Lie group actions and metric criticality in ropelength-filtered knot spaces

We examine the role of the compact Lie group $SO(3)\times O(2)$ in the metric critical-point theory of normalized thick knots. The quotient by this group is part of the construction of ropelength-filtered knot spaces, but passage to an orbit space need not preserve the weak-slope notion of criticality. We establish elementary structural results for the action, including finiteness of isotropy for nontrivial knot types and exact invariance of the descending strong slope under a compact isometric quotient. We give counterexamples, including one with the connected group $SO(2)$, showing that the analogous assertion for the weak slope fails in general. On the other hand, we prove that it does hold, up to a multiplicative constant, at every point where the orbit map admits a Lipschitz local section together with a continuous gauge, and we construct such data for normalized thick knots from finitely many Fourier modes. Consequently, on the open stratum of thick knots with trivial isotropy, weak-slope criticality on the parametrized model and on the quotient coincide; at a configuration with nontrivial finite isotropy $\Gamma$ we show that finitely many Fourier modes recover $\Gamma$ exactly, and that the comparison problem is equivalent, up to constants, to the same problem for the finite group $\Gamma$ acting on a slice. For the unknot the round circles, whose isotropy is $O(2)$, remain outside this reduction. We also show that configurations whose thickness is not saturated have descending strong slope at least $2\pi/(\pi+1)$. For closed subgroups we define symmetry-constrained connected and path minimax levels and prove attainment of finite connected merge levels from compact sublevels; a finite-dimensional example, computed exactly, demonstrates a strict symmetry-breaking minimax gap. Finally we distinguish these static statements from Noether conservation laws for a separately specified dynamics.