## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Gromov hyperbolicity and a variation of the Gordian complex

#### Abstract

We introduce new simplicial complexes by using various invariants and local moves for knots, which give generalizations of the Gordian complex defined by Hirasawa and Uchida. In particular, we focus on the simplicial complex defined by using the Alexander-Conway polynomial and the Delta-move, and show that the simplicial complex is Gromov hyperbolic and quasi-isometric to the real line.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 87, Number 2 (2011), 17-21.

Dates
First available in Project Euclid: 1 February 2011

http://projecteuclid.org/euclid.pja/1296570389

Digital Object Identifier
doi:10.3792/pjaa.87.17

Mathematical Reviews number (MathSciNet)
MR2797579

Zentralblatt MATH identifier
1218.57006

#### Citation

Ichihara, Kazuhiro; Jong, In Dae. Gromov hyperbolicity and a variation of the Gordian complex. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 2, 17--21. doi:10.3792/pjaa.87.17. http://projecteuclid.org/euclid.pja/1296570389.

#### References

• J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275–306.
• M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, Berlin, 1999.
• J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, in Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), 329–358, Pergamon, Oxford.
• H. Fujii, Geometric indices and the Alexander polynomial of a knot, Proc. Amer. Math. Soc. 124 (1996), no. 9, 2923–2933.
• J.-M. Gambaudo and É. Ghys, Braids and signatures, Bull. Soc. Math. France 133 (2005), no. 4, 541–579.
• M. N. Goussarov, Knotted graphs and a geometrical technique of $n$-equivalences, POMI Sankt Petersburg preprint, circa (1995). (in Russian).
• M. Gromov, Hyperbolic groups, in Essays in group theory, Math. Sci. Res. Inst. Publ., 8 Springer, New York, 1987, 75–263.
• K. Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000), 1–83.
• U. Hamenstädt, Geometry of the complex of curves and of Teichmüller space, in Handbook of Teichmüller theory. Vol. I, 447–467, IRMA Lect. Math. Theor. Phys., 11 Eur. Math. Soc., Zürich.
• W. J. Harvey, Boundary structure of the modular group, in Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), 245–251, Ann. of Math. Stud., 97 Princeton Univ. Press, Princeton, NJ.
• M. Hirasawa and Y. Uchida, The Gordian complex of knots, J. Knot Theory Ramifications 11 (2002), no. 3, 363–368.
• I. D. Jong, Alexander polynomials of alternating knots of genus two, Osaka J. Math. 46 (2009), no. 2, 353–371.
• I. D. Jong, Alexander polynomials of alternating knots of genus two II, J. Knot Theory Ramifications 19 (2010), no. 8, 1075–1092.
• A. Kawauchi, A survey of knot theory, translated and revised from the 1990 Japanese original by the author, Birkhäuser, Basel, 1996.
• A. Kawauchi, On the Alexander polynomials of knots with Gordian distance one. (Preprint).
• H. Kondo, Knots of unknotting number 1 and their Alexander polynomials, Osaka J. Math. 16 (1979), no. 2, 551–559.
• J. Levine, A characterization of knot polynomials, Topology 4 (1965), 135–141.
• H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149.
• S. V. Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987), no. 2, 268–278.
• H. Murakami and Y. Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989), no. 1, 75–89.
• T. Nakamura, Braidzel surfaces for fibered knots with given Alexander polynomials, Kobe J. Math. 26 (2009), no. 1–2, 17–28.
• Y. Nakanishi and Y. Ohyama, Local moves and Gordian complexes, J. Knot Theory Ramifications 15 (2006), no. 9, 1215–1224.
• Y. Ohyama, The $C_{k}$-Gordian complex of knots, J. Knot Theory Ramifications 15 (2006), no. 1, 73–80.
• Y. Ohyama and H. Yamada, A $C_{n}$-move for a knot and the coefficients of the Conway polynomial, J. Knot Theory Ramifications 17 (2008), no. 7, 771–785.
• M. Okada, Delta-unknotting operation and the second coefficient of the Conway polynomial, J. Math. Soc. Japan 42 (1990), no. 4, 713–717.
• D. Rolfsen, Knots and links, Publish or Perish, Berkeley, CA, 1976.
• T. Sakai, A remark on the Alexander polynomials of knots, Math. Sem. Notes Kobe Univ. 5 (1977), no. 3, 451–456.
• H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1935), no. 1, 571–592.