Difference between revisions of "Kirby calculus"
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''Kirby moves'' | ''Kirby moves'' | ||
| − | A set of moves between different surgery presentations of a | + | 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 | + | 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, | + | 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 | + | 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 | + | 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 | + | 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==== | ====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 | + | <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> |
Latest revision as of 08:53, 25 August 2014
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 |
How to Cite This Entry:
Kirby calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kirby_calculus&oldid=33131
Kirby calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kirby_calculus&oldid=33131
This article was adapted from an original article by Joanna Kania-Bartoszyńska (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article