The Ropelengths of Alternating Montesinos Links Are Proportional to Their Crossing Numbers

It has long been conjectured that the ropelength of any alternating link is bounded below by a constant multiple of its crossing number. While this conjecture was recently confirmed for alternating knots (i.e., alternating links with a single component), it remains open for alternating links with multiple components. In this paper, we establish that the conjecture holds for all alternating Montesinos links. Specifically, we prove the existence of positive constants \( 1/36 < b_0 < b_1 < 36 \) such that the ropelength \( R(L) \) of any alternating Montesinos link \( L \) satisfies \[ b_0 Cr(L) ≤ R(L) ≤ b_1 Cr(L), \] where \( Cr(L) \) denotes the crossing number of \( L \).