ArticleOriginal scientific text
Title
Homfly polynomials as vassiliev link invariants
Authors 1, 2
Affiliations
- Department of Mathematics, Osaka City University%, Sumiyoshi-ku, Osaka 558-8585, Japan
- Department of Mathematics, Yamaguchi University, Yamaguchi 753-8512, Japan
Abstract
We prove that the number of linearly independent Vassiliev invariants for an r-component link of order n, which derived from the HOMFLY polynomial, is greater than or equal to min{n,[(n+r-1)/2]}.
Keywords
link, Conway polynomial, Jones polynomial, HOMFLY polynomial, Vassiliev link invariant
Bibliography
- [BN] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472.
- [B1] J. S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. 28 (1993), 253-287.
- [B2] J. S. Birman, On the combinatorics of Vassiliev invariants, Braid group, knot theory and statistical mechanics II (M. L. Ge and C. N. Yang, eds.), Advanced Series in Mathematical Physics, World Scientific, Singapore-New Jersey-London-Hong Kong 1994, pp. 1-19.
- [BL] J. S. Birman and X.-S. Lin, Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993), 225-270.
- [CD] S. V. Chmutov and S. V. Duzhin, An upper bound for the number of Vassiliev knot invariants, J. Knot Theory Ramifications 3 (1994), 141-151.
- [C] J. H. Conway, An enumeration of knots and links, Computational Problems in Abstract Algebra (J. Leech, ed.), Pergamon Press, New York, 1969, pp. 329-358.
- [FYHLMO] P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985), 239-246.
- [G] M. N. Gusarov, A new form of the Conway-Jones polynomial of oriented links, Zap. Nauchn. Sem. Len. Otdel Mat. Inst. Steklov (LOMI) 193 (1991), Geom. i Topol. 1, 4-9; English translation: Topology of manifolds and varieties, Advances in Soviet Mathematics. Vol. 18, Amer. Math. Soc., Providence, RI, 1994, pp. 167-172.
- [H] J. Hoste, The first coefficient of the Conway polynomial, Proc. Amer. Math. Soc. 95 (1985), 299-302.
- [J] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126 (1987), 335-388.
- [Kf] L. H. Kauffman On Knots, Ann. of Math. Studies 115, Princeton Univ. Press, Princeton, 1987.
- [Kw] A. Kawauchi On coefficient polynomials of the skein polynomial of an oriented link, Kobe J. Math. 11 (1994), 49-68.
- [L] J. Lannes Sur les invariants de Vassiliev de degré inférieur ou égal à 3, Enseign. Math. (2) 39 (1993), 295-316.
- [LM] W. B. R. Lickorish and K. C. Millett, A polynomial invariant of oriented links, Topology 26 (1987), 107-141.
- [MM] P. Melvin and H. R. Morton, The coloured Jones function, Commun. Math. Phys. 169 (1995), 501-520.
- [Me] G. Meng, Bracket models for weight systems and the universal Vassiliev invariants, Topology Appl. 76 (1997), 47-60.
- [Mi] Y. Miyazawa, The third derivative of the Jones polynomial, J. Knot Theory Ramifications, (to appear).
- [Mu1] H. Murakami, On derivatives of the Jones polynomial, Kobe J. Math. 3 (1986), 61-64.
- [Mu2] H. Murakami, Vassiliev type invariant of order two for a link, Proc. Amer. Math. Soc. 124 (1996), 3889-3896.
- [N] K. Y. Ng, Groups of ribbon knots, preprint.
- [PT] J. H. Przytycki and P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1987), 115-139.
- [R] D. Rolfsen Knots and Links, Lecture Series no. 7, Publish or Perish, Berkeley, 1976.
- [S1] T. Stanford, Finite-type invariants of knots, links, and graphs, Topology 35 (1996), 1027-1050.
- [S2] T. Stanford, The functoriality of Vassiliev-type invariants of links, braids, and knotted graphs, J. Knot Theory Ramifications 3 (1994) 247-262.
- [V] V. A. Vassiliev, Cohomology of knot spaces, Theory of Singularities and its Applications (V. I. Arnold, ed.), Advances in Soviet Mathematics, Vol. 1, Amer. Math. Soc., Providence, RI, 1990, pp. 23-69.