Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (appr
oaches a geometric limit), but a corresponding checkerboard surface increases in complexity with crossing number. In this paper, we generalize checkerboard surfaces to certain immersed surfaces, called twisted checkerboard surfaces, whose geometry better reflects that of the alternating link in many cases. We describe the surfaces, show that they are essential in the complement of an alternating link, and discuss their properties, including an analysis of homotopy classes of arcs on the surfaces in the link complement.
We prove that any diagram of the unknot with c crossings may be reduced to the trivial diagram using at most (236 c)^{11} Reidemeister moves. Moreover, every diagram in this sequence has at most (7 c)^2 crossings. We also prove a similar theorem for
split links, which provides a polynomial upper bound on the number of Reidemeister moves required to transform a diagram of the link into a disconnected diagram.
We show that the crossing number of a satellite knot is at least 10^{-13} times the crossing number of its companion knot.
We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.
We show there exist tunnel number one hyperbolic 3-manifolds with arbitrarily long unknotting tunnel. This provides a negative answer to an old question of Colin Adams.
