Tokyo Journal of Mathematics

$SC_n$-moves and the $(n+1)$-st Coefficients of the Conway Polynomials of Links

Haruko Aida MIYAZAWA

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Abstract

A local move called a $C_n$-move is related to Vassiliev invariants. It is known that two knots are related by $C_n$-moves if and only if they have the same values of Vassiliev invariants of order less than $n$. In the link case, it is shown that a $C_n$-move does not change the values of any Vassiliev invariants of order less than $n$. It is also known that, if two links can be transformed into each other by a $C_n$-move, then the $n$-th coefficients of the Conway polynomials of them, which are Vassiliev invariants of order $n$, are congruent to each other modulo $2$. An $SC_n$-move is defined as a special $C_n$-move. It is shown that an $SC_n$-move does not change the values of any Vassiliev invariants of links of order less than $n+1$. In this paper, we consider the difference of the $(n+1)$-st coefficients of the Conway polynomials of two links which can be transformed into each other by an $SC_n$-move.

Article information

Source
Tokyo J. Math., Volume 32, Number 2 (2009), 395-408.

Dates
First available in Project Euclid: 22 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1264170238

Digital Object Identifier
doi:10.3836/tjm/1264170238

Mathematical Reviews number (MathSciNet)
MR2589951

Zentralblatt MATH identifier
1197.57010

Citation

MIYAZAWA, Haruko Aida. $SC_n$-moves and the $(n+1)$-st Coefficients of the Conway Polynomials of Links. Tokyo J. Math. 32 (2009), no. 2, 395--408. doi:10.3836/tjm/1264170238. https://projecteuclid.org/euclid.tjm/1264170238


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References

  • Bar-Natan, D., On the Vassiliev knot invariants, Topology, 34 (1995), 423–472.
  • Birman, J. S., Lin, X,-S., Knot polynomials and Vassiliev's invariants, Invent. Math., 111 (1993), 225–270.
  • Goussarov, M. N., On $n$-equivalence of knots and invariants of finite degree, in, Topology of Manifold and Varieties (Viro, O. ed.), Amer. Math. Soc., Province (1994), 173–192.
  • Goussarov, M. N., Knotted graphs and a geometrical tequnique of $n$-equivalence, POMI Sankt Petersburg preprint, circa (1995), in, Russian.
  • Habiro, K., Master Thesis, University of Tokyo (1994).,
  • Habiro, K., Clasp-pass moves on knots, preprint.
  • Habiro, K., Claspers and finite type invariants of links, Geom. Topol., 4 (2000), 1–83.
  • Miyazawa, H. A., $C_n$-moves and polynomial invariants for links, Kobe J. Math., 17 (2000), 99–117.
  • Miyazawa, H. A., $C_n$-moves and $V_n$-equivalence for links, Tokyo J. Math. (to, appear).
  • Murakami, H., Nakanishi, Y., On a certain move generating link-homology, Math. Ann., 284 (1989), 75–89.
  • Ng, K. Y., Stanford, T., On Gusarov's groups of knots, Math. Proc. Camb. Phil. Soc., 126 (1999), 63–76.
  • Ohyama, Y., Vassiliev invariants and similarity of knots, Proc. Amer. Math. Soc., 123 (1995), 287–291.
  • Ohyama, Y., Remarks on $C_n$-moves for links and Vassiliev invariants of order $n$, J. Knot Theory Ramifications, 11 (2002), 507–514.
  • Ohyama, Y., Tsukamoto, T., On Habiro's $C_n$-moves and Vassiliev invariants, J. Knot Theory Ramifications, 8 (1999), 15–26.
  • Ohyama, Y., Yamada, H., A $C_n$-move for a knot and the coefficients of the Conway polynomial, J. Knot Theory Ramifications, 17 (2008), 771–785.
  • Stanford, T., Braid commutators and Vassiliev invariants, Pacific J. Math., 174 (1996), 269–276.
  • Vassiliev, V. A., Cohomology of knot space, in, Theory of Singularities and its Applications (ed. Arnold, V. I.), Adv. Soviet Math., Vol.1, Amer. Math. Soc. (1990).