Heegaard splittings of twisted torus knots

Abstract

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting.Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain "twisted torus knots" originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. We also conjecture that these manifolds possess irreducible yet weakly reducible splittings of genus three. There are no known examples of such Heegaard splittings.

Bridge Spectra of Twisted Torus Knots

“…Applying the bound of Proposition 5.2 to these knots, we see that they have arbitrarily large genus one bridge number. We use the argument of Proposition 4.3 from [20] to show that these knots are hyperbolic: Because K n is primitive on at least one side of Σ, it is a tunnel number one knot. Toroidal tunnel one knots are classified by Morimoto and Sakuma [22] and have genus one bridge number one.…”

“…Lee [19] has characterized twisted torus knots which are actually the unknot. Moriah and Sedgwick have shown that certain hyperbolic twisted torus knots have minimal genus Heegaard splittings which are unique up to isotopy [20]. It is also known that twisted torus knots have arbitrarily large volume [7].…”

Bounding the stable genera of Heegaard splittings from below

“…Moriah and Sedgwick [12] have asked whether there is a closed 3-manifold with a weakly reducible Heegaard splitting of nonminimal genus. In these examples, both Heegaard splittings are weakly reducible.…”

TWISTED TORUS KNOTS T(p, q; 3, s) ARE TUNNEL NUMBER ONE

“…So jq = 1 (mod p) and we can see that j > k from (1). Adding equations, we get −(j − k)q = 1 (mod p) and this contradicts equations (2).…”