A new criterion for knots with free periods. (Q1890434)

For integers \(p\geq 2\) and \(q\neq 0\), a knot in the 3-sphere is called a \((p,q)\)-lens knot if it covers a link in the lens space \(L(p,q)\). The author gives a necessary condition for a knot to be a \((p,q)\)-lens knot, using the second coefficient of the HOMFLY polynomial. As an application, it is shown that this criterion rules out the possibility of being \((5,1)\)-lens for 80 among the 84 knots with less than 9 crossings.