Signature functions of knots
HTML articles powered by AMS MathViewer
- by Charles Livingston
- Proc. Amer. Math. Soc. 146 (2018), 4513-4520
- DOI: https://doi.org/10.1090/proc/14102
- Published electronically: May 4, 2018
- PDF | Request permission
Abstract:
The signature function of a knot is an integer-valued step function on the unit circle in the complex plane. Necessary and sufficient conditions for a function to be the signature function of a knot are presented.References
- Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
- Jae Choon Cha and Charles Livingston, Knot signature functions are independent, Proc. Amer. Math. Soc. 132 (2004), no. 9, 2809–2816. MR 2054808, DOI 10.1090/S0002-9939-04-07378-2
- T. D. Cochran and W. B. R. Lickorish, Unknotting information from $4$-manifolds, Trans. Amer. Math. Soc. 297 (1986), no. 1, 125–142. MR 849471, DOI 10.1090/S0002-9947-1986-0849471-4
- Hisako Kondo, Knots of unknotting number $1$ and their Alexander polynomials, Osaka Math. J. 16 (1979), no. 2, 551–559. MR 539606
- J. Levine, A characterization of knot polynomials, Topology 4 (1965), 135–141. MR 180964, DOI 10.1016/0040-9383(65)90061-3
- J. Levine, Invariants of knot cobordism, Invent. Math. 8 (1969), 98–110; addendum, ibid. 8 (1969), 355. MR 253348, DOI 10.1007/BF01404613
- J. P. Levine, Metabolic and hyperbolic forms from knot theory, J. Pure Appl. Algebra 58 (1989), no. 3, 251–260. MR 1004605, DOI 10.1016/0022-4049(89)90040-6
- W. B. Raymond Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997. MR 1472978, DOI 10.1007/978-1-4612-0691-0
- Charles Livingston, Computations of the Ozsváth-Szabó knot concordance invariant, Geom. Topol. 8 (2004), 735–742. MR 2057779, DOI 10.2140/gt.2004.8.735
- Takao Matumoto, On the signature invariants of a non-singular complex sesquilinear form, J. Math. Soc. Japan 29 (1977), no. 1, 67–71. MR 437456, DOI 10.2969/jmsj/02910067
- John W. Milnor, Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967) Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 115–133. MR 0242163
- Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422. MR 171275, DOI 10.1090/S0002-9947-1965-0171275-5
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
- Tsuyoshi Sakai, A remark on the Alexander polynomials of knots, Math. Sem. Notes Kobe Univ. 5 (1977), no. 3, 451–456. MR 494064
- H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1935), no. 1, 571–592 (German). MR 1512955, DOI 10.1007/BF01448044
- Laurence R. Taylor, On the genera of knots, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 144–154. MR 547461
- A. G. Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251–264. MR 248854, DOI 10.1017/s0305004100044947
- H. F. Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76 (1962), 464–498. MR 143201, DOI 10.2307/1970369
Bibliographic Information
- Charles Livingston
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 193092
- Email: livingst@indiana.edu
- Received by editor(s): November 18, 2017
- Received by editor(s) in revised form: January 15, 2018
- Published electronically: May 4, 2018
- Additional Notes: This work was supported by a grant from the National Science Foundation, NSF-DMS-1505586.
- Communicated by: David Futer
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4513-4520
- MSC (2010): Primary 57M25
- DOI: https://doi.org/10.1090/proc/14102
- MathSciNet review: 3834675