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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d1101101.png" /> be a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d1101102.png" />-dimensional [[Manifold|manifold]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d1101103.png" /> be a solid torus in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d1101104.png" />. Remove <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d1101105.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d1101106.png" /> and glue in instead of it another solid torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d1101107.png" /> by a [[Homeomorphism|homeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d1101108.png" />. One says that the resulting new <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d1101109.png" />-dimensional manifold
+
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$#C+1 = 127 : ~/encyclopedia/old_files/data/D110/D.1100110 Dehn surgery
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011010.png" /></td> </tr></table>
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is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011011.png" /> by a Dehn surgery. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011012.png" /> is determined by the following data: 1) a knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011013.png" /> (a core circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011014.png" /> of the solid torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011015.png" />, cf. also [[Knot theory|Knot theory]]); and 2) a non-trivial simple closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011016.png" /> (the image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011017.png" /> of a meridian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011018.png" />). The Dehn surgery is called integer if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011019.png" /> is a longitude of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011020.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011021.png" /> intersects a meridional curve of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011022.png" /> transversally in a single point.
+
Let  $  M $
 +
be a closed  $  3 $-
 +
dimensional [[Manifold|manifold]] and let  $  N $
 +
be a solid torus in the interior of $  M $.  
 +
Remove  $  { \mathop{\rm Int} } N $
 +
from  $  M $
 +
and glue in instead of it another solid torus  $  N _ {1} $
 +
by a [[Homeomorphism|homeomorphism]]  $  h : {\partial  N _ {1} } \rightarrow {\partial  N } $.  
 +
One says that the resulting new  $  3 $-
 +
dimensional manifold
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011023.png" />, then among all longitudes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011024.png" /> there is a preferred one, which bounds a surface in the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011025.png" />. The preferred longitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011026.png" /> forms together with a meridian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011028.png" /> a coordinate system on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011029.png" />. Therefore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011030.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011032.png" /> are coprime integers, and is determined by the rational number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011033.png" />. The Dehn surgery is integer if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011034.png" /> is an integer. This explains the terminology.
+
$$
 +
M _ {1} = ( M \setminus  { \mathop{\rm Int} } N ) \cup _ {h} N _ {1}  $$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011035.png" /> be two handle-bodies having the same genus (cf. [[Handle theory|Handle theory]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011036.png" /> be a [[Homeomorphism|homeomorphism]]. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011037.png" /> the closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011038.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011039.png" /> obtained by gluing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011041.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011042.png" />. Choose a simple closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011043.png" /> and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011044.png" /> the Dehn twist along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011045.png" />. To be more precise, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011046.png" /> is a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011047.png" /> obtained by cutting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011048.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011049.png" />, isotopically rotating one side of the cut by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011050.png" />, and gluing back. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011051.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011053.png" /> coincide outside a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011054.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011057.png" /> do actually coincide outside regular neighbourhoods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011058.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011060.png" />, respectively. It follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011061.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011062.png" /> by a Dehn surgery along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011063.png" />. One can easily show that the surgery is integer.
+
is obtained from  $  M $
 +
by a Dehn surgery. Note that  $  M _ {1} $
 +
is determined by the following data: 1) a knot  $  K \subset  M $(
 +
a core circle  $  \{ * \} \times S  ^ {1} $
 +
of the solid torus  $  N = D  ^ {2} \times S  ^ {1} $,
 +
cf. also [[Knot theory|Knot theory]]); and 2) a non-trivial simple closed curve $  l \subset  \partial  N $(
 +
the image under  $  h $
 +
of a meridian of  $  N _ {1} $).  
 +
The Dehn surgery is called integer if  $  l $
 +
is a longitude of  $  N $,  
 +
i.e., $  l $
 +
intersects a meridional curve of $  N $
 +
transversally in a single point.
  
