 4 (2002), 313-335.
• [Pr-4] J. H. Przytycki, From 3-moves to Lagrangian tangles and cubic skein modules, in: Advances in Topological Quantum Field Theory (Kananaskis Village, 2001), Kluwer, 2004, 71-125.
• [Pr-5] J. H. Przytycki, The Trieste look at knot theory, in: Introductory Lectures on Knot Theory, Ser. Knots and Everything 46, World Sci., 2012, 407-441.
• [Pr-6] J. H. Przytycki, Symplectic structure on colorings and Lagrangian tangles, Abstracts Amer. Math. Soc. 21 (2000), 545.
• [Ron] M. Ronan, Lectures on Buildings, Perspectives in Math. 7, Academic Press, Boston, MA, 1989.
• [Tra] P. Traczyk, Conway polynomial and oriented rotant links, Geom. Dedicata 110 (2005), 49-61.
• [Tur] V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds, de Gruyter Stud. Math. 18, de Gruyter, Berlin, 1994.
• [Wa-1] B. Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (1983), 157-174.
• [Wa-2] B. Wajnryb, Markov classes in certain finite symplectic representations of braid groups, in: Braids (Santa Cruz, CA, 1986), Contemp. Math. 78, Amer. Math. Soc., Providence, RI, 1988, 687-695.
• [Wa-3] B. Wajnryb, A braidlike presentation of Sp(n; p), Israel J. Math. 76 (1991), 265-288.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).