Floer homology and splicing knot complements

Eftekhary, Eaman

doi:10.2140/agt.2015.15.3155

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Floer homology and splicing knot complements

Eaman Eftekhary

Algebraic & Geometric Topology 15 (2015) 3155–3213

DOI: 10.2140/agt.2015.15.3155

Abstract

We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y (K_{1}, K_{2})$ obtained by splicing the complements of the knots $K_{i} \subset Y_{i}$ , $i = 1, 2$ , in terms of the knot Floer homology of $K_{1}$ and $K_{2}$ . We also present a few applications. If $h_{n}^{i}$ denotes the rank of the Heegaard Floer group $\hat{HFK}$ for the knot obtained by $n$ –surgery over $K_{i}$ , we show that the rank of $\hat{HF} (Y (K_{1}, K_{2}))$ is bounded below by

| (h_{\infty}^{1} - h_{1}^{1}) (h_{\infty}^{2} - h_{1}^{2}) - (h_{0}^{1} - h_{1}^{1}) (h_{0}^{2} - h_{1}^{2}) | .

We also show that if splicing the complement of a knot $K \subset Y$ with the trefoil complements gives a homology sphere $L “$ –space, then $K$ is trivial and $Y$ is a homology sphere $L “$ –space.

Keywords

Floer homology, splicing, essential torus

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 57R58

References

Publication

Received: 4 November 2013

Revised: 19 February 2015

Accepted: 1 March 2015

Published: 12 January 2016

Authors

Eaman Eftekhary
School of Mathematics Institute for Research in Fundamental Sciences (IPM) PO Box 19395-5746 Tehran 19395 Iran
http://math.ipm.ir/~eftekhary

Volume 15, issue 6 (2015)