Hostname: page-component-857557d7f7-nhjpk Total loading time: 0 Render date: 2025-11-22T06:50:10.570Z Has data issue: false hasContentIssue false
  • English
  • Français

On the genera of symmetric unions of knots

Part of: Low-dimensional topology Differential topology

Published online by Cambridge University Press:  03 November 2025

Michel Boileau ,
Teruaki Kitano Open the ORCID record for Teruaki Kitano [Opens in a new window]  and
Yuta Nozaki Open the ORCID record for Yuta Nozaki [Opens in a new window]
Michel Boileau
Affiliation:
Aix-Marseille University , CNRS, Centrale Marseille, I2M, UMR 7373, France e-mail: michel.boileau@univ-amu.fr
Teruaki Kitano
Affiliation:
Department of Information Systems Science, Faculty of Science and Engineering, Soka University , Japan e-mail: kitano@soka.ac.jp
Yuta Nozaki*
Affiliation:
Faculty of Environment and Information Sciences, Yokohama National University , Yokohama, Japan WPI-SKCM2, Hiroshima University , Japan
*
e-mail: nozaki-yuta-vn@ynu.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful condition to constrain possible symmetric union presentations of a given ribbon knot. It is an open question whether every ribbon knot is a symmetric union.

Keywords

Symmetric unionknot groupgenustwisted Alexander polynomial

MSC classification

Secondary: 57M05: Fundamental group, presentations, free differential calculus 57R18: Topology and geometry of orbifolds

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 26
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This study was supported in part by JSPS KAKENHI Grant Numbers JP19K03505, JP20K14317, and JP23K12974. The first and second authors were supported by the Soka University International Collaborative Research Grant.

