Unconstrained p-densities degenerate to the unknot for p ≤ 2 while ropelength-windowed p-densities avoid this and admit minimizers.
O’Hara,Energy functionals of knots II, Topol
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Unconstrained and Ropelength-Windowed $p$-densities of Knot Types
Unconstrained p-densities degenerate to the unknot for p ≤ 2 while ropelength-windowed p-densities avoid this and admit minimizers.