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Quantum invariants can provide sharp Heegaard genus bounds

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  • Computer Science
  • Mathematics
  • Physics


Wong, H. Osaka J. Math. 48 (2011), 709–717 QUANTUM INVARIANTS CAN PROVIDE SHARP HEEGAARD GENUS BOUNDS HELEN WONG (Received May 29, 2009, revised March 3, 2010) Abstract Using Seifert fibered three-manifold examples of Boileau and Zieschang, we demonstrate that the Reshetikhin–Turaev quantum invariants may be used to provide a sharp lower bound on the Heegaard genus which is strictly larger than the rank of the fundamental group. 1. Introduction For a closed oriented, connected three-manifold M , the Heegaard genus g(M) is defined to be the smallest integer so that M has a Heegaard splitting of that genus. Classically studied, the Heegaard genus is notoriously difficult to compute. In this paper, we investigate the effectiveness of a lower bound on g(M) deriving from the Reshetikhin–Turaev quantum invariants, as was discovered in [6] and in [18]. The Reshetikhin–Turaev quantum invariants for three-manifolds were originally con- ceived by Witten in [20] as a generalization of the Jones polynomial for knots and links. As such, they allow an algorithmic and combinatorial definition, though the actual calcu- lation is often computationally expensive. Of their known topological applications, the lower bound on g(M) deriving from the quantum invariants may be one of the most powerful and useful. Until the advent of the quantum invariants, the best known bounds on Heegaard genus came from algebraic topology. For a group G, let its rank r (G) be the mini- mal number of elements required to generate G. The rank r (�1 M) of the fundamental group of a three-manifold is a lower bound on g(M). By studying the Seifert fibered space examples of Boileau and Zieschang in [4], we show that quantum invariants may be used to provide a lower bound on g(M) which is strictly larger than r (�1 M). Fur- ther, in this particular case, the calculation of the quantum invariant is significantly sim- pler and shorter than the determination of r (�1 M), as appears in [4]. It is shown in [5] that for a rand

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