Toroidally alternating knots and links

Alternating distances of knots and links

“…Alternating genus. Following his work on almost alternating links, Adams [Ada94] defined toroidally alternating links as those links L that have an alternating projection π to a Heegaard torus Σ such that the complement of the projection in the Heegaard torus, i.e. Σ−π(L), is a disjoint union of disks.…”

Totally geodesic surfaces and homology

“…The Menasco-Reid conjecture has been shown true for many other classes of knots, including almost alternating knots [2], Montesinos knots [13], toroidally alternating knots [1], 3-bridge and double torus knots [5] and knots of braid index 3 [8] and 4 [9]. For a knot in one of the above families, any closed essential surface in its complement has a topological feature which obstructs it from being even quasi-Fuchsian.…”

“…F (or more precisely .f; F /) is said to be totally geodesic if f . 1 .F // is conjugate into PSL 2 ‫./ޒ.‬ Thurston and Bonahon have described the geometry of surface groups in hyperbolic 3-manifolds as falling into three classes: doubly degenerate groups, quasi-Fuchsian groups and groups with accidental parabolics. The class of totally geodesic surface groups is a "positive codimension" subclass of the quasi-Fuchsian groups, so one may expect that hyperbolic 3-manifolds containing totally geodesic surface groups are special.…”

Closed quasi-Fuchsian surfaces in hyperbolic knot complements

“…1 .S// is the entire S 2 1 ; or (c) essential with accidental parabolics if it is essential and some non-peripheral element of 1 .S/ has a parabolic image in f . 1 .S //…”

“…M in a hyperbolic knot manifold M is said to be (a) quasi-Fuchsian if it is essential and f . 1 .S// is a quasi-Fuchsian subgroup of Isom C .H 3 /; or (b) geometrically infinite if it is essential and the limit set of f . 1 .S// is the entire S 2 1 ; or (c) essential with accidental parabolics if it is essential and some non-peripheral element of 1 .S/ has a parabolic image in f .…”