Hi Beamer,
as bicycle rubber is black and so very good absorpting, you can cut it with a 'weak' pigtailed diode-laser of maybe 1 Watt and a single focussing lens - higher powers and/or a CO2-laser will do better and much faster.
But you have to filter and exhaust the smoke, as it's corrosive and highly toxic!
And again - don't work with or use lasers with this powers without proper laser-goggles and safety precautions!
Viktor