Kirby calculus

Kirby moves

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

W.B.R. Lickorish [a3] and A.D. Wallace [a4] showed that any orientable $3$-manifold may be obtained as the result of 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 [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 [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

[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 $S^3$" Invent. Math. , 45 (1978) pp. 35–56
[a3]	W.B.R. Lickorish, "A representation of orientable combinatorial $3$-manifolds" Ann. Math. , 76 (1962) pp. 531–540
[a4]	A.H. Wallace, "Modification and cobounding manifolds" Canad. J. Math. , 12 (1960) pp. 503–528