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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
⊂ 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
HFK ̂ for the knot
obtained by
n –surgery
over
K i , we show
that the rank of
HF ̂ ( Y ( K 1 , K 2 ) )
is bounded below by
| ( h ∞ 1
− h
1 1 ) ( h
∞ 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
⊂
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
Publication
Received: 4 November 2013
Revised: 19 February 2015
Accepted: 1 March 2015
Published: 12 January 2016