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''Kirby moves''

A set of moves between different surgery presentations of a $3$-manifold.

W.B.R. Lickorish [[#References|[a3]]] and A.D. Wallace [[#References|[a4]]] showed that any orientable $3$-manifold may be obtained as the result of [[Surgery|surgery]] on some framed link in the $3$-sphere.

A framed link is a finite, disjoint collection of smoothly embedded circles, with an integer (framing) assigned to each circle. R. Kirby [[#References|[a2]]] described two operations (the calculus) on a framed link and proved that two different framed links, $L$ and $L'$, yield the same $3$-manifold if and only if one can pass from $L$ to $L'$ by a sequence of these operations.

1) Blow-up: One may add or subtract from $L$ an unknotted circle with framing $1$ or $-1$, which is separated from the other circles by an embedded $2$-sphere.

2) Handle slide: Given two circles $\gamma_i$ and $\gamma_j$ in $L$, one may replace $\gamma_j$ with $\gamma_j'$ obtained as follows. First, push $\gamma_i$ off itself (missing $L$) using the framing to get $\gamma_i'$. Then, let $\gamma_j'$ be a band sum of $\gamma_i'$ with $\gamma_j$. Framing on $\gamma_j$ is changed by taking the sum of framings on $\gamma_i$ and on $\gamma_j$ with $\pm$ algebraic linking number of $\gamma_i$ with $\gamma_j$.

R.P. Fenn and C.P. Rourke [[#References|[a1]]] proved that these operations are equivalent to a $K$-move, where links $L$ and $L'$ are identical except in a part where an arbitrary number of parallel strands of $L$ are passing through an unknot $\gamma_0$ with framing $-1$ (or $+1$). In the link $L'$ the unknot $\gamma_0$ disappears and the parallel strands of $L$ are given one full right-hand (respectively, left-hand) twist.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.P. Fenn,   C.P. Rourke,   "On Kirby's calculus of links"  ''Topology'' , '''18'''  (1979)  pp. 1–15</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Kirby,   "A calculus for framed links in $S^3$"  ''Invent. Math.'' , '''45'''  (1978)  pp. 35–56</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.B.R. Lickorish,   "A representation of orientable combinatorial $3$-manifolds"  ''Ann. Math.'' , '''76'''  (1962)  pp. 531–540</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.H. Wallace,   "Modification and cobounding manifolds"  ''Canad. J. Math.'' , '''12'''  (1960)  pp. 503–528</TD></TR></table>