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We study properties of the signature function of the torus knot Tp,g. First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.
Słowa kluczowe
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Rocznik
Tom
Strony
167--177
Opis fizyczny
Bibliogr. 15 poz.
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autor
autor
- Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland, mcboro@mimuw.edu.pl
Bibliografia
- [Bo] M. Borodzik, A ρ invariant of iterated torus knots, preprint, arXiv:0906.3660 (2009).
- [Br] E. Brieskorn, Beispiele zur Differenzialtopologie von Singularitaten, Invent. Math. 2 (1966), 1-14.
- [COT1] T. Cochran, K. Orr and P. Teichner, Knot concordance, Whitney towers and L2-signatures, Ann. of Math. 157 (2003), 433-519.
- [COT2] T. Cochran, K. Orr and P. Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004), 105-123.
- [Co] J. Collins, L2 signature of torus knots, preprint (2010), arXiv:1001.1329.
- [GLM] C. McA. Gordon, R. Litherland and K. Murasugi, Signatures of covering links, Canad. J. Math. 33 (1981), 381-394.
- [HZ] F. Hirzebruch and D. Zagier, The Atiyah-Singer Theorem and Elementary Number Theory, Publish or Perish, Boston, USA, 1974.
- [Kau] L. Kauffman, On Knots, Ann. of Math. Stud. 115, Princeton Univ. Press, Princeton, 1987.
- [KM] R. Kirby and P. Melvin, Dedekind sums, μ-invariants, and the signature cocycle, Math. Ann. 29 (1994), 231-267.
- [Li] R. A. Litherland, Signatures of iterated torus knots, in: Topology of Low-Dimensional Manifolds (Chelwood Gate, 1977), Lecture Notes in Math. 722, Springer, Berlin, 1979, 71-84.
- [Nem] A. Nemethi, On the spectrum of curve singularities, in: Singularity (Oberwolfach, 1996), Progr. Math. 162, Birkhäuser, 1998, 93-102.
- [NY] M. A. Nouh and A. Yasuhara, Torus knots that cannot be untied by twisting, Rev. Mat. Complut. 14 (2001), 423-437.
- [RG] H. Rademacher and E. Grosswald, Dedekind Sums, Carus Math. Monogr. 16, Math. Assoc. Amer., 1972.
- [Ro] K. Rosen, Dedekind-Rademacher sums and lattice points in triangles and tetrahedra, Acta Arith. 39 (1981), 59-75.
- [Vi] O. Viro, Branched coverings of manifold with boundary and link invariants, J, Math. USSR-Izv. 7 (1973), 1239-1256.
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Bibliografia
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bwmeta1.element.baztech-article-BAT5-0053-0007