Algebraic & Geometric Topology

The Thurston polytope for four-stranded pretzel links

Joan Licata

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In this paper we use Heegaard Floer link homology to determine the dual Thurston polytope for pretzel links of the form P(2r11,2q1,2q2,2r2+1),ri,qi+. We apply this result to determine the Thurston norms of spanning surfaces for the individual link components, and we explicitly construct norm-realizing surfaces for the homology classes which are vertices of the Thurston polytope.

Article information

Algebr. Geom. Topol., Volume 8, Number 1 (2008), 211-243.

Received: 4 October 2006
Revised: 16 August 2007
Accepted: 4 December 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 53D99: None of the above, but in this section 57R58: Floer homology 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Thurston norm pretzel link Heegaard Floer Seifert surface


Licata, Joan. The Thurston polytope for four-stranded pretzel links. Algebr. Geom. Topol. 8 (2008), no. 1, 211--243. doi:10.2140/agt.2008.8.211.

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  • D Gabai, Foliations and the topology of 3–manifolds, J. Differential Geom. 18 (1983) 445–503
  • R Lipshitz, A cylindrical reformulation of Heegaard Floer homology
  • C T McMullen, The Alexander polynomial of a 3–manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. $(4)$ 35 (2002) 153–171
  • Y Ni, A note on knot Floer homology of links
  • P S Ozsváth, Z Szabó, Holomorphic disks, link invariants, and the multi-variable Alexander polynomial
  • P S Ozsváth, Z Szabó, Link Floer homology and the Thurston norm
  • P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58–116
  • P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. $(2)$ 159 (2004) 1027–1158
  • S Sarkar, J Wang, A combinatorial description of some Heegaard Floer homologies
  • W P Thurston, A norm for the homology of 3–manifolds, Mem. Amer. Math. Soc. 59 (1986) i–vi and 99–130