Aitchison, I. R.; Rubinstein, J. H. 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. Reviewer: D.McCullough (Norman) Cited in 4 Documents 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) Keywords:unsplittable alternating link projection; 2-coloring; 3-manifolds; Tait conjectures; knot invariants; fundamental group Citations:Zbl 0747.00039 PDF BibTeX XML Cite \textit{I. R. Aitchison} and \textit{J. H. Rubinstein}, in: Knots 90. Proceedings of the international conference on knot theory and related topics, held in Osaka, Japan, August 15-19, 1990. Berlin etc.: Walter de Gruyter. 543--558 (1992; Zbl 0765.57005) OpenURL