## 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.

### 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)

Zbl 0747.00039