A FINITE SET OF LOCAL MOVES FOR KIRBY CALCULUS

We exhibit a finite set of local moves that connect any two surgery presentations of the same 3-manifold via framed links in S3. The moves are handle-slides and blow-downs/ups of a particular simple kind.

FOUR-DIMENSIONAL MANIFOLDS CONSTRUCTED BY LENS SPACE SURGERIES ALONG TORUS KNOTS

A framed knot with an integral coefficient determines a simply-connected 4-manifold by 2-handle attachment. Its boundary is a 3-manifold obtained by Dehn surgery along the framed knot. For a pair of such Dehn surgeries along distinct knots whose results are homeomorphic, it is a natural problem: Determine the closed 4-manifold obtained by pasting the 4-manifolds along their boundaries. We determine the complete list (set) of pairs of integral surgeries along distinct torus knots whose resulting manifolds are orientation preserving/reversing homeomorphic lens spaces, and study the closed 4-manifolds constructed as above. The list consists of five sequences. All framed links and Kirby calculus are indexed by integers. As a bi-product, some sequences of embeddings of lens spaces into the standard 4-manifolds are constructed.

SURGERY DESCRIPTIONS AND VOLUMES OF BERGE KNOTS II: DESCRIPTIONS ON THE MINIMALLY TWISTED FIVE CHAIN LINK

Using Kirby calculus, we explicitly pass from Berge's R-R descriptions of ten families of knots with lens space surgeries to surgery descriptions on the minimally twisted five chain link (MT5C). Since the MT5C admits a strong inversion, we also give the corresponding tangle descriptions.