References

Problems in low-dimensional topology . In: Kirby, R. and Kazez, William H. (eds.), Geometric topology (Athens, GA, 1993), volume 2 of AMS/IP Studies in Advanced Mathematics, Amer. Math. Soc., Providence, RI, 1997, pp. 35–473.Google Scholar
Boileau, M., Boyer, S., Reid, A. W., and Wang, S., Simon’s conjecture for two-bridge knots . Comm. Anal. Geom. 18(2010), no. 1, 121–143.Google Scholar
Boileau, M., Boyer, S., Rolfsen, D., and Wang, S. C., One-domination of knots . Illinois J. Math. 60(2016), no. 1, 117–139.Google Scholar
Boileau, M., Kitano, T., and Nozaki, Y., A preorder on the set of links defined via orbifolds. Preprint, 2025. arXiv:2510.12372v1.Google Scholar
Boileau, M. and Porti, J., Geometrization of 3-orbifolds of cyclic type . Astérisque (2001), no. 272, 208. Appendix A by Michael Heusener and Porti.Google Scholar
Boileau, M., Rubinstein, J. H., and Wang, S., Finiteness of 3-manifolds associated with non-zero degree mappings . Comment. Math. Helv. 89(2014), no. 1, 33–68.Google Scholar
Boileau, M. and Zieschang, H., Heegaard genus of closed orientable Seifert $3$ -manifolds . Invent. Math. 76(1984), no. 3, 455–468.Google Scholar
Boileau, M. and Zimmermann, B., The $\pi$ -orbifold group of a link . Math. Z. 200(1989), no. 2, 187–208.Google Scholar
Brejevs, V. and Feller, P., The twisting number of a ribbon knot is bounded below by its doubly slice genus. Preprint, 2024. arXiv:2404.07619v1.Google Scholar
Burde, G., Zieschang, H., and Heusener, M., Knots, volume 5 of De Gruyter Studies in Mathematics, De Gruyter, Berlin, extended edition, 2014.Google Scholar
Casson, A. and Jungreis, D., Convergence groups and Seifert fibered $3$ -manifolds . Invent. Math. 118(1994), no. 3, 441–456.Google Scholar
Crowell, R. H. and Fox, R. H., Introduction to knot theory, Graduate Texts in Mathematics, 57, Springer-Verlag, New York, NY, 1977. Reprint of the 1963 original.Google Scholar
Culler, M. and Shalen, P. B., Varieties of group representations and splittings of $3$ -manifolds . Ann. of Math. (2) 117(1983), no. 1, 109–146.Google Scholar
Feustel, C. D., Some applications of Waldhausen’s results on irreducible surfaces . Trans. Amer. Math. Soc. 149(1970), 575–583.Google Scholar
Friedl, S., Centralizers in 3-manifold groups (twisted topological invariants and topology of low-dimensional manifolds) . RIMS Kôkyûroku 1747(2011), 23–34.Google Scholar
Friedl, S. and Lück, W., ${L}^2$ -Euler characteristics and the Thurston norm . Proc. Lond. Math. Soc. (3) 118(2019), no. 4, 857–900.Google Scholar
Friedl, S. and Nagel, M., Twisted Reidemeister torsion and the Thurston norm: Graph manifolds and finite representations . Illinois J. Math. 59(2015), no. 3, 691–705.Google Scholar
Friedl, S. and Vidussi, S.. A survey of twisted Alexander polynomials . In: M. Banagl and D. Vogel (eds.), The mathematics of knots, volume 1 of Contributions in Mathematical and Computational Sciences, Springer, Heidelberg, 2011, 45–94.Google Scholar
Friedl, S. and Vidussi, S., The Thurston norm and twisted Alexander polynomials . J. Reine Angew. Math. 707(2015), 87–102.Google Scholar
Gabai, D., Foliations and the topology of $3$ -manifolds . J. Differential Geom. 18(1983), no. 3, 445–503.Google Scholar
Gabai, D., Genera of the arborescent links . Mem. Amer. Math. Soc. 59(1986), no. 339, 1–98.Google Scholar
Gabai, D., Foliations and the topology of $3$ -manifolds. III . J. Differential Geom. 26(1987), no. 3, 479–536.Google Scholar
Gabai, D., Convergence groups are Fuchsian groups . Ann. of Math. (2) 136(1992), no. 3, 447–510.Google Scholar
Goda, H. and Sakasai, T., Homology cylinders and sutured manifolds for homologically fibered knots . Tokyo J. Math. 36(2013), no. 1, 85–111.Google Scholar
Horie, K., Kitano, T., Matsumoto, M., and Suzuki, M., A partial order on the set of prime knots with up to 11 crossings . J. Knot Theory Ramif. 20(2011), no. 2, 275–303.Google Scholar
Jaco, W., Constructing $3$ -manifolds from group homomorphisms . Bull. Amer. Math. Soc. 74(1968), 936–940.Google Scholar
Jaco, W., Lectures on three-manifold topology, volume 43 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 1980.Google Scholar
Jaco, W. H. and Shalen, P. B., Seifert fibered spaces in $3$ -manifolds . Mem. Amer. Math. Soc. 21(1979), no. 220, viii+192.Google Scholar
Jankins, M. and Neumann, W. D., Lectures on Seifert manifolds, volume 2 of Brandeis Lecture Notes, Brandeis University, Waltham, MA, 1983.Google Scholar
Kinoshita, S. and Terasaka, H., On unions of knots . Osaka Math. J. 9(1957), 131–153.Google Scholar
Kitano, T. and Suzuki, M., A partial order in the knot table . Experiment. Math. 14(2005), no. 4, 385–390.Google Scholar
Kitano, T. and Suzuki, M., A partial order in the knot table. II . Acta Math. Sin. (Engl. Ser.) 24(2008), no. 11, 1801–1816.Google Scholar
Kitano, T. and Suzuki, M., Twisted Alexander polynomials and a partial order on the set of prime knots . In: N. Iwase, T. Kohno, R. Levi, D. Tamaki, and J. Wu (eds.), Groups, homotopy and configuration spaces, volume 13 of Geometry & Topology Monographs, Geometry & Topology Publications, Coventry, 2008, pp. 307–321.Google Scholar
Knapp, A. W., Doubly generated Fuchsian groups . Michigan Math. J. 15(1968), 289–304.Google Scholar
Köse, F. C.. Symmetric union of knots, double branched covers and Dehn surgery. Ph.D. thesis, The University of Texas at Austin, 2022.Google Scholar
Kronheimer, P. B. and Mrowka, T. S., Dehn surgery, the fundamental group and SU $(2)$ . Math. Res. Lett. 11(2004), nos. 5–6, 741–754.Google Scholar
Lamm, C., Symmetric unions and ribbon knots . Osaka J. Math. 37(2000), no. 3, 537–550.Google Scholar
Lamm, C., The search for nonsymmetric ribbon knots . Exp. Math. 30(2021), no. 3, 349–363.Google Scholar
Lamm, C., Symmetric union presentations for 2-bridge ribbon knots . J. Knot Theory Ramif. 30(2021), no. 12, Article no. 2141009, 11 pp.Google Scholar
Montesinos, J. M., Seifert manifolds that are ramified two-sheeted cyclic coverings . Bol. Soc. Mat. Mexicana 2(1973), no. 18, 1–32.Google Scholar
Moore, A. H., Symmetric unions without cosmetic crossing changes . In: G. Letzter, K. Lauter, E. Chambers, N. Flournoy, J. Elisenda Grigsby, C. Martin, K. Ryan, and K. Trivisa (eds.), Advances in the mathematical sciences, volume 6 of Association for Women in Mathematics Series, Springer, Cham, 2016, pp. 103–116.Google Scholar
Oertel, U., Closed incompressible surfaces in complements of star links . Pacific. J. Math. 111(1984), no. 1, 209–230.Google Scholar
Orlik, P., Seifert manifolds, Lecture Notes in Mathematics, 291, Springer-Verlag, Berlin, 1972.Google Scholar
Rong, Y. W., Degree one maps between geometric $3$ -manifolds . Trans. Amer. Math. Soc. 332(1992), no. 1, 411–436.Google Scholar
Samelson, H. (ed.), Queries. American Mathematical Society, Providence, RI, 1976, pp. 410–411. Notices of the AMS.Google Scholar
Seeliger, A., Symmetrische Vereinigungen als Darstellungen von Bandknoten bis 14 Kreuzungen (symmetric union presentations for ribbon knots up to 14 crossings). Diploma thesis, Stuttgart University, 2014.Google Scholar
Simon, J., Roots and centralizers of peripheral elements in knot groups . Math. Ann. 222(1976), no. 3, 205–209.Google Scholar
Wada, M., Twisted Alexander polynomial for finitely presentable groups . Topology 33(1994), no. 2, 241–256.Google Scholar
Waldhausen, F., Eine Klasse von $3$ -dimensionalen Mannigfaltigkeiten. I, II . Invent. Math. 3(1967), 308–333. ibid. 4 (1967), 87–117.Google Scholar
Wall, C. T. C., In: Ranicki, A. A. (ed.), Surgery on compact manifolds. 2nd ed., volume 69 of Mathematical surveys and Monographs, American Mathematical Society, Providence, RI, 1999.Google Scholar