Kirby calculus changes framed-link diagrams by handle slides and creation or cancellation of standard handle pairs without changing the represented four-manifold up to diffeomorphism.
For an m-component integral framed surgery link, normalize its full colored bracket by the displayed expression, where is the signature of its linking matrix. The Kirby color handle-slide identity handles slides; cancels disjoint plus- or minus-one-framed unknots. Kirby calculus therefore makes this an oriented 3-manifold invariant. This convention has .
// Target: module-theory.bigb
A Kirby diagram depicts a handle decomposition of a four-dimensional manifold using a framed link for the two-handles and dotted circles for the one-handles. The framing specifies the attaching normal directions; it controls the resulting intersection form and the boundary Dehn surgery.
If a meridian of a knot component in a surgery link links only one other component, filling it with coefficient removes it and replaces the other coefficient by . This gives the negative continued-fraction chain description of rational Dehn filling.
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Kirby calculus is a mathematical technique used in the field of low-dimensional topology, particularly in the study of 3-manifolds. It is named after Rob Kirby, who introduced this concept in a series of papers in the 1970s. The main focus of Kirby calculus is on the manipulation and understanding of 3-manifolds via the use of specific types of diagrams called Kirby diagrams or handlebody diagrams.