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**Canonical surgery on alternating link diagrams.**
*(English)*
Zbl 0765.57005

Knots 90, Proc. Int. Conf. Knot Theory Rel. Topics, Osaka/Japan 1990, 543-558 (1992).

[For the entire collection see Zbl 0747.00039.]

Associated to an unsplittable alternating link projection is a graph \(G\) with no separating edges, whose vertices correspond to the regions of one color in a \(2\)-coloring of the complementary regions, and whose edges correspond to the vertices at which two such regions meet. The regions of the second color determine the dual graph \(G^*\). The link projection determines two “checkerboard” surfaces, not necessarily orientable, bounding the link. The authors describe two \(3\)-manifolds \(M_ G\) and \(M_{G^*}\) obtained by surgery using framings determined by these surfaces. When the alternating link is a knot, the recently resolved Tait conjectures imply that these \(3\)-manifolds are independent of the projection and hence are knot invariants. Under some additional hypotheses, the authors also construct a singular Euclidean metric arising from a cubical decomposition of \(M_ G\), such that \(M_ G\) contains an immersed \(\pi_ 1\)-injective surface totally geodesic with respect to the metric. This immersed surface satisfies the \(4\)-plane, \(1\)-line condition of Hass and Scott, showing that \(M_ G\) is determined up to homeomorphism by its fundamental group.

Associated to an unsplittable alternating link projection is a graph \(G\) with no separating edges, whose vertices correspond to the regions of one color in a \(2\)-coloring of the complementary regions, and whose edges correspond to the vertices at which two such regions meet. The regions of the second color determine the dual graph \(G^*\). The link projection determines two “checkerboard” surfaces, not necessarily orientable, bounding the link. The authors describe two \(3\)-manifolds \(M_ G\) and \(M_{G^*}\) obtained by surgery using framings determined by these surfaces. When the alternating link is a knot, the recently resolved Tait conjectures imply that these \(3\)-manifolds are independent of the projection and hence are knot invariants. Under some additional hypotheses, the authors also construct a singular Euclidean metric arising from a cubical decomposition of \(M_ G\), such that \(M_ G\) contains an immersed \(\pi_ 1\)-injective surface totally geodesic with respect to the metric. This immersed surface satisfies the \(4\)-plane, \(1\)-line condition of Hass and Scott, showing that \(M_ G\) is determined up to homeomorphism by its fundamental group.

Reviewer: D.McCullough (Norman)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57M50 | General geometric structures on low-dimensional manifolds |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |