Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

LOG-CONCAVITY AND ZEROS OF THE ALEXANDER POLYNOMIAL

  • Received : 2013.01.03
  • Published : 2014.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

We show that roots of log-concave Alexander knot polynomials are dense in C. This in particular implies that the log-concavity and Hoste's conjecture on the Alexander polynomial of alternating knots are (essentially) independent.

Keywords

References

  1. G. Burde, Das Alexanderpolynom der Knoten mit zwei Brucken, Arch. Math. (Basel) 44 (1985), no. 2, 180-189. https://doi.org/10.1007/BF01194083
  2. R. Crowell, Genus of alternating link types, Ann. of Math. 69 (1959), no. 2, 258-275. https://doi.org/10.2307/1970181
  3. S. Garoufalidis, Does the Jones polynomial determine the signature of a knot?, preprint, math.GT/0310203.
  4. R. I. Hartley, On two-bridged knot polynomials, J. Austral. Math. Soc. Ser. A 28 (1979), no. 2, 241-249. https://doi.org/10.1017/S1446788700015743
  5. X. Jin, F. Zhang, F. Dong, and E. G. Tay, Zeros of the Jones polynomial are dense in the complex plane, Electron. J. Combin. 17 (2010), no. 1, Research Paper 94, 10 pp.
  6. I. D. Jong, Alexander polynomials of alternating knots of genus two, Osaka J. Math. 46 (2009), no. 2, 353-371.
  7. I. D. Jong, Alexander polynomials of alternating knots of genus two II, J. Knot Theory Ramifications 19 (2010), no. 8, 1075-1092. https://doi.org/10.1142/S0218216510008285
  8. L. Lyubich and K. Murasugi, On zeros of the Alexander polynomial of an alternating knot, Topology Appl. 159 (2012), no. 1, 290-303. https://doi.org/10.1016/j.topol.2011.09.035
  9. W. W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37-44. https://doi.org/10.1016/0040-9383(84)90023-5
  10. K. Murasugi, On the genus of the alternating knot. I. II., J. Math. Soc. Japan 10 (1958), 94-105, 235-248. https://doi.org/10.2969/jmsj/01010094
  11. K. Murasugi, On the Alexander polynomial of alternating algebraic knots, J. Austral. Math. Soc. Ser. A 39 (1985), no. 3, 317-333. https://doi.org/10.1017/S1446788700026094
  12. P. S. Ozsvath and Z. Szabo, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58-116. https://doi.org/10.1016/j.aim.2003.05.001
  13. A. Stoimenow, Diagram genus, generators and applications, preprint, arXiv:1101.3390.
  14. A. Stoimenow, Newton-like polynomials of links, Enseign. Math. (2) 51 (2005), no. 3-4, 211-230.
  15. A. Stoimenow, Knots of (canonical) genus two, Fund. Math. 200 (2008), no. 1, 1-67. https://doi.org/10.4064/fm200-1-1
  16. A. Stoimenow, Realizing Alexander polynomials by hyperbolic links, Expos. Math. 28 (2010), no. 2, 133-178. https://doi.org/10.1016/j.exmath.2009.06.003

Cited by

  1. Hoste’s Conjecture and Roots of Link Polynomials vol.22, pp.2, 2018, https://doi.org/10.1007/s00026-018-0389-x