Algebraic & Geometric Topology

On the slice-ribbon conjecture for pretzel knots

Ana G Lecuona

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots P(p1,,pn) with one pi even. The 3–stranded case yields two interesting families of examples: The first consists of knots for which the nonsliceness is detected by the Alexander polynomial while several modern obstructions to sliceness vanish. The second family has the property that the correction terms from Heegaard–Floer homology of the double branched covers of these knots do not obstruct the existence of a rational homology ball; however, the Casson–Gordon invariants show that the double branched covers do not bound rational homology balls.

Article information

Source
Algebr. Geom. Topol., Volume 15, Number 4 (2015), 2133-2173.

Dates
Received: 23 April 2014
Revised: 27 October 2014
Accepted: 31 October 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510840999

Digital Object Identifier
doi:10.2140/agt.2015.15.2133

Mathematical Reviews number (MathSciNet)
MR3402337

Zentralblatt MATH identifier
1331.57012

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
Slice-ribbon conjecture pretzel knots rational homology balls

Citation

Lecuona, Ana G. On the slice-ribbon conjecture for pretzel knots. Algebr. Geom. Topol. 15 (2015), no. 4, 2133--2173. doi:10.2140/agt.2015.15.2133. https://projecteuclid.org/euclid.agt/1510840999


Export citation

References

  • O Ahmadi, G Vega, On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields, Finite Fields Appl. 14 (2008) 124–131
  • R,E Bedient, Double branched covers and pretzel knots, Pacific J. Math. 112 (1984) 265–272
  • G Burde, H Zieschang, Knots, 2nd edition, Studies in Math. 5, de Gruyter (2003)
  • A,J Casson, C,M Gordon, On slice knots in dimension three, from: “Algebraic and geometric topology”, (R,J Milgram, editor), Proc. Sympos. Pure Math. 32, Amer. Math. Soc. (1978) 39–53
  • A,J Casson, C,M Gordon, Cobordism of classical knots, from: “À la recherche de la topologie perdue”, Progr. Math. 62, Birkhäuser, Boston, MA (1986) 181–199
  • D Cimasoni, V Florens, Generalized Seifert surfaces and signatures of colored links, Trans. Amer. Math. Soc. 360 (2008) 1223–1264
  • S,K Donaldson, The orientation of Yang–Mills moduli spaces and $4$–manifold topology, J. Differential Geom. 26 (1987) 397–428
  • E Eftekhary, Heegaard Floer homologies for pretzel knots
  • R,H Fox, Some problems in knot theory, from: “Topology of $3$–manifolds and related topics”, Prentice-Hall, Englewood Cliffs, NJ (1962) 168–176
  • R,H Fox, J,W Milnor, Singularities of $2$–spheres in $4$–space and cobordism of knots, Osaka J. Math. 3 (1966) 257–267
  • R,E Gompf, A,I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Math. 20, Amer. Math. Soc. (1999)
  • J Greene, S Jabuka, The slice-ribbon conjecture for $3$–stranded pretzel knots, Amer. J. Math. 133 (2011) 555–580
  • C Herald, P Kirk, C Livingston, Metabelian representations, twisted Alexander polynomials, knot slicing and mutation, Math. Z. 265 (2010) 925–949
  • S Jabuka, Rational Witt classes of pretzel knots, Osaka J. Math. 47 (2010) 977–1027
  • P Kirk, C Livingston, Twisted Alexander invariants, Reidemeister torsion and Casson–Gordon invariants, Topology 38 (1999) 635–661
  • P Kirk, C Livingston, Twisted knot polynomials: Inversion, mutation and concordance, Topology 38 (1999) 663–671
  • J Levine, Invariants of knot cobordism, Invent. Math. 8 (1969) 98–110
  • J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229–244
  • P Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007) 429–472
  • P Lisca, Sums of lens spaces bounding rational balls, Algebr. Geom. Topol. 7 (2007) 2141–2164
  • J,M Montesinos, Seifert manifolds that are ramified two-sheeted cyclic coverings, Bol. Soc. Mat. Mexicana 18 (1973) 1–32
  • K Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965) 387–422
  • W,D Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981) 299–344
  • W,D Neumann, F Raymond, Seifert manifolds, plumbing, $\mu $–invariant and orientation reversing maps, from: “Algebraic and geometric topology”, (K,C Millett, editor), Lecture Notes in Math. 664, Springer, Berlin (1978) 163–196
  • P Ozsváth, Z Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003) 185–224
  • R von Randow, Zur Topologie von dreidimensionalen Baummannigfaltigkeiten, Bonn. Math. Schr. No. 14 (1962) 131
  • K Reidemeister, Knotentheorie, Springer, Berlin (1974)
  • N Saveliev, A surgery formula for the $\bar\mu$–invariant, Topology Appl. 106 (2000) 91–102
  • H,F Trotter, Noninvertible knots exist, Topology 2 (1963) 275–280