Periodic knots and Heegaard Floer correction terms
The paper under review gives new obstructions to periodicity of knots by Heegaard Floer correction terms of cyclic branched covers. Assume that a knot \(K\) admits a period of prime \(q\). That is, there is an orientation-preserving diffeomorphism \(f:S^3\to S^3\) such that \(f(K)=K\) and the order of \(f\) is \(q\). By the positive solution of the Smith conjecture, the fixed point set of \(f\) is the unknot disjoint from \(K\). Let \(l\) be a prime distinct from \(q\) and \(n\) be a power of some prime number. Consider the \(n\)-fold cyclic branched covers \(Y\) and \(\bar{Y}\) of the \(3\)-sphere \(S^3\) branched over \(K\) and its quotient knot \(\bar{K}\), respectively. Suppose that the \(l\)-primary subgroup of \(H_1(\bar{Y})\) is trivial. Then the Heegaard Floer correction terms \(d(Y,s)\) corresponding to \(\mathrm{spin}^c\)-structures \(s\in H_1(Y)_l-\{0\}\) occur with multiplicities divisible by \(q\). Here, \(H_1(Y)_l\) is the \(l\)-primary subgroup of \(H_1(Y)\), and the set of \(\mathrm{spin}^c\)-structures \(\mathrm{Spin}^c(Y)\) is identified with \(H_1(Y)\) under the diffeomorphism \(F:Y\to Y\) induced by \(f\). The case where \(H_1(\bar{Y})_l\) is non-trivial is also treated. Then a weaker obstruction is presented. The authors give many examples of knots for which the new obstructions are successful but classical homological ones fail. Periodicity of knots with fewer than 12 crossings has been examined extensively. Some of those results are included in this paper. As an application of the new obstructions, the authors give a nearly complete classification of 12-crossing alternating knots with odd prime period and a complete list of all alternating knots with 13, 14, 15 crossings that admit a period of odd prime order bigger than 3.
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