- Volume 33 Issue 2
DOI QR Code
CLASSIFICATION OF A FAMILY OF RIBBON 2-KNOTS WITH TRIVIAL ALEXANDER POLYNOMIAL
- Kanenobu, Taizo (Department of Mathematics Osaka City University) ;
- Sumi, Toshio (Faculty of Arts and Science Kyushu University)
- Received : 2017.05.25
- Accepted : 2017.11.02
- Published : 2018.04.30
We consider a family of ribbon 2-knots with trivial Alexander polynomial. We give nonabelian SL(2, C)-representations from the groups of these knots, and then calculate the twisted Alexander polynomials associated to these representations, which allows us to classify this family of knots.
ribbon 2-knot;Alexander polynomial;knot group;twisted Alexander polynomial
Supported by : JSPS KAKENHI
- K. Habiro, T. Kanenobu, and A. Shima, Finite type invariants of ribbon 2-knots, in Low-dimensional topology (Funchal, 1998), 187-196, Contemp. Math., 233, Amer. Math. Soc., Providence, RI, 1999.
- T. Kanenobu and S. Komatsu, Enumeration of ribbon 2-knots presented by virtual arcs with up to four crossings, J. Knot Theory Ramifications 26 (2017), no. 8, 1750042, 41 pp.
- T. Kanenobu and T. Sumi, Classification of ribbon 2-knots presented by virtual arcs with up to four crossings, preprint, 2017.
- S. Kinoshita, On the Alexander polynomials of 2-spheres in a 4-sphere, Ann. of Math. (2) 74 (1961), 518-531. https://doi.org/10.2307/1970296
- T. Kitano and T. Morifuji, Divisibility of twisted Alexander polynomials and fibered knots, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 1, 179-186.
- X. S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.) 17 (2001), no. 3, 361-380.
- Y. Marumoto, On ribbon 2-knots of 1-fusion, Math. Sem. Notes Kobe Univ. 5 (1977), no. 1, 59-68.
- R. Riley, Nonabelian representations of 2-bridge knot groups, Quart. J. Math. Oxford Ser. (2) 35 (1984), no. 138, 191-208. https://doi.org/10.1093/qmath/35.2.191
- R. Riley, Holomorphically parameterized families of subgroups of SL(2,C), Mathematika 32 (1985), no. 2, 248-264. https://doi.org/10.1112/S0025579300011037
- S. Satoh, Virtual knot presentation of ribbon torus-knots, J. Knot Theory Ramifications 9 (2000), no. 4, 531-542.
- S. Suzuki, Knotting problems of 2-spheres in 4-sphere, Math. Sem. Notes Kobe Univ. 4 (1976), no. 3, 241-371.
- K. Takahashi, Classification of ribbon 2-knot groups by using twisted Alexander polynomial, Master's thesis, Osaka City University, 2014, (in Japanese).
- M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994), no. 2, 241-256. https://doi.org/10.1016/0040-9383(94)90013-2
T. Yajima, On simply knotted spheres in
$R^4$, Osaka J. Math. 1 (1964), 133-152.
- T. Yanagawa, On ribbon 2-knots. The 3-manifold bounded by the 2-knots, Osaka J. Math. 6 (1969), 447-464.
- T. Yasuda, Crossing and base numbers of ribbon 2-knots, J. Knot Theory Ramifications 10 (2001), no. 7, 999-1003. https://doi.org/10.1142/S021821650100130X
- T. Yasuda, Ribbon 2-knots with ribbon crossing number four, Research reports of Nara Technical College (2008), no. 44, 69-72, (in Japanese).
- T. Yasuda, Ribbon 2-knots with ribbon crossing number four. II, Research reports of Nara Technical College (2009), no. 45, 59-62, (in Japanese).
- T. Yasuda, Ribbon 2-knots with ribbon crossing number four. III, Research reports of Nara Technical College (2010), no. 46, 45-48, (in Japanese).
- T. Yasuda, Ribbon 2-knots with ribbon crossing number four. IV, Research reports of Nara Technical College (2011), no. 47, 37-40, (in Japanese).