Osaka Journal of Mathematics

Casson-Walker invariants of cyclic covers branched along satellite knots

Yukihiro Tsutsumi

Full-text: Open access

Abstract

We express the Casson-Walker invariants for the cyclic covering spaces of the three-dimensional sphere branched along satellite knots in terms of companions, patterns, and winding numbers.

Article information

Source
Osaka J. Math., Volume 45, Number 1 (2008), 149-157.

Dates
First available in Project Euclid: 14 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1205503562

Mathematical Reviews number (MathSciNet)
MR2416654

Zentralblatt MATH identifier
1146.57023

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

Citation

Tsutsumi, Yukihiro. Casson-Walker invariants of cyclic covers branched along satellite knots. Osaka J. Math. 45 (2008), no. 1, 149--157. https://projecteuclid.org/euclid.ojm/1205503562


Export citation

References

  • S. Akbulut and J.D. McCarthy: Casson's Invariant for Oriented Homology $3$-Spheres, Princeton Univ. Press, Princeton, NJ, 1990.
  • R. Sturm Beiss: The Arf and Sato link concordance invariants, Trans. Amer. Math. Soc. 322 (1990), 479--491.
  • S. Boyer and A. Nicas: Varieties of group representations and Casson's invariant for rational homology $3$-spheres, Trans. Amer. Math. Soc. 322 (1990), 507--522.
  • T.D. Cochran: Concordance invariance of coefficients of Conway's link polynomial, Invent. Math. 82 (1985), 527--541.
  • O. Collin and N. Saveliev: Equivariant Casson invariants via gauge theory, J. Reine Angew. Math. 541 (2001), 143--169.
  • G. Fujita: A splicing formula for Casson-Walker's invariant, Math. Ann. 296 (1993), 327--338.
  • S. Fukuhara and N. Maruyama: A sum formula for Casson's $\lambda$-invariant, Tokyo J. Math. 11 (1988), 281--287.
  • S. Garoufalidis and A. Kricker: Finite type invariants of cyclic branched covers, Topology 43 (2004), 1247--1283.
  • J. Hoste: A formula for Casson's invariant, Trans. Amer. Math. Soc. 297 (1986), 547--562.
  • K. Ishibe: The Casson-Walker invariant for branched cyclic covers of $S^3$ branched over a doubled knot, Osaka J. Math. 34 (1997), 481--495.
  • A. Kawauchi: Almost identical imitations of $(3,1)$-dimensional manifold pairs, Osaka J. Math. 26 (1989), 743--758.
  • S. Kojima and M. Yamasaki: Some new invariants of links, Invent. Math. 54 (1979), 213--228.
  • C. Lescop: Global Surgery Formula for the Casson-Walker Invariant, Ann. of Math. Stud. 140, Princeton Univ. Press, Princeton, NJ, 1996.
  • D. Mullins: The generalized Casson invariant for $2$-fold branched covers of $S^3$ and the Jones polynomial, Topology 32 (1993), 419--438.
  • R. Myers: Excellent $1$-manifolds in compact $3$-manifolds, Topology Appl. 49 (1993), 115--127.
  • W.D. Neumann and D. Zagier: Volumes of hyperbolic three-manifolds, Topology 24 (1985), 307--332.
  • T. Sakai and K. Takeda: Fujita's formula for Casson-Walker's invariant and Seifert's formula for satellite knots, Kobe J. Math. 15 (1998), 165--171.
  • N. Saveliev: Lectures on the Topology of $3$-Manifolds, de Gruyter, Berlin, 1999.
  • H. Seifert: On the homology invariants of knots, Quart. J. Math., Oxford Ser. (2) 1 (1950), 23--32.
  • Y. Shinohara: On the signature of knots and links, Trans. Amer. Math. Soc. 156 (1971), 273\nobreakdash--285.
  • Yasuyoshi Tsutsumi: The Casson invariant of the cyclic covering branched over some satellite knot, J. Knot Theory Ramifications 14 (2005), 1029--1044.
  • Yukihiro Tsutsumi: Excellent non-orientable spanning surfaces with distinct boundary slopes, Topology Appl. 139 (2004), 199--210.
  • K. Walker: An Extention of Casson's Invariant, Ann. of Math. Stud. 126, Princeton Univ. Press, Princeton, NJ, 1992.