School Seminars and Colloquia

Using train tracks to distinguish knots

Geometry/Topology Seminar

by Iain Aitchison


Institution: University of Melbourne
Date: Mon 15th August 2005
Time: 4:15 PM
Location: Room 107, Richard Berry Building, University of Melbourne

Abstract: Given Danny Calegari's lectures on surface homeomorphisms, we will
give a talk describing an application of train tracks to knot theory:

It is known that there are exactly two fibred knots with genus 1
fibre, the trefoil and the figure 8 knots, distinguished by their
Alexander polynomials. At one time it was conjectured that there
might be only finitely many fibred knots of any given genus with the
same Alexander polynomial. The actual situation for genus > 1 is
essentially as
complicated as one could expect:

We give a simple construction producing infinitely many knots K_n
which simultaneously
(a) are prime;
(b) are fibred;
(c) all have genus 2 fibre;
(d) have hyperbolic complements;
(e) are ribbon knots;
(f) have the same Alexander module structure;
(g) have monodromy obtained by a simple isotopy of a standard surface
in the 3-sphere;
(h) are intersections of a spun figure-8 knot in 4-space with the 3-sphere;
(i) are intersections of an unknotted 2 sphere in 4-space with the 3-sphere;
(j) are obtained by Dehn twisting the monodromy of 4_1#4_1

These are distinguished by having distinct invariant weight systems
on the same underlying train track. This simple train track will be
described. [So far: older, unpublished work.] Generalizing this
construction gives examples of "distinct" genus 2 fibred knots whose
monodromy differ by elements arbitrarily deep in the lower central or
derived series of the genus 2 Torelli group (the subgroup of the
mapping class group whose elements act trivially on homology).

For More Information: Lawrenec Reeves email: l.reeves@ms.unimelb.edu.au