Dichromatic link invariants

Hoste, Jim; Kidwell, Mark E.

doi:10.1090/S0002-9947-1990-0961623-2

Dichromatic link invariants
HTML articles powered by AMS MathViewer

by Jim Hoste and Mark E. Kidwell

Trans. Amer. Math. Soc. 321 (1990), 197-229

DOI: https://doi.org/10.1090/S0002-9947-1990-0961623-2

PDF | Request permission

Abstract:

We investigate the skein theory of oriented dichromatic links in ${S^3}$. We define a new chromatic skein invariant for a special class of dichromatic links. This invariant generalizes both the two-variable Alexander polynomial and the twisted Alexander polynomial. Alternatively, one may view this new invariant as an invariant of oriented monochromatic links in ${S^1} \times {D^2}$, and as such it is the exact analog of the twisted Alexander polynomial. We discuss basic properties of this new invariant and applications to link interchangeability. For the full class of dichromatic links we show that there does not exist a chromatic skein invariant which is a mutual extension of both the two-variable Alexander polynomial and the twisted Alexander polynomial.

References

Robert D. Brandt, W. B. R. Lickorish, and Kenneth C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986), no. 3, 563–573. MR 837528, DOI 10.1007/BF01388747
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014
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. (N.S.) 12 (1985), no. 2, 239–246. MR 776477, DOI 10.1090/S0273-0979-1985-15361-3
Richard Hartley, The Conway potential function for links, Comment. Math. Helv. 58 (1983), no. 3, 365–378. MR 727708, DOI 10.1007/BF02564642

A new polynomial for knots and links—preliminary report

6

Jim Hoste, A polynomial invariant of knots and links, Pacific J. Math. 124 (1986), no. 2, 295–320. MR 856165
Jim Hoste and Józef H. Przytycki, An invariant of dichromatic links, Proc. Amer. Math. Soc. 105 (1989), no. 4, 1003–1007. MR 989100, DOI 10.1090/S0002-9939-1989-0989100-0
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403
Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/0040-9383(87)90009-7
Louis H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly 95 (1988), no. 3, 195–242. MR 935433, DOI 10.2307/2323625
Mark E. Kidwell, On the two-variable Conway potential function, Proc. Amer. Math. Soc. 98 (1986), no. 3, 485–494. MR 857947, DOI 10.1090/S0002-9939-1986-0857947-4
W. B. R. Lickorish and K. C. Millett, The new polynomial invariants of knots and links, Math. Mag. 61 (1988), no. 1, 3–23. MR 934822, DOI 10.2307/2690324
H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 107–109. MR 809504, DOI 10.1017/S0305004100063982

Fox’s congruence classes and Conway’s potential function of knots and links

Skein modules of

-manifolds

Survey on recent invariants in classical knot theory

Józef H. Przytycki and PawełTraczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1988), no. 2, 115–139. MR 945888
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
V. G. Turaev, Reidemeister torsion in knot theory, Uspekhi Mat. Nauk 41 (1986), no. 1(247), 97–147, 240 (Russian). MR 832411