Narrow Results

We show that any tunnel number one knot group has a two generator one relator presentation in which the relator is a palindrome in the generators. We use this fact to compute the character variety for this knot groups and we show that it is an affine algebraic set .

There are exactly four mutually non-isotopic unknotting tunnels τ_i, i = 1,2,3,4 for the pretzel knot P(-2,3,7). Moreover, there are at most 3 non-stabilized genus 3 Heegaard splittings.

We show that every tunnel number 1 genus 1 knot can be changed into a genus 3 fibered knot by a crossing change.