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Twisted cohomology pairings of knots I; diagrammatic computation

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Abstract

We provide a diagrammatic computation for the bilinear form, which is defined as the pairing between the (relative) cup products with every local coefficients and every integral homology 2-class of every link in the 3-sphere. As a corollary, we construct bilinear forms on the twisted Alexander modules of links.

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Notes

  1. See [3, Sects. 1–2 ] for details.

References

  1. Blanchfield, R.: Intersection theory of manifolds with operators with applications to knot theory. Ann. Math. 65, 340–356 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brown, K.S.: Cohomology of Groups, Graduate Texts in Mathematics, 87. Springer, New York (1994)

    Google Scholar 

  3. Bieri, R., Eckmann, B.: Relative homology and Poincaré duality for group pairs. J. Pure Appl. Algebra 13, 277–319 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Camacho, L., Dionísio, F. M., Picken, R.: Colourings and the Alexander Polynomial, arXiv:1303.5019

  5. Carter, J.S., Jelsovsky, D., Kamada, S., Langford, L., Saito, M.: Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Am. Math. Soc. 355, 3947–3989 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carter, J.S., Kamada, S., Saito, M.: Geometric interpretations of quandle homology. J. Knot Theory Ramif. 10, 345–386 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cochran, T.D., Orr, K., Teichner, P.: Knot concordance, whitney towers and \(L_2\)-signatures. Ann. Math. 157, 433–519 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Friedl, S., Vidussi, S.: A survey of twisted Alexander polynomials, The mathematics of knots, 45–94, Contrib. Math. Comput. Sci., 1. Springer, Heidelberg (2011)

  9. Cappell, S., Shaneson, J.L.: The codimension two placement problem and homology equivalent manifolds. Ann. Math. 99, 227–348 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Goldman, W.: Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85, 263–302 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hillman, J.: Algebraic invariants of links, second edition, Series on Knots and everything. 32 World Scientific (2002)

  12. Ishii, A., Iwakiri, M., Jang, Y., Oshiro, K.: A \(G\)-family of quandles and handlebody-knots. Ill. J. Math. 57, 817–838 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23, 37–65 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lin, X.S.: Representations of knot groups and twisted Alexander polynomials. Acta Math. Sin. (Engl. Ser.) 17, 361–380 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nosaka, T.: Quandle cocycles from invariant theory. Adv. Math. 245, 423–438 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nosaka, T.: Bilinear-form invariants of Lefschetz fibrations over the 2-sphere, preprint, arXiv:1611.04405

  17. Nosaka, T.: Twisted cohomology pairings of knots II; to classical invariants, preprint, arXiv:1602.01131

  18. Nosaka, T.: Twisted cohomology pairings of knots III, in preparation

  19. Milnor, J.W.: Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), Prindle, pp. 115–133. Weber & Schmidt, Boston, Mass., (1968)

  20. Milnor, J.W.: On isometries of inner product spaces. Invent. Math. 8, 83–97 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  21. Trotter, H.F.: Homology of group systems with applications to knot theory. Ann. Math. 76, 464–498 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wada, M.: Twisted Alexander polynomial for finitely presentable groups. Topology 33, 241–256 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zickert, C.: The volume and Chern-Simons invariant of a representation. Duke Math. J. 150, 489–532 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author sincerely expresses his gratitude to Akio Kawauchi for many useful discussions on the classical Blanchfield pairing. He also thanks Takahiro Kitayama and Masahico Saito for valuable comments. The work is partially supported by JSPS KAKENHI Grant Number 00646903.

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Nosaka, T. Twisted cohomology pairings of knots I; diagrammatic computation. Geom Dedicata 189, 139–160 (2017). https://doi.org/10.1007/s10711-017-0221-5

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