Tying up the loose ends of tight knots

The ropelength of a physical knot refers to the minimum contour length it can have while respecting a no-overlap constraint. There are proven upper and lower bounds to the scaling of the ropelength with respect to the crossing number of the knot, as well as numerical estimates based on knot-tightening algorithms. The proven bounds are widely separated from numerical estimates, and the knots that have been numerically tightened are typically not complex enough to constrain stronger conjectured bounds. In this work I discuss some recent results examining the ropelength of torus and satellite knots that are significantly more complex than those previously analyzed, with the goal of constraining conjectured ropelength bounds and providing more insight on the crossing-ropelength relationshp. I will also discuss the convex hull volume of tight knots, which has a nontrivial relationship with the contour length.