Knots of (canonical) genus two
Tom 200 / 2008
Fundamenta Mathematicae 200 (2008), 1-67
MSC: Primary 57M25; Secondary 57M27, 57M50, 22E10, 20F36.
DOI: 10.4064/fm200-1-1
Streszczenie
We give a description of all knot diagrams of canonical genus 2 and 3, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3- and 4-move conjectures, and the calculation of the maximal hyperbolic volume for canonical (weak) genus 2 knots. We also study the values of the link polynomials at roots of unity, extending denseness results of Jones. Using these values, examples of knots with non-sharp Morton (canonical genus) inequality are found. Several results are generalized to arbitrary canonical genus.
