Low Dimensional Topology

January 22, 2009

Advanced curves in genus two Heegaard surfaces

Filed under: 3-manifolds,Curve complexes,Knot theory — Jesse Johnson @ 9:59 am

Given a loop in the boundary of a genus two handlebody, it is occasionally useful to characterize the curve in terms of what you get when you attach a 2-handle along that curve.  In particular, such a loop is called primitive if attaching a 2-handle produces a solid torus and is called Seifert if attaching the 2-handles produces a (small) Seifert fibered space. 

 Given a knot in the 3-sphere that happens to sit in a genus two Heegaard surface, we can characterize that loop in terms of how it sits with respect to the two handlebodies bounded by the Heegaard surface.  For example, if it is primitive in both handlebodies (i.e. double primitive) then we call the knot a Berge knot after John Berge, who classified such knots.  (A very simple construction shows that every Berge knot has a Dehn filling producing a lens space.  The converse of this statement is the Berge conjecture.)  Knots that are primitive to one side and Seifert on the other are called Dean knots after John Dean, who studied them in his dissertation at UT Austin.  (Also, Michael Williams has shown [1] that all primitive/Seifert loops satisfy the Berge conjecture.)

What I’d like to know is how many different positions on a genus-two Heegaard surface it is possible for a given knot type to have.  Two loops have the same knot type if there is an ambient isotopy of the 3-sphere taking one to the other.  We will say that two loops in a genus two surface  are equivalent if there is an ambient isotopy of the 3-sphere that takes the genus two surface to itself and takes one of the loops onto the other.  So, how many inequivalent loops in a genus two Heegaard surface can we have with  the same, given knot type?

For primitive knots there many different positions.  In particular, consider the slope on the knot defined by the Heegaard surface, i.e. the intersection of the Heegaard surface with the boundary of a regular neighborhood of the knot.   Given a primitive loop in a handlebody, if we push it into the handlebody then the complement of that loop will be a compression body.  We can then push it back into the boundary of the handlebody to produce a other primitive loops with any integral slope on the knot.  Thus for primitive loops we should ask how many inequivalent primitive positions there are with the same slope.

Another way to get inequivalent positions of a given knot type is to look for non-isotopic genus two Heegaard splittings.  Given a genus two Heegaard splitting of a knot complement, the knot can be pushed into the Heegaard surface (and this can be done so the resulting loops has any slope we like).  In the 3-sphere the Heegaard surface is itotopic to a standard Heegaard surface, so this defines a primitive position for the knot.  From this position there is a unique way to reconstruct the Heegaard surface by pushing the knot back into a handlebody.  (This isn’t quite true for Berge knots, but lets ignore that for now.)  So, if we do this with two different Heegaard surfaces for the knot complement, the resulting positions of the knot will be different.

Brandy Guntel (a grad. student at UT Austin) has used slightly different methods to find two primitive/Seifert loops that have the same knot type.  These are twisted torus knots:  A knot K(p,q,r,n) is what one gets by taking a (p,q) torus knot, picking up r consecutive strands and twisting them around n times.  It is fairly straightforward to make such a knot sit in a genus two Heegaard surface in a canonical way.  Brandy has shown that the knots K(kq+(q-1)/2, q, (q-1)/2, -1) and K(kq+(q+1)/2, q, (q+1)/2, -1), where k is an integer greater than or equal to 2 and q is an odd integer greater than or equal to 5, have the same knot type and the same slope in the genus two surface.  (She tells me that the proof is a complicated braid argument and I believe her.)  These two different positions correspond to two different Heegaard splittings of the knot complement which, after the Dehn surgery become different Heegaard splittings of the small Seifert fibered space.  A small Seifert fibered space admits up to three genus two Heegaard splittings so one might ask:  Can a knot have three distinct primitive/Seifert positions?  Can it have more then three?

So these primitive knots are quite interesting, but something that doesn’t seem to have been studied is non-primitive knots, or what I’ll call advanced knots.  (The opposite of primitive is advanced, right?)  These are loops in the Heegaard surface that are not primitive on either side.  In particular, here’s what I’d like to know:  Can a knot type that has a primitive position also have a non-primitive position?  (My guess is yes.)  In particular, can a torus knot be isotoped to a non-primitive position?  Can a knot that does not have a primitive position have two different non-primitive positions?  If a loop is sufficiently far in the curve complex from one or both of the handlebody sets, then must it have a unique position?  (This would be analogous to a result for Heegaard splittings of primitive loops.)

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