Heegaard Floer correction terms and rational genus bounds

Abstract

Given an element in the first homology of a rational homology 3-sphere $Y$, one can consider the minimal rational genus of all knots in this homology class. This defines a function $\Theta$ on $H_1(Y;\mathbb{Z})$, which was introduced by Turaev as an analogue of Thurston norm. We will give a lower bound for this function using the correction terms in Heegaard Floer homology. As a corollary, we show that Floer simple knots in L-spaces are genus minimizers in their homology classes, hence answer questions of Turaev and Rasmussen about genus minimizers in lens spaces.

21 pages. V2: corrects a mistake in the proof of Proposition 1.5, incorporates the referee's comments

References in corpus (4)

• Floer homology and knot complements
• Lens space surgeries and L-space homology spheres
• Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology
• Cosmetic surgeries on knots in $S^3$

Cited by in corpus (2)

• Four-ball genus bounds and a refinement of the Ozsvath-Szabo tau-invariant
• Non-simple genus minimizers in lens spaces