## Algebraic & Geometric Topology

### Floer homology and splicing knot complements

Eaman Eftekhary

#### Abstract

We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y (K1,K2)$ obtained by splicing the complements of the knots $Ki ⊂ Y i$, $i = 1,2$, in terms of the knot Floer homology of $K1$ and $K2$. We also present a few applications. If $hni$ denotes the rank of the Heegaard Floer group $HFK̂$ for the knot obtained by $n$–surgery over $Ki$, we show that the rank of $HF̂(Y (K1,K2))$ is bounded below by

$|(h∞1 − h 11)(h ∞2 − h 12) − (h 01 − h 11)(h 02 − h 12)|.$

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

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3155-3213.

Dates
Revised: 19 February 2015
Accepted: 1 March 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841064

Digital Object Identifier
doi:10.2140/agt.2015.15.3155

Mathematical Reviews number (MathSciNet)
MR3450759

Zentralblatt MATH identifier
1335.57020

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R58: Floer homology

#### Citation

Eftekhary, Eaman. Floer homology and splicing knot complements. Algebr. Geom. Topol. 15 (2015), no. 6, 3155--3213. doi:10.2140/agt.2015.15.3155. https://projecteuclid.org/euclid.agt/1510841064

#### References

• A Alishahi, E Eftekhary, A refinement of sutured Floer homology, J. Symplectic Geom. 13 (2015) 609–743
• E Eftekhary, Longitude Floer homology and the Whitehead double, Algebr. Geom. Topol. 5 (2005) 1389–1418
• E Eftekhary, A combinatorial approach to surgery formulas in Heegaard Floer homology, Algebr. Geom. Topol. 9 (2009) 2225–2246
• E Eftekhary, Bordered Floer homology and existence of incompressible tori in homology spheres, preprint (2015)
• M Hedden, A,S Levine, Splicing knot complements and bordered Floer homology, preprint To appear in J. Reine Angew. Math.
• R Lipshitz, O Ozsváth, D Thurston, Bordered Heegaard Floer homology: Invariance and pairing, preprint (2014)
• R Lipshitz, O Ozsváth, D Thurston, Notes on bordered Floer homology, preprint (2014)
• C Manolescu, P Ozsváth, S Sarkar, A combinatorial description of knot Floer homology, Ann. of Math. 169 (2009) 633–660
• C Manolescu, P Ozsváth, Z Szabó, D Thurston, On combinatorial link Floer homology, Geom. Topol. 11 (2007) 2339–2412
• P Ozsváth, A,I Stipsicz, Contact surgeries and the transverse invariant in knot Floer homology, J. Inst. Math. Jussieu 9 (2010) 601–632
• 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. 159 (2004) 1027–1158
• P Ozsváth, Z Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005) 1–33
• P,S Ozsváth, Z Szabó, Knot Floer homology and integer surgeries, Algebr. Geom. Topol. 8 (2008) 101–153
• S Sarkar, Maslov index formulas for Whitney $n$–gons, J. Symplectic Geom. 9 (2011) 251–270
• S Sarkar, J Wang, An algorithm for computing some Heegaard Floer homologies, Ann. of Math. 171 (2010) 1213–1236