Kirby calculus

Builds on Link Dehn surgery Handle decomposition Lickorish–Wallace theorem

Definition

A framed link in $S^3$ is a link $L = K_1 \cup \cdots \cup K_r$ in $S^3$ with an integer $n_i$ , its framing, attached to each component $K_i$ . It determines a closed oriented $3$ -manifold $M_L$ by integral Dehn surgery on $L$ . It also determines a compact $4$ -manifold $W_L$ , obtained from $D^4$ by attaching one $2$ -handle along each $K_i$ with framing $n_i$ . This is a handle decomposition of $W_L$ , and $\partial W_L = M_L$ . Kirby calculus is the system of the following moves on framed links, together with isotopy of links:

Blow-up and blow-down: add to $L$ , or remove from $L$ , an unknot with framing $\pm 1$ inside a ball disjoint from the rest of $L$ .
Handle slide: replace a component $K_i$ by a band sum of $K_i$ with a parallel copy of another component $K_j$ . The parallel copy is taken with respect to the framing $n_j$ , and the framing of the new component changes accordingly.

Remarks

By the Lickorish–Wallace theorem, every closed connected oriented $3$ -manifold is $M_L$ for some framed link $L$ . Kirby's theorem states that $M_L$ and $M_{L'}$ are orientation-preserving diffeomorphic if and only if $L$ and $L'$ are related by a finite sequence of the moves above. A handle slide does not change $W_L$ up to diffeomorphism, while a blow-up replaces $W_L$ by its connected sum with $\pm \C P^2$ ; both leave $M_L$ unchanged. Example: the unknot with framing $+1$ describes $S^3$ itself, since a blow-down turns it into the empty link, which describes $S^3$ . Non-example: the unknot with framing $0$ cannot be blown down, because move 1 needs framing $\pm 1$ ; indeed it describes $S^1 \times S^2$ , which is not $S^3$ .

Reading path

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3-manifold concept
Dehn surgery concept
Handle decomposition concept
Link concept
Lickorish–Wallace theorem statement
Kirby calculus definition

Tag	`ZFNZ0M7X41`
URI	`m2://ZFNZ0M7X41`
Kind	definition
Writing	W0 stub
MSC	57K40, 57K30, 57R65
Defines	Kirby calculus
Uses	Link, Dehn surgery, Handle decomposition, Lickorish–Wallace theorem
Changed	2 November 2026

Kirby calculus

Definition

Remarks

Reading path

Further reading

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