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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm200-1-1,
author = {A. Stoimenow},
title = {Knots of (canonical) genus two},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {1-67},
zbl = {1190.57008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm200-1-1}
}
A. Stoimenow. Knots of (canonical) genus two. Fundamenta Mathematicae, Tome 201 (2008) pp. 1-67. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm200-1-1/