ScalingStacks

Remark 4.16. A strong version of the so-called knight move conjecture posited that the complex of any long knot decomposes (up to homotopy equivalence) into one shifted copy of $C_{0}$ and some number of copies of $C_{1}$; see [19, Conjecture 1]. The argument in the previous proof shows that this can fail only due to the presence of more than one shifted copy of $C_{0}$. Three copies of $C_{0}$ can be detected in the counterexample to the knight move conjecture found by Manolescu–Marengon [24].