Journal of the Mathematical Society of Japan

The Gordian distance of handlebody-knots and Alexander biquandle colorings


Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandle colorings. We construct handlebody-knots with Gordian distance $n$ and unknotting number $n$ for any positive integer $n$.

Article information

J. Math. Soc. Japan, Volume 70, Number 4 (2018), 1247-1267.

Received: 28 February 2017
First available in Project Euclid: 27 July 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M15: Relations with graph theory [See also 05Cxx] 57M27: Invariants of knots and 3-manifolds

handlebody-knot biquandle Gordian distance unknotting number


MURAO, Tomo. The Gordian distance of handlebody-knots and Alexander biquandle colorings. J. Math. Soc. Japan 70 (2018), no. 4, 1247--1267. doi:10.2969/jmsj/77417741.

Export citation


  • J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA, 9 (1923), 93–95.
  • W. E. Clark, M. Elhamdadi, M. Saito and T. Yeatman, Quandle colorings of knots and applications, J. Knot Theory Ramifications, 23 (2014), 1450035, 29 pp.
  • R. Fenn, C. Rourke and B. Sanderson, Trunks and classifying spaces, Appl. Categ. Structures, 3 (1995), 321–356.
  • A. Ishii, Moves and invariants for knotted handlebodies, Algebr. Geom. Topol., 8 (2008), 1403–1418.
  • A. Ishii, The Markov theorems for spatial graphs and handlebody-knots with Y-orientations, Internat. J. Math., 26 (2015), 1550116, 23 pp.
  • A. Ishii and M. Iwakiri, Quandle cocycle invariants for spatial graphs and knotted handlebodies, Canad. J. Math., 64 (2012), 102–122.
  • A. Ishii, M. Iwakiri, Y. Jang and K. Oshiro, A $G$-family of quandles and handlebody-knots, Illinois J. Math., 57 (2013), 817–838.
  • A. Ishii, M. Iwakiri, S. Kamada, J. Kim, S. Matsuzaki and K. Oshiro, A multiple conjugation biquandle and handlebody-links, Hiroshima Math. J., 48 (2018), 89–117.
  • A. Ishii and K. Kishimoto, The IH-complex of spatial trivalent graphs, Tokyo. J. Math., 33 (2010), 523–535.
  • A. Ishii, K. Kishimoto, H. Moriuchi and M. Suzuki, A table of genus two handlebody-knots up to six crossings, J. Knot Theory Ramifications, 21 (2012), 1250035, 9 pp.
  • A. Ishii and S. Nelson, Partially multiplicative biquandles and handlebody-knots, Contemp. Math., 689 (2017), 159–176.
  • M. Iwakiri, Unknotting numbers for handlebody-knots and Alexander quandle colorings, J. Knot Theory Ramifications, 24 (2015), 1550059, 13 pp.
  • D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg., 23 (1982), 37–65.
  • S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.), 119(161) (1982), 78–88.
  • T. Murao, On bind maps for braids, J. Knot Theory Ramifications, 25 (2016), 1650004, 25 pp.
  • Y. Nakanishi, A note on unknotting number, Math. Sem. Notes Kobe Univ., 9 (1981), 99–108.
  • J. Przytycki, 3-coloring and other elementary invariants of knots, Banach Center Publ., 42 (1998), 275–295.