Geometrically and diagrammatically maximal knots
  The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We show that many families of alternating knots and links simultaneously maximize both ratios. 
Abhijit Champanerkar, Ilya Kofman, Jessica S. Purcell
math.GT