We present a polynomial space Monte Carlo algorithm that given a directed graph on $n$ vertices and average outdegree $��$, detects if the graph has a Hamiltonian cycle in $2^{n-��(\frac{n}��)}$ time. This asymptotic scaling of the savings in the running time matches the fastest known exponential space algorithm by Bj��rklund and Williams ICALP 2019. By comparison, the previously best polynomial space algorithm by Kowalik and Majewski IPEC 2020 guarantees a $2^{n-��(\frac{n}{2^��})}$ time bound. Our algorithm combines for the first time the idea of obtaining a fingerprint of the presence of a Hamiltonian cycle through an inclusion--exclusion summation over the Laplacian of the graph from Bj��rklund, Kaski, and Koutis ICALP 2017, with the idea of sieving for the non-zero terms in an inclusion--exclusion summation by listing solutions to systems of linear equations over $\mathbb{Z}_2$ from Bj��rklund and Husfeldt FOCS 2013.