# Numerical computation of the second Vassiliev invariant, and the permutation $(1 3 4 2)$

$$\DeclareMathOperator\SLL{SLL}$$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:

1. Configuration Space Integrals
2. Using the alternating self-linking number
3. 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:

1. #### Configuration space integrals

$$\nu_2(\gamma)=\rho_1(\gamma)+\rho_2(\gamma)$$ where

$$\Delta_4=\{(t_1,t_2,t_3,t_4)|0 and $$\Delta_3=\{(t_1,t_2,t_3,\textbf{z})|0

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

1. #### 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 $$$$\begin{split} \SLL(\gamma)&=\frac{1}{8\pi}\int_{0}^1\int_{0}^{j_1}\int_{0}^{j_2}\int_{0}^{j_3}(\dot{\gamma}(j_1)\times\dot{\gamma}(j_3))\cdot\frac{\gamma(j_1)-\gamma(j_3)}{|\gamma(j_1)-\gamma(j_3)|^3}(\dot{\gamma}(j_2)\times\dot{\gamma}(j_4))\cdot\frac{\gamma(j_2)-\gamma(j_4)}{|\gamma(j_2)-\gamma(j_4)|^3}\\ &\chi(j_1,j_2,j_3,j_4)dj_4dj_3dj_2dj_1, %&\chi(\Gamma(j_1,j_3)+\Gamma(j_2,j_4))dj_4dj_3dj_2dj_1\\ \end{split}$$$$

where $$\Gamma(s,t)=\frac{\gamma(s)-\gamma(t)}{|\gamma(s)-\gamma(t)|}$$, for $$s,t\in[0,1]$$, $$0\leq j_4 and where $$\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, $$\Gamma(j_1,j_3)=-\Gamma(j_2,j_4)$$ .

This integral is of course very similar to $$\rho_2$$ above.

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 $$t_1 and the line making the points $$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.

• My preferred way of describing the type-2 invariant for closed knots is illustrated in this web app, created by Sean Lee (a summer student at U.Vic) sean564.github.io/top Click on the "i" tag in the top-right for some documentation. Commented Jun 5 at 4:54

This is not a full answer to your question, but it gives some information.

The expression (1) is an integral version of the Polyak-Viro formula for the type-2 invariant, described here: https://academic.oup.com/imrn/article-abstract/1994/11/445/790435?redirectedFrom=PDF

Misha Polyak gave a cute argument showing why (1) and (3) are equivalent.

My understanding is he intends to publish a fairly general result of this form in the future. He has a presentation on the topic available here: https://polyak.net.technion.ac.il/files/2021/05/Enumerative-geometry-and-finite-type-invariants.pdf and I think if you do a little further Googling you can find a video recording of this presentation, as well.

There is also a more geometric version of this argument in my paper "On the automorphism groups of hyperbolic manifolds" (with D.Gabai). But here the argument is in high dimensions, not dimension 3, and we're not studying knots in the 3-sphere but embedded arcs in $$S^1 \times D^{n-1}$$ for $$n \geq 4$$. But it's a similar argument. The main difference is Polyak's argument takes place inside the configuration space of 4 points, while ours is in the configuration space of 3 points.

Regarding your question on why the (1342) permutation appears, it can be thought of as an encoding of the "X" chord diagram, i.e. if you think about the spectral sequence for the homology of the space of knots, this is one of the preferred book-keeping devices to keep track of the combinatorics of this Vassiliev invariant. Chord diagrams describe the types of double points one needs to keep track of to compute these types of invariants.

• Brills, thank you! The work by Polyak is exactly what I was looking for, and I think I found the video you had mentioned on youtube here. Separately, I will also check out your other paper you mentioned. Amazing that an author of 3/ responded! C: I accepted this answer because I think 2/ is very similar to counting alternating quadrisecants with signs Commented Jun 6 at 1:28
• That looks like an appropriate video. I watched a slightly different one, I think it was recorded in Paris. Commented Jun 6 at 1:44