The Vassilev invariant of degree 2, which i'll denote as $v_2$ may be computed as the sum of two multiple integrals,
$v_2=\rho_1(\gamma)+\rho_2(\gamma)$
for a smooth embedding $\gamma(t):\mathbb{S}^1\rightarrow\mathbb{R}^3$, by integrating over the configuration spaces:
$\Delta_4=\{(t_1,t_2,t_3,t_4)|0<t_1<t_2<t_3<t_4<1\}$ and $\Delta_3=\{(t_1,t_2,t_3,\textbf{z})|0<t_1<t_2<t_3<1,\textbf{z}\in\mathbb{R}^3-\{(\gamma(t_1),\gamma(t_2),\gamma(t_3))\}\}$
via:
\begin{eqnarray}
&\rho_1(\gamma)=-\frac{1}{32\pi^3}\int_{(t_1,t_2,t_3,\textbf{z})\in\Delta_3(\gamma)}\mathrm{Det}\big[E(\textbf{z},
t_1),E(\textbf{z},t_2),E(\textbf{z},t_3)\big]d\textbf{z}dt_1dt_2dt_3\\
&\textrm{ with }E(\textbf{z},t)=\frac{(\textbf{z}-\gamma(t))\times\gamma'(t)}{\|\textbf{z}-\gamma(t)\|^3}\nonumber
\end{eqnarray}
and
\begin{eqnarray*}
&\rho_2(\gamma)=\frac{1}{8\pi^2}\int_{(t_1,t_2,t_3,t_4)\in\Delta_4}\frac{\mathrm{Det}\big[\gamma(t_3)-\gamma(t_1)
,\gamma'(t_3),\gamma'(t_1)\big]}{\|\gamma(t_3)-\gamma(t_1)
\|^3}\frac{\mathrm{Det}\big[\gamma(t_4)-\gamma(t_2)
,\gamma'(t_4),\gamma'(t_2)\big]}{\|\gamma(t_4)-\gamma(t_2)\|^3}dt_1dt_2dt_3dt_4\nonumber
\end{eqnarray*}
My question: is the smoothness of the embedding $\gamma(t)$ required here? If I were to instead consider a PL/polygonal knot, made up of parametrized line segments, would there be an issue with the integrals converging or being being ill-defined?
Thanks!