A criterion for knots of period 3 (Q919315)

The criterion uses the two-variable polynomial P(m,\(\ell)\); \(P_ 0\) denotes its terms of degree zero in m. Theorem 1. Assume a knot K has period 3. Let \(P_ 0(K)=\sum a_ k\ell^{2k}\). Then for every k, \(a_{3k+1}+a_{3k+2}\equiv 0 mod 3.\) The criterion can be successfully applied to a number of knots whose period 3 was doubtful. The proof reduces the 3-periodic knot diagram to that of a 3-periodic closed braid and finally to that of a torus knot t(k,3). A direct computation concludes the proof using a formula of V. Jones.