Abstract
Twisted torus knots are a generalization of torus knots obtained by in- troducing additional full twists to adjacent strands of torus knots. In this talk, we present an explicit formula for the Alexander polynomial of twisted torus knots. We use a presentation of the knot group of twisted torus knots and Fox’s free differential calculus. We further explore the applications of our computations, including a determination of the genus for certain families of twisted torus knots. This is joint work with Adnan.
Content
Definition
Twisted Torus knots are like torus knots but there is an extra twist of parallel strands.
Results
- Tisted torus knots are simple hyperbolic knots, dehn surgery probels (?), We know its Heegard splittings
- We have a closed form for the Alexander polynomial (Morton 06)
- Classification of twisted torusknots that are unknots (Lee, ‘12, …)
Invariants
We calculate invariants for twisted torus knots. We want to calculate the alexander polynomial. For that we want to calculate the Knotengruppe. (This was given by Adnan-P.). When we have that, we can use Fox Free Calculus to get the Alexander polynomial.
Uses
The degree of the polynomial can be used as a lower bound of the knot genus. Actually, we can calculate an upper bound too.
Next steps
Calculate the Floer Homology of the knot.