disk with $n$ marked points, and $\widetilde{Kh}$ denotes reduced Khovanov homology.

The organization of this article is as follows. In the next section we give a proof of our main results, Theorems \[thm1\] and \[thm2\], based on a surgery exact triangle for Heegaard Floer homology with twisted coefficients. Section \[proofsec\] also contains the proof of Corollary \[cor:fiberedgrowth\]. In Section \[braidsec\], we spell out the connection between twist numbers and Khovanov homology. Then in Section \[gradedsec\] we revisit our proof of Theorems \[thm1\] and \[thm2\] to refine our estimates on $N_Y$ and give the proof of Theorem \[thm3\], making use of an absolute grading on Heegaard Floer homology by homotopy classes of oriented plane fields on $Y$ due to Gripp and Huang [@GH]. In the final section we provide more details on the construction of the twisted surgery triangle that plays a primary role in the proof of our main theorems.

[**Acknowledgements:**]{} The authors would like to thank the Tokyo Institute of Technology, and Tamas Kalman, Hitoshi Murakami, and Yuanyuan Bao for organizing a conference held there in 2012, where the idea for this project originated. We are indebted to John Baldwin, Will Kazez, David Krcatovich, Gordana Matic, Olga Plamenevskaya, and 