Define a framed link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011064.png" /> to be a [[Link|link]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011065.png" /> such that every component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011066.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011067.png" /> is supplied with an integer number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011068.png" />, called a framing. If one performs Dehn surgeries along all components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011069.png" />, taking for each component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011070.png" /> the framing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011071.png" /> as the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011072.png" /> of the surgery, one obtains a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011073.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011074.png" />. Since any orientation-preserving homeomorphism of the boundary of a handle-body is isotopic to a product of Dehn twists [[#References|[a1]]], it follows from the above relation between Dehn twists and integer Dehn surgeries that for every closed orientable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011075.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011076.png" /> there exists a framed link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011077.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011078.png" />.
+
If  $  M = S  ^ {3} $,
 +
then among all longitudes of  $  N $
 +
there is a preferred one, which bounds a surface in the complement of  $  N $.  
 +
The preferred longitude  $  l _ {0} $
 +
forms together with a meridian  $  m $
 +
of $  N $
 +
a coordinate system on  $  \partial  N $.  
 +
Therefore, $  l $
 +
has the form  $  l = m  ^ {p} l _ {0}  ^ {q} $,
 +
where  $  p,q $
 +
are coprime integers, and is determined by the rational number  $  r = {p / q } $.  
 +
The Dehn surgery is integer if and only if  $  r $
 +
is an integer. This explains the terminology.
  
The following question naturally arises: When do two framed links determine homeomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011079.png" />-dimensional manifolds? In 1978 R. Kirby answered this question by showing that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011080.png" /> if and only if one can pass from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011081.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011082.png" /> by a sequence of the following moves and their inverses [[#References|[a2]]]:
+
Let  $  H,H  ^  \prime  $
 +
be two handle-bodies having the same genus (cf. [[Handle theory|Handle theory]]) and let  $  h : {\partial  H } \rightarrow {\partial  H  ^  \prime  } $
 +
be a [[Homeomorphism|homeomorphism]]. Denote by  $  M $
 +
the closed  $  3 $-
 +
dimensional manifold  $  H \cap _ {h} H  ^  \prime  $
 +
obtained by gluing  $  H $
 +
and  $  H  ^  \prime  $
 +
along  $  h $.  
 +
Choose a simple closed curve  $  s \subset  \partial  H $
 +
and denote by $  \tau _ {s} $
 +
the Dehn twist along  $  s $.  
 +
To be more precise,  $  \tau _ {s} $
 +
is a homeomorphism  $  \partial  H \rightarrow \partial  H $
 +
obtained by cutting  $  \partial  H $
 +
along  $  s $,
 +
isotopically rotating one side of the cut by  $  2 \pi $,
 +
and gluing back. Let  $  M _ {1} = H \cap _ {h \tau _ {s}  } H  ^  \prime  $.  
 +
Since  $  h $
 +
and  $  h \tau _ {s} $
 +
coincide outside a neighbourhood of  $  s $
 +
in  $  \partial  H $,
 +
$  M $
 +
and  $  M _ {1} $
 +
do actually coincide outside regular neighbourhoods of  $  s $
 +
in  $  M $
 +
and  $  M _ {1} $,
 +
respectively. It follows that  $  M _ {1} $
 +
is obtained from  $  M $
 +
by a Dehn surgery along  $  s $.
 +
One can easily show that the surgery is integer.
  
1) replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011083.png" /> by the link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011084.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011085.png" /> is a new unknotted component with framing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011086.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011087.png" /> is contained in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011088.png" />-dimensional ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011090.png" />;
+
Define a framed link $  {\mathsf L} $
 +
to be a [[Link|link]]  $  L \subset  S  ^ {3} $
 +
such that every component $  K $
 +
of  $  L $
 +
is supplied with an integer number  $  \varphi ( K ) $,
 +
called a framing. If one performs Dehn surgeries along all components of  $  L $,
 +
taking for each component  $  K \subset  L $
 +
the framing  $  \varphi ( K ) $
 +
as the parameter  $  r $
 +
of the surgery, one obtains a  $  3 $-
 +
dimensional manifold  $  \chi ( {\mathsf L} ) $.  
 +
Since any orientation-preserving homeomorphism of the boundary of a handle-body is isotopic to a product of Dehn twists [[#References|[a1]]], it follows from the above relation between Dehn twists and integer Dehn surgeries that for every closed orientable  $  3 $-
 +
dimensional manifold  $  M $
 +
there exists a framed link  $  {\mathsf L} \subset  S  ^ {3} $
 +
such that  $  M = \chi ( {\mathsf L} ) $.
  
