# Using train tracks to distinguish knots

*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 [So far: older, unpublished work.]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