Let $\mathcal{K}$ be the commutative monoid whose elements are (isotopy) equivalence classes of knots with composition under the connected knot sum, and $\mathbb{Z}\mathcal{K}$ be the corresponding monoid algebra. Consider the 'tautological knot invariant' which sends a knot to itself, and whose extension under the Vassiliev skein relation acts as a linear function $\mathcal{K}\rightarrow\mathbb{Z}\mathcal{K}$.

Chmutov (Introduction to Vassiliev knot invariants) claims that,

(i) The images of of knots with $n$ double points span a subalgebra $\mathcal{K}_{n}$ of $\mathbb{Z}\mathcal{K}$

(ii) $\mathcal{K}_{n}$ is an ideal of $\mathbb{Z}\mathcal{K}$

Could anybody shed some light on how this is so? I can't immediately see how the required properties are preserved under the connected knot sum. Thanks!

the imageof a knot with $n$ singularities)$\in\mathcal{K}_{n}$ ? $\endgroup$ – Meths Feb 14 '19 at 18:12