Summary: BERGE'S DISTANCE 3 PAIRS OF GENUS 2
Abstract. Following an example discovered by John Berge [Be2],
we show that there is a 4-component link L (S1
so that, generically, the result of Dehn surgery on L is a 3-manifold
with two inequivalent genus 2 Heegaard splittings, and each of
these Heegaard splittings is of Hempel distance 3.
In [Be2] John Berge introduces a criterion which, if satisfied by a
genus two Heegaard splitting of a 3-manifold, ensures that the Heegaard
splitting is of distance 3 or greater. Furthermore, he gives an example
of a pair of such Heegaard splittings of the same 3-manifold, splittings
which he shows to be inequivalent. Such an example demonstrates
conclusively that the list in [RS] of possible manifolds with two or
more inequivalent genus 2 Heegaard splittings is incomplete.