Abstract
Let D be a general connected reduced alternating link diagram, C be the set of crossings of D and C′ be the nonempty subset of C. In this paper we first define a multiple crossing-twisted link family {D n(C′)|n=1,2,…} based on D and C′, which produces (2,2n+1)-torus knot family, the link family A n defined in Chang and Shrock (Physica A 301:196–218, 2001) and the pretzel link family P(n,n,n) as special cases. Then by applying Beraha-Kahane-Weiss’s Theorem we prove that limits of zeros of Jones polynomials of {D n(C′)|n=1,2,…} are the unit circle |z|=1 (It is independent of the selections of D and C′) and several isolated limits, which can be determined by computing flow polynomials of subgraphs of G corresponding to D. Furthermore, we use the method of Brown and Hickman (Discrete Math. 242:17–30, 2002) to prove that, for any ε>0, all zeros of Jones polynomial of the link D n(C) lie inside the circle |z|=1+ε, provided that n is large enough. Our results extend results of F.Y. Wu, J. Wang, S.-C. Chang, R. Shrock and the present authors and refine partial result of A. Champanerkar and L. Kofman.
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References
Beraha, S., Kahane, J., Weiss, N.J.: Limits of zeros of recursively defined families of polynomials. Studies in Foundations and Combinatorics. Advances in Mathematics Supplementary Studies, vol. 1, pp. 212–232 (1978)
Bondy, J.A., Murty, U.S.R.: Graph Theory and Its Applications. Macmillan, London (1976)
Brown, J.I., Hickman, C.A.: On chromatic roots of large subdivisions of graphs. Discrete Math. 242, 17–30 (2002)
Champanerkar, A., Kofman, L.: On the Mahler measure of Jones polynomials under twisting. Algebraic Geom. Topol. 5, 1–22 (2005)
Chang, S.-C., Shrock, R.: Zeros of Jones polynomials for families of knots and links. Physica A 301, 196–218 (2001)
Jin, X., Zhang, F.: Zeros of the Jones polynomials for families of pretzel links. Physica A 328, 391–408 (2003)
Jin, X., Zhang, F.: The Kauffman brackets for equivalence classes of links. Adv. Appl. Math. 34, 47–64 (2005)
Jin, X.: Jones polynomials and the distribution of their zeros of links. Doctoral Dissertation, Xiamen University (2004)
Jones, V.F.R.: A polynomial invariant for knots via Von Neumann algebras. Bull. Am. Math. Soc. 12, 103–112 (1985)
Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126, 335–388 (1987)
Jones, V.F.R.: Knot theory and statistical mechanics. Sci. Am. 263, 98–103 (1990)
Kauffman, L.H.: State models and the Jones polynomial. Topology 26, 395–407 (1987)
Kauffman, L.H.: A Tutte polynomial for signed graphs. Discrete Appl. Math. 25, 105–127 (1989)
Lee, T.D., Yang, C.N.: Statistical theory of equations of states and phase transitions, II: lattice gas and Ising model. Phys. Rev. 87, 410–419 (1952)
Lin, X.-S.: Zeros of the Jones polynomial. http://math.ucr.edu/~xl/abs-jk.pdf
Read, R.C., Whitehead, E.G. Jr.: Chromatic polynomials of homeomorphism classes of graphs. Discrete Math. 204, 337–356 (1999)
Shahmohamad, H.: A survey on flow polynomial. Util. Math. 62, 13–32 (2002)
Wu, F.Y.: Jones polynomial as a Potts model partition function. J. Knot Theory Ramif. 1(1), 47–57 (1992)
Wu, F.Y., Wang, J.: Zeros of the Jones polynomial. Physica A 296, 483–494 (2001)
Yang, C.N., Lee, T.D.: Statistical theory of equations of states and phase transitions, I: theory of condensation. Phys. Rev. 87, 404–409 (1952)
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This work is supported by NSFC Grant No. 10831001.
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Jin, X., Zhang, F. Zeros of the Jones Polynomial for Multiple Crossing-Twisted Links. J Stat Phys 140, 1054–1064 (2010). https://doi.org/10.1007/s10955-010-0027-4
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DOI: https://doi.org/10.1007/s10955-010-0027-4