Research ArticlesNo Access

On pretzel knots and Conjecture $ℤ$

Jesús Rodríguez-Viorato

http://orcid.org/0000-0003-4320-6677

Instituto de Matemáticas, UNAM, México

E-mail Address: jesus.rodriguez.viorato@gmail.com

Search for more papers by this author

and

Francisco Gonzaléz Acuña

Instituto de Matemáticas, UNAM, México

E-mail Address: ficomx@yahoo.com.mx

Search for more papers by this author

https://doi.org/10.1142/S0218216516500127Cited by:1 (Source: Crossref)

Abstract

Conjecture $ℤ$ is a knot theoretical equivalent form of the Kervaire conjecture. We show that Conjecture $ℤ$ is true for all the pretzel knots of the form $P (p, q, - r)$ where $p$ , $q$ and $r$ are odd positive integers.

Keywords:

AMSC: 57M07, 57M25, 57M99

References

1. M. Eudave-Muñoz, On knots with icon surfaces, Osaka J. Math. 50(1) (2013) 271–285. Web of Science, Google Scholar
2. F. González-Acuña and A. Ramírez, A knot-theoretic equivalent of the Kervaire conjecture, J. Knot Theory Ramifications 15(4) (2006) 471–478. Link, Web of Science, Google Scholar
3. A. E. Hatcher, Measured lamination spaces for surfaces, from the topological viewpoint, Topology Appl. 30(1) (1988) 63–88. Crossref, Google Scholar
4. A. Hatcher and U. Oertel, Boundary slopes for Montesinos knots, Topology 28(4) (1989) 453–480. Crossref, Web of Science, Google Scholar
5. M. Hirasawa and K. Murasugi, Genera and fibredness of Montesinos knots, Pacific J. Math. 225(1) (2006) 53–83. Crossref, Web of Science, Google Scholar
6. J. M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs, Vol. 12 (The Clarendon Press, Oxford, 1995). Crossref, Google Scholar
7. E. Klimenko and M. Sakuma, Two-generator discrete subgroups of Isom(H²) containing orientation-reversing elements, Geom. Dedicata 72(3) (1998) 247–282. Crossref, Web of Science, Google Scholar
8. R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Classics in Mathematics (Springer-Verlag, Berlin, 2001). Reprint of the 1977 edition. Crossref, Google Scholar