The Kontsevich integral on parenthesized braids

Use coordinates $(z,t)\in \mathbb{C}\times \mathbb{R}$ on oriented ${\mathbb{R}}^3$, with $t$ increasing upward. A knot is a smooth embedding $K:S^1\hookrightarrow \mathbb{C}\times \mathbb{R}$ with the orientation of $S^1$. A strict Morse presentation is one for which $t{\mathrm{\mid }}_K$ has nondegenerate critical points, meaning its second derivative in a local coordinate is nonzero there, and pairwise distinct critical values. Write $t_{\min }:={\min }_{x\in K}t(x)$, $t_{\max }:={\max }_{x\in K}t(x)$, and $r(K)$ for the number of local maxima. A parenthesized $q$-braid is a smooth braid with $q$ strands, each strictly increasing in $t$, together with an ordered binary parenthesization of its lower endpoints and an ordered binary parenthesization of its upper endpoints.

For $n\ge 0$, a degree-$n$ chord diagram is an oriented circle with $n$ disjoint unordered pairs of distinct points, called chords. Let ${\mathcal{A}}_n^{\mathrm{f}\mathrm{r}}$ be the rational vector space on these diagrams modulo 4T, and let ${\mathcal{A}}_n^{\mathrm{u}\mathrm{n}}$ be its quotient by 1T as well. A 4T relation fixes one chord $pq$ and another chord endpoint $r$, moves the other endpoint of the second chord to just before or after $p$ and just before or after $q$, and takes the alternating sum of those four diagrams. A chord is isolated when it intersects no other chord, where intersection means that chord endpoints alternate around the circle; 1T sets every such diagram to zero. Put ${\widehat{\mathcal{A}}}_{\mathbb{C}}^{\mathrm{f}\mathrm{r}\mathrm{/}\mathrm{u}\mathrm{n}}:={\prod }_{n\ge 0}({\mathcal{A}}_n^{\mathrm{f}\mathrm{r}\mathrm{/}\mathrm{u}\mathrm{n}}{\otimes }_{\mathbb{Q}}\mathbb{C})$, with multiplication induced by connected sum of diagrams and unit the empty diagram. For $q$ ordered upward strands, let ${\widehat{\mathcal{A}}}_{\mathbb{C}}(q)$ be the completion of the associative algebra generated by $t_{ij}=t_{ji}$ for $1\le i<j\le q$, with relations $[t_{ij},t_{kl}]=0$ when $i,j,k,l$ are distinct and $[t_{ij},t_{ik}+t_{jk}]=0$ when $i,j,k$ are distinct. The generator $t_{ij}$ is a horizontal chord joining strands $i$ and $j$; multiplication stacks diagrams from bottom to top.

More generally, a compact strict Morse tangle $T$ is a compact oriented smooth one-dimensional submanifold of $\mathbb{C}\times [a,b]$, with distinct boundary points on the planes $t=a$ and $t=b$, transverse to those planes, and with nondegenerate interior critical points of height having distinct critical values. A tangle chord diagram is a finite set of unordered pairs of distinct interior points