Non-slice 3-stranded pretzel knots. (English) Zbl 1500.57006
The paper studies the Alexander Polynomials of pretzel knots of the form \(P(a, -a-2, -\frac{(a+1)^2}{2})\) and derives conditions for when these knots are topologically slice using the Fox-Milnor factorization criterion [R. H. Fox and J. W. Milnor, Osaka J. Math. 3, 257–267 (1966; Zbl 0146.45501)]. The main result is that, for \(a\) an integer whose congruence class mod 120 is not 1 or 97, the 3-stranded pretzel knot \(P(a, -a-2, -\frac{(a+1)^2}{2})\) is ribbon if and only if it is topologically slice. In particular, it follows that the Slice-Ribbon Conjecture [R. H. Fox, in: Topology of 3-manifolds and related topics. Proceedings of the University of Georgia Institute 1961. Englewood Cliffs, N.J.: Prentice-Hall, Inc.. 168–176 (1962; Zbl 1246.57011)] holds for all 3-stranded pretzel knots not excluded by the above statement. The latter conclusion combines the results of this paper with prior work by A. G. Lecuona [Algebr. Geom. Topol. 15, No. 4, 2133–2173 (2015; Zbl 1331.57012)] and A. N. Miller [ibid. 17, No. 5, 3057–3079 (2017; Zbl 1376.57010)].
Reviewer: Alexandra Kjuchukova (Notre Dame)
MSC:
| 57K10 | Knot theory |
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
| 57K40 | General topology of 4-manifolds |
| 57N70 | Cobordism and concordance in topological manifolds |
References:
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