$\DeclareMathOperator\SLL{SLL}$For a smooth embedding $\gamma(t):\mathbb{S}^1\rightarrow\mathbb{R}^3$, the Vassilev invariant of degree 2, which I will denote as $\nu_2(\gamma)$, may be computed numerically via several different methods:
- Configuration Space Integrals
- Using the alternating self-linking number
- As a signed count of alternating quadrisecants
I've provided a quick summary on each of the above and references below my question for convenience.
In each of these approaches, the permutation $(1 3 4 2)$ is present in one way or another and I would like to understand better why that might be. Similar integrands also appear (since we're computing degrees of maps ultimately, I think) among each of the approaches.
My question: Since all these approaches calculate $\nu_2(\gamma)$, can we derive the result of 3 from the result of 2 or 1...or even the result of approach 2 from 1? (A bit unclear where the $\rho_1$ integral would come from).
Thanks!
Calculation references:
Configuration space integrals
$\nu_2(\gamma)=\rho_1(\gamma)+\rho_2(\gamma)$ where
$\Delta_4=\{(t_1,t_2,t_3,t_4)|0<t_1<t_2<t_3<t_4<1\}$ and $\Delta_3=\{(t_1,t_2,t_3,\textbf{z})|0<t_1<t_2<t_3<1,\textbf{z}\in\mathbb{R}^3-\{(\gamma(t_1),\gamma(t_2),\gamma(t_3))\}\}$
and:
\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*}
See here, and further computational advancements here
The alternating self-linking number
$\nu_2(\gamma)=\frac{1}{4}+6\cdot \SLL(\gamma)$ where $\SLL(\gamma)$ is the alternating self-linking number first defined here by Panagiotou and Kauffman. Here is the definition \begin{equation} \begin{split} \SLL(\gamma)&=\frac{1}{8\pi}\int_{0}^1\int_{0}^{j_1}\int_{0}^{j_2}\int_{0}^{j_3}(\dot{\gamma}(j_1)\times\dot{\gamma}(j_3))\cdot\frac{\gamma(j_1)-\gamma(j_3)}{|\gamma(j_1)-\gamma(j_3)|^3}(\dot{\gamma}(j_2)\times\dot{\gamma}(j_4))\cdot\frac{\gamma(j_2)-\gamma(j_4)}{|\gamma(j_2)-\gamma(j_4)|^3}\\ &\chi(j_1,j_2,j_3,j_4)dj_4dj_3dj_2dj_1, %&\chi(\Gamma(j_1,j_3)+\Gamma(j_2,j_4))dj_4dj_3dj_2dj_1\\ \end{split} \end{equation}
where $\Gamma(s,t)=\frac{\gamma(s)-\gamma(t)}{|\gamma(s)-\gamma(t)|}$, for $s,t\in[0,1]$, $0\leq j_4<j_3<j_2<j_1\leq1$ and where $\chi(j_1,j_2,j_3,j_4)=1$, when $(j_1,j_2,j_3,j_4)\in E$ and $\chi(j_1,j_2,j_3,j_4)=0$, otherwise, where $E\subset[0,1]^4$, such that $0\leq j_1<j_2<j_3<j_4\leq 1$, $\Gamma(j_1,j_3)=-\Gamma(j_2,j_4)$ .
This integral is of course very similar to $\rho_2$ above.
Counting alternating quadrisecants
See section 6 of the beautiful approach here by Budney, Conant, Scannell and Sinha. There are quite a bit of details which I will attempt to summarize and cast in a similar notation as above. This paper considers proper embeddings $f : \mathbb{I} \to \mathbb{I}^3$ where $\mathbb{I}$ is an interval, and "knots" are identified with the image of such embeddings (and therefore have fixed endpoints). Let $C_4$ denote the subspace of $C_4 (\mathbb{R}^3)$ of collinear configurations of points labeled by the $4$-cycle $(1 3 4 2)$ - by this it is meant that $t_i\in\mathbb{I}$ and $t_1<t_2<t_3<t_4$ and the line making the points $f(t_1), f(t_2), f(t_3), f(t_4)$ collinear runs through the points in the order given by the aforementioned $4$-cycle.
A sign $\epsilon_x$ is associated with a quadruple $x=(f(t_1),f(t_2),f(t_3),f(t_4)) \in C_4$ by defining it to be the sign of the determinant of the $2 \times 2$ matrix:
$$\begin{bmatrix} |f(t_3) - f(t_2)|\cdot \det[v, f'(t_1),f'(t_3)] & |f(t_3)-f(t_1)| \cdot \det[v, f'(t_2),f'(t_3)] \\ |f(t_4) - f(t_2)|\cdot \det[v, f'(t_4),f'(t_1)] & |f(t_4)-f(t_1)| \cdot \det[v, f'(t_2),f'(t_4)] \\ \end{bmatrix} $$
where $v=f(t_2)-f(t_1)$.
With this sign convention, a self-linking invariant is defined as:
Let $K = \mathrm{Im}(f)$ be a knot $K \subseteq \mathbb{I}^3$. If $C_4[K]$ is transverse to $C_4$, then $$ \sum_{x \in C_4(K) \cap C_4} \epsilon_x$$
When the knot is closed, a bit more work can be done to show that this count also equals $\nu_2(\gamma)$, see section 6.3 for the details.
