# 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)
• Accepted : 2017.11.02
• Published : 2018.04.30
• 206 17

#### Abstract

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.

#### Keywords

ribbon 2-knot;Alexander polynomial;knot group;twisted Alexander polynomial

#### Acknowledgement

Supported by : JSPS KAKENHI

#### References

1. 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.
2. 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.
3. T. Kanenobu and T. Sumi, Classification of ribbon 2-knots presented by virtual arcs with up to four crossings, preprint, 2017.
4. 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
5. 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.
6. X. S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.) 17 (2001), no. 3, 361-380.
7. Y. Marumoto, On ribbon 2-knots of 1-fusion, Math. Sem. Notes Kobe Univ. 5 (1977), no. 1, 59-68.
8. 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
9. R. Riley, Holomorphically parameterized families of subgroups of SL(2,C), Mathematika 32 (1985), no. 2, 248-264. https://doi.org/10.1112/S0025579300011037
10. S. Satoh, Virtual knot presentation of ribbon torus-knots, J. Knot Theory Ramifications 9 (2000), no. 4, 531-542.
11. S. Suzuki, Knotting problems of 2-spheres in 4-sphere, Math. Sem. Notes Kobe Univ. 4 (1976), no. 3, 241-371.
12. K. Takahashi, Classification of ribbon 2-knot groups by using twisted Alexander polynomial, Master's thesis, Osaka City University, 2014, (in Japanese).
13. 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
14. T. Yajima, On simply knotted spheres in $R^4$, Osaka J. Math. 1 (1964), 133-152.
15. T. Yanagawa, On ribbon 2-knots. The 3-manifold bounded by the 2-knots, Osaka J. Math. 6 (1969), 447-464.
16. 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
17. T. Yasuda, Ribbon 2-knots with ribbon crossing number four, Research reports of Nara Technical College (2008), no. 44, 69-72, (in Japanese).
18. T. Yasuda, Ribbon 2-knots with ribbon crossing number four. II, Research reports of Nara Technical College (2009), no. 45, 59-62, (in Japanese).
19. T. Yasuda, Ribbon 2-knots with ribbon crossing number four. III, Research reports of Nara Technical College (2010), no. 46, 45-48, (in Japanese).
20. T. Yasuda, Ribbon 2-knots with ribbon crossing number four. IV, Research reports of Nara Technical College (2011), no. 47, 37-40, (in Japanese).