Kirby calculus
Kirby moves
A set of moves between different surgery presentations of a -manifold.
W.B.R. Lickorish [a3] and A.D. Wallace [a4] showed that any orientable -manifold may be obtained as the result of surgery on some framed link in the
-sphere.
A framed link is a finite, disjoint collection of smoothly embedded circles, with an integer (framing) assigned to each circle. R. Kirby [a2] described two operations (the calculus) on a framed link and proved that two different framed links, and
, yield the same
-manifold if and only if one can pass from
to
by a sequence of these operations.
1) Blow-up: One may add or subtract from an unknotted circle with framing
or
, which is separated from the other circles by an embedded
-sphere.
2) Handle slide: Given two circles and
in
, one may replace
with
obtained as follows. First, push
off itself (missing
) using the framing to get
. Then, let
be a band sum of
with
. Framing on
is changed by taking the sum of framings on
and on
with
algebraic linking number of
with
.
R.P. Fenn and C.P. Rourke [a1] proved that these operations are equivalent to a -move, where links
and
are identical except in a part where an arbitrary number of parallel strands of
are passing through an unknot
with framing
(or
). In the link
the unknot
disappears and the parallel strands of
are given one full right-hand (respectively, left-hand) twist.
References
[a1] | R.P. Fenn, C.P. Rourke, "On Kirby's calculus of links" Topology , 18 (1979) pp. 1–15 |
[a2] | R. Kirby, "A calculus for framed links in ![]() |
[a3] | W.B.R. Lickorish, "A representation of orientable combinatorial ![]() |
[a4] | A.H. Wallace, "Modification and cobounding manifolds" Canad. J. Math. , 12 (1960) pp. 503–528 |
Kirby calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kirby_calculus&oldid=12516