2) replace a component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011091.png" /> by a geometric sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011093.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011094.png" /> with another component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011095.png" /> (see [[#References|[a2]]] for the exact definition of the geometric sum).
+
The following question naturally arises: When do two framed links determine homeomorphic  $  3 $-
 +
dimensional manifolds? In 1978 R. Kirby answered this question by showing that  $  \chi ( { {\mathsf L} _ {1} } ) = \chi ( { {\mathsf L} _ {2} } ) $
 +
if and only if one can pass from  $  { {\mathsf L} _ {1} } $
 +
to  $  { {\mathsf L} _ {2} } $
 +
by a sequence of the following moves and their inverses [[#References|[a2]]]:
  
This result became broadly known as the Kirby calculus for framed links, thanks to its convenience for presenting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011096.png" />-dimensional manifolds.
+
1) replace  $  {\mathsf L} $
 +
by the link  $  {\mathsf L} \cup {\mathsf O} $,  
 +
where  $  {\mathsf O} $
 +
is a new unknotted component with framing  $  \pm  1 $
 +
such that  $  {\mathsf O} $
 +
is contained in a  $  3 $-
 +
dimensional ball  $  B  ^ {3} \subset  S  ^ {3} $,
 +
$  B  ^ {3} \cap {\mathsf L} = \emptyset $;
  
Recall that a framing of a knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011097.png" /> determines a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011098.png" /> of the standard solid torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d11011099.png" /> onto a regular neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110100.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110101.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110102.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110103.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110104.png" /> obtained by attaching a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110105.png" />-dimensional handle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110106.png" /> of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110107.png" /> to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110108.png" />-dimensional ball via the homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110109.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110111.png" />. It follows from the definition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110112.png" />. Similarly, for any framed link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110113.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110114.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110115.png" /> is the boundary of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110116.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110117.png" /> obtained by attaching handles of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110118.png" /> to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110119.png" />-dimensional ball. Move 1) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110120.png" /> corresponds to replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110121.png" /> by a connected sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110122.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110123.png" />. Move 2) corresponds to a sliding of one handle of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110124.png" /> over another and does not change <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110125.png" />. One can show that any framed link in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110126.png" /> can be transformed by moves 1), 2) and their inverses to a link with even framings [[#References|[a3]]]. In the latter case the tangent bundle of the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110110/d110110127.png" />-dimensional manifold is trivial.
+
2) replace a component  $  l _ {i} \subset  {\mathsf L} $
 +
by a geometric sum  $  l _ {i} + l _ {j} $,
 +
$  i \neq j $,
 +
of  $  l _ {i} $
 +
with another component  $  l _ {j} \subset  {\mathsf L} $(
 +
see [[#References|[a2]]] for the exact definition of the geometric sum).
 +
 
 +
This result became broadly known as the Kirby calculus for framed links, thanks to its convenience for presenting  $  3 $-
 +
dimensional manifolds.
 +
 
 +
Recall that a framing of a knot $  K \subset  S  ^ {3} $
 +
determines a homeomorphism $  h $
 +
of the standard solid torus $  D  ^ {2} \times S  ^ {1} $
 +
onto a regular neighbourhood $  N $
 +
of $  K $.  
 +
Denote by $  W  ^ {4} ( {\mathsf K} ) $
 +
the $  4 $-
 +
dimensional manifold $  B  ^ {4} \cup _ {h} H  ^ {4} $
 +
obtained by attaching a $  4 $-
 +
dimensional handle $  H $
 +
of index $  2 $
 +
to the $  4 $-
 +
dimensional ball via the homeomorphism $  h $
 +
between $  D  ^ {2} \times S  ^ {1} \subset  D  ^ {2} \times D  ^ {2} = H  ^ {4} $
 +
and $  N \subset  S  ^ {3} = \partial  D  ^ {4} $.  
 +
It follows from the definition that $  \partial  W  ^ {4} ( {\mathsf K} ) = \chi ( {\mathsf K} ) $.  
 +
Similarly, for any framed link $  {\mathsf L} $
 +
the $  3 $-
 +
dimensional manifold $  \chi ( {\mathsf L} ) $
 +
is the boundary of the $  4 $-
 +
dimensional manifold $  W  ^ {4} ( {\mathsf L} ) $
 +
obtained by attaching handles of index $  2 $
 +
to the $  4 $-
 +
dimensional ball. Move 1) on $  {\mathsf L} $
 +
corresponds to replacing $  W ( {\mathsf L} ) $
 +
by a connected sum of $  W ( {\mathsf L} ) $
 +
with $  \pm  CP  ^ {2} $.  
 +
Move 2) corresponds to a sliding of one handle of index $  2 $
 +
over another and does not change $  W ( {\mathsf L} ) $.  
 +
One can show that any framed link in $  S  ^ {3} $
 +
can be transformed by moves 1), 2) and their inverses to a link with even framings [[#References|[a3]]]. In the latter case the tangent bundle of the corresponding $  4 $-
 +
dimensional manifold is trivial.
  
