Verify candidate Murasugi-sum decompositions of knots

Verify whether the 510 knots with at most 12 crossings having top Links–Gould coefficient a power of $q$ times $1+3q^2$, and the 360 knots with at most 12 crossings having top coefficient a power of $q$ times $3+q^2$, are Murasugi-sums of the indicated Hopf-link factor with a collection of Hopf links.

Background

The authors identify families of small knots whose top Links–Gould coefficients factor in the same way as those of a twisted Hopf link and Hopf links. This factorization makes the knots candidates for Murasugi-sum decompositions. The proposed decompositions were not verified in the paper, leaving the concrete classification of these knots unresolved.

References

The link $H_2=\p A_2$ ($A_2$ as above) has $top(H_2)=3q+q3$ so these 510 knots (resp. 360) are candidates to be Murasugi-sums of the mirror image $m(H_2)$ (resp. $H_2$) with a bunch of Hopf links, but we haven't checked this.

— The top coefficient of the Links-Gould invariant is Murasugi-sum multiplicative  (2610.09354 - Anghel et al., 7 Oct 2026) in Section 1, Introduction