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On the Polyak–Viro Vassiliev Invariant of Degree 4

Published online by Cambridge University Press:  20 November 2018

A. Stoimenow
A. Stoimenow*
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan e-mail: [email protected]
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Abstract

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Using the Polyak–Viro Gauss diagram formula for the degree-4 Vassiliev invariant, we extend some previous results on positive knots and the non-triviality of the Jones polynomial of untwisted Whitehead doubles.

Keywords

57M2757M2505C3057M15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 4 , 01 December 2006 , pp. 609 - 623
Copyright
Copyright © Canadian Mathematical Society 2006

References

[A] A’Campo, N., Generic immersions of curves, knots, monodromy and gordian number. Inst. Hautes études Sci. Publ. Math. 88(1999), 151–169.Google Scholar
[BoW] Boileau, M. and Weber, C., Le problème de J. Milnor sur le nombre gordien des noeuds algébriques. Enseign. Math. 30(1984), no. 3–4, 173–222.Google Scholar
[Cr] Cromwell, P. R., Homogeneous links. J. London Math. Soc. (2) 39(1989), no. 2, 535–552.Google Scholar
[CM] Cromwell, P. R. and Morton, H. R., Positivity of knot polynomials on positive links. J. Knot Theory Ramifications 1(1992), no. 2, 203–206.Google Scholar
[Fi] Fiedler, T., A small state sum for knots. Topology 32(1993), no. 2, 281–294.Google Scholar
[Fi2] Fiedler, T., Gauss Diagram Invariants for Knots and Links. Mathematics and Its Applications 532, Kluwer Academic Publishers, Dordrecht, 2001.Google Scholar
[FS] Fiedler, T. and Stoimenow, A., New knot and link invariants. In: Knots in Hellas, 98, Ser. Knots Everything 24, World Scientific, River Edge, NJ, 2000, pp. 59–79.Google Scholar
[J] Jones, V. F. R., A polynomial invariant of knots and links via von Neumann algebras. Bull. Amer. Math. Soc. 12(1985), no. 1, 103–111.Google Scholar
[Kr] Kreimer, D., Knots and Feynman Diagrams. Cambridge Lecture Notes in Physics 13, Cambridge University Press, Cambridge, 2000.Google Scholar
[Ng] Ng, K. Y., Groups of ribbon knots. Topology 37(1998), no. 2, 441–458.Google Scholar
[PV] Polyak, M. and Viro, O., Gauss diagram formulas for Vassiliev invariants. Int. Math. Res. Notices 1994, no. 11, 445ff. (electronic)Google Scholar
[Ro] Rolfsen, D., Knots and Links. Mathematics Lecture Series 7. Publish or Perish, Berkeley, CA, 1976.Google Scholar
[Ru] Rudolph, L., Braided surfaces and Seifert ribbons for closed braids. Comment. Math. Helv. 58(1983), no. 1, 1–37.Google Scholar
[Ru2] Rudolph, L., Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. 29(1993), no. 1, 51–59.Google Scholar
[S] Stanford, T., Braid commutators and Vassiliev invariants. Pacific J. Math. 174(1996), no. 1, 269–276.Google Scholar
[S2] Stanford, T., Some computational results on mod 2 finite-type invariants of knots and string links. In: Invariants of Knots and 3-Manifolds, Geom. Topol. Monogr. 4, Geom. Topol. Publ., Coventry, 2002, pp. 363–376.Google Scholar
[St] Stoimenow, A., Positive knots, closed braids, and the Jones polynomial. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2(2003), no. 2, 237–285.Google Scholar
[St2] Stoimenow, A., Gauss sums on almost positive knots. Compositio Math. 140(2004), no. 1, 228–254.Google Scholar
[St3] Stoimenow, A., The skein polynomial of closed 3-braids. J. Reine Angew.Math. 564(2003), 167–180.Google Scholar
[St4] Stoimenow, A., Knots of genus two, preprint math.GT/0303012.Google Scholar
[Th] Thistlethwaite, M. B., On the Kauffman polynomial of an adequate link. Invent. Math. 93(1988), no. 2, 285–296.Google Scholar
[Va] Vassiliev, V. A., Cohomology of knot spaces. In: Theory of Singularities and its Applications, Adv. Soviet Math. 1, American Mathematical Society, Providence, RI, 1990, pp. 23–69.Google Scholar
[Wi] Williams, R. F., Lorenz knots are prime. Ergodic Theory Dynam. Systems 4(1984), no. 1, 147–163.Google Scholar
[Yo] Yokota, Y., Polynomial invariants of positive links. Topology 31(1992), no. 4, 805–811.Google Scholar