 
See [[#References|[a4]]] for more details.
 
See [[#References|[a4]]] for more details.

Revision as of 17:32, 5 June 2020


Let $ M $ be a closed $ 3 $- dimensional manifold and let $ N $ be a solid torus in the interior of $ M $. Remove $ { \mathop{\rm Int} } N $ from $ M $ and glue in instead of it another solid torus $ N _ {1} $ by a homeomorphism $ h : {\partial N _ {1} } \rightarrow {\partial N } $. One says that the resulting new $ 3 $- dimensional manifold

$$ M _ {1} = ( M \setminus { \mathop{\rm Int} } N ) \cup _ {h} N _ {1} $$

is obtained from $ M $ by a Dehn surgery. Note that $ M _ {1} $ is determined by the following data: 1) a knot $ K \subset M $( a core circle $ \{ * \} \times S ^ {1} $ of the solid torus $ N = D ^ {2} \times S ^ {1} $, cf. also Knot theory); and 2) a non-trivial simple closed curve $ l \subset \partial N $( the image under $ h $ of a meridian of $ N _ {1} $). The Dehn surgery is called integer if $ l $ is a longitude of $ N $, i.e., $ l $ intersects a meridional curve of $ N $ transversally in a single point.

If $ M = S ^ {3} $, then among all longitudes of $ N $ there is a preferred one, which bounds a surface in the complement of $ N $. The preferred longitude $ l _ {0} $ forms together with a meridian $ m $ of $ N $ a coordinate system on $ \partial N $. Therefore, $ l $ has the form $ l = m ^ {p} l _ {0} ^ {q} $, where $ p,q $ are coprime integers, and is determined by the rational number $ r = {p / q } $. The Dehn surgery is integer if and only if $ r $ is an integer. This explains the terminology.

Let $ H,H ^ \prime $ be two handle-bodies having the same genus (cf. Handle theory) and let $ h : {\partial H } \rightarrow {\partial H ^ \prime } $ be a homeomorphism. Denote by $ M $ the closed $ 3 $- dimensional manifold $ H \cap _ {h} H ^ \prime $ obtained by gluing $ H $ and $ H ^ \prime $ along $ h $. Choose a simple closed curve $ s \subset \partial H $ and denote by $ \tau _ {s} $ the Dehn twist along $ s $. To be more precise, $ \tau _ {s} $ is a homeomorphism $ \partial H \rightarrow \partial H $ obtained by cutting $ \partial H $ along $ s $, isotopically rotating one side of the cut by $ 2 \pi $, and gluing back. Let $ M _ {1} = H \cap _ {h \tau _ {s} } H ^ \prime $. Since $ h $ and $ h \tau _ {s} $ coincide outside a neighbourhood of $ s $ in $ \partial H $, $ M $ and $ M _ {1} $ do actually coincide outside regular neighbourhoods of $ s $ in $ M $ and $ M _ {1} $, respectively. It follows that $ M _ {1} $ is obtained from $ M $ by a Dehn surgery along $ s $. One can easily show that the surgery is integer.

Define a framed link $ {\mathsf L} $ to be a link $ L \subset S ^ {3} $ such that every component $ K $ of $ L $ is supplied with an integer number $ \varphi ( K ) $, called a framing. If one performs Dehn surgeries along all components of $ L $, taking for each component $ K \subset L $ the framing $ \varphi ( K ) $ as the parameter $ r $ of the surgery, one obtains a $ 3 $- dimensional manifold $ \chi ( {\mathsf L} ) $. Since any orientation-preserving homeomorphism of the boundary of a handle-body is isotopic to a product of Dehn twists [a1], it follows from the above relation between Dehn twists and integer Dehn surgeries that for every closed orientable $ 3 $- dimensional manifold $ M $ there exists a framed link $ {\mathsf L} \subset S ^ {3} $ such that $ M = \chi ( {\mathsf L} ) $.

The following question naturally arises: When do two framed links determine homeomorphic $ 3 $- dimensional manifolds? In 1978 R. Kirby answered this question by showing that $ \chi ( { {\mathsf L} _ {1} } ) = \chi ( { {\mathsf L} _ {2} } ) $ if and only if one can pass from $ { {\mathsf L} _ {1} } $ to $ { {\mathsf L} _ {2} } $ by a sequence of the following moves and their inverses [a2]:

1) replace $ {\mathsf L} $ by the link $ {\mathsf L} \cup {\mathsf O} $, where $ {\mathsf O} $ is a new unknotted component with framing $ \pm 1 $ such that $ {\mathsf O} $ is contained in a $ 3 $- dimensional ball $ B ^ {3} \subset S ^ {3} $, $ B ^ {3} \cap {\mathsf L} = \emptyset $;

2) replace a component $ l _ {i} \subset {\mathsf L} $ by a geometric sum $ l _ {i} + l _ {j} $, $ i \neq j $, of $ l _ {i} $ with another component $ l _ {j} \subset {\mathsf L} $( see [a2] for the exact definition of the geometric sum).

This result became broadly known as the Kirby calculus for framed links, thanks to its convenience for presenting $ 3 $- dimensional manifolds.

Recall that a framing of a knot $ K \subset S ^ {3} $ determines a homeomorphism $ h $ of the standard solid torus $ D ^ {2} \times S ^ {1} $ onto a regular neighbourhood $ N $ of $ K $. Denote by $ W ^ {4} ( {\mathsf K} ) $ the $ 4 $- dimensional manifold $ B ^ {4} \cup _ {h} H ^ {4} $ obtained by attaching a $ 4 $- dimensional handle $ H $ of index $ 2 $ to the $ 4 $- dimensional ball via the homeomorphism $ h $ between $ D ^ {2} \times S ^ {1} \subset D ^ {2} \times D ^ {2} = H ^ {4} $ and $ N \subset S ^ {3} = \partial D ^ {4} $. It follows from the definition that $ \partial W ^ {4} ( {\mathsf K} ) = \chi ( {\mathsf K} ) $. Similarly, for any framed link $ {\mathsf L} $ the $ 3 $- dimensional manifold $ \chi ( {\mathsf L} ) $ is the boundary of the $ 4 $- dimensional manifold $ W ^ {4} ( {\mathsf L} ) $ obtained by attaching handles of index $ 2 $ to the $ 4 $- dimensional ball. Move 1) on $ {\mathsf L} $ corresponds to replacing $ W ( {\mathsf L} ) $ by a connected sum of $ W ( {\mathsf L} ) $ with $ \pm CP ^ {2} $. Move 2) corresponds to a sliding of one handle of index $ 2 $ over another and does not change $ W ( {\mathsf L} ) $. One can show that any framed link in $ S ^ {3} $ can be transformed by moves 1), 2) and their inverses to a link with even framings [a3]. In the latter case the tangent bundle of the corresponding $ 4 $- dimensional manifold is trivial.

See [a4] for more details.

References

[a1] W.B.R. Lickorish, "A representation of orientable combinatorial 3-manifolds" Ann. Math , 76 (1962) pp. 531–540
[a2] R. Kirby, "A calculus for framed links in " Invent. Math. , 45 (1978) pp. 35–56
[a3] S. Kaplan, "Constructing framed 4-manifolds with given almost framed boundaries" Trans. Amer. Math. Soc. , 254 (1979) pp. 237–263
[a4] A.T. Fomenko, S.V. Matveev, "Algorithmic and computer methods in three dimensional topology" , Kluwer Acad. Publ. (1997)
How to Cite This Entry:
Dehn surgery. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dehn_surgery&oldid=13147
This article was adapted from an original article by S.V. Matveev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article