The HOMFLY–PT–Kauffman criterion for factorised HZ transforms of 2-component links

Let $\mathcal{L}$ be a 2-component link, and let its HOMFLY–PT and Kauffman polynomials be related by the condition referred to as

$$in the source. A **factorised HZ-transform criterion** asserts that this relation is sufficient for the HZ transform of the HOMFLY–PT polynomial to admit a factorised form, such as the form referred to as$$

.

HOMFLY–PT–Kauffman factorisation criterion. If the HOMFLY–PT and Kauffman polynomials for a 2-component link $\mathcal{L}$ satisfy

$$, then the HZ transform of its HOMFLY–PT polynomial admits a factorised form, such as$$

.

The paper presents this as a sufficient criterion rather than a necessary one: the link $L10n42\{1\}$ admits HZ factorisability but does not satisfy the stated relation. The criterion therefore does not give a one-to-one correspondence with HZ factorisability, and the general mechanism behind the exceptional example remains open.