Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 032, 29 pages      math.GT/0605164      https://doi.org/10.3842/SIGMA.2010.032

A Euclidean Geometric Invariant of Framed (Un)Knots in Manifolds

Jérôme Dubois a, Igor G. Korepanov b and Evgeniy V. Martyushev b
a) Institut de Mathématiques de Jussieu, Université Paris Diderot-Paris 7, UFR de Mathématiques, Case 7012, Bâtiment Chevaleret, 2, place Jussieu, 75205 Paris Cedex 13, France
b) South Ural State University, 76 Lenin Avenue, Chelyabinsk 454080, Russia

Received October 09, 2009, in final form April 07, 2010; Published online April 15, 2010

Abstract
We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. Our invariant is related to a finite-dimensional fermionic topological quantum field theory.

Key words: Pachner moves; Reidemeister torsion; framed knots; differential relations in Euclidean geometry; topological quantum field theory.

pdf (463 kb)   ps (361 kb)   tex (203 kb)

References

  1. Atiyah M.F., Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. (1988), no. 68, 175-186.
  2. Bel'kov S.I., Korepanov I.G., Martyushev E.V., A simple topological quantum field theory for manifolds with triangulated boundary, arXiv:0907.3787.
  3. Boden H.U., Herald C.M., Kirk P.A., Klassen E.P., Gauge theoretic invariants of Dehn surgeries on knots, Geom. Topol. 5 (2001), 143-226, math.GT/9908020.
  4. Dubois J., Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups, Ann. Inst. Fourier (Grenoble) 55 (2005), 1685-1734, math.GT/0403470.
  5. Korepanov I.G., Invariants of PL-manifolds from metrized simplicial complexes. Three-dimensional case, J. Nonlinear Math. Phys. 8 (2001), 196-210, math.GT/0009225.
  6. Korepanov I.G., Euclidean 4-simplices and invariants of four-dimensional manifolds. I. Moves 3→3, Theoret. and Math. Phys. 131 (2002), 765-774, math.GT/0211165.
  7. Korepanov I.G., Euclidean 4-simplices and invariants of four-dimensional manifolds. II. An algebraic complex and moves 2↔4, Theoret. and Math. Phys. 133 (2002), 1338-1347, math.GT/0211166.
  8. Korepanov I.G., Euclidean 4-simplices and invariants of four-dimensional manifolds. III. Moves 1↔5 and related structures, Theoret. and Math. Phys. 135 (2003), 601-613, math.GT/0211167.
  9. Korepanov I.G., SL(2)-solution of the pentagon equation and invariants of three-dimensional manifolds, Theoret. and Math. Phys. 138 (2004), 18-27, math.AT/0304149.
  10. Korepanov I.G., Invariants of three-dimensional manifolds from four-dimensional Euclidean geometry, math.GT/0611325.
  11. Korepanov I.G., Geometric torsions and invariants of manifolds with a triangulated boundary, Theoret. and Math. Phys. 158 (2009), 82-95, arXiv:0803.0123.
  12. Korepanov I.G., Geometric torsions and an Atiyah-style topological field theory, Theoret. and Math. Phys. 158 (2009), 344-354, arXiv:0806.2514.
  13. Korepanov I.G., Algebraic relations with anticommuting variables for four-dimensional Pachner moves 3→3 and 2↔4, arXiv:0911.1395.
  14. Korepanov I.G., Kashaev R.M., Martyushev E.V., A finite-dimensional TQFT for three-manifolds based on group PSL(2,C) and cross-ratios, arXiv:0809.4239.
  15. Korepanov I.G., Martyushev E.V., Distinguishing three-dimensional lens spaces L(7,1) and L(7,2) by means of classical pentagon equation, J. Nonlinear Math. Phys. 9 (2002), 86-98, math.GT/0210343.
  16. Lickorish W.B.R., Simplicial moves on complexes and manifolds, in Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., Vol. 2, Geom. Topol. Publ., Coventry, 1999, 299-320, math.GT/9911256.
  17. Martyushev E.V., Euclidean geometric invariants of links in 3-sphere, Izv. Chelyabinsk. Nauchn. Tsentra 2004 (2004), no. 4 (26), 1-5, math.GT/0409241.
  18. Martyushev E.V., Geometric invariants of three-dimensional manifolds, knots and links, Ph.D. Thesis, South Ural State University, 2007 (in Russian).
  19. Ponzano G., Regge T., Semiclassical limit of Racah coefficients, in Spectroscopic and group Theoretical Methods in Physics, Editor F. Bloch, North-Holland Publ. Co., Amsterdam, 1968, 1-58.
  20. Regge T., General relativity without coordinates, Nuovo Cimento (10) 19 (1961), 558-571.
  21. Reidemeister K., Homotopieringe und Linsenräume, Abh. Math. Semin. Hamb. Univ. 11 (1935), 102-109.
  22. Turaev V., Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001.

Previous article   Next article   Contents of Volume 6 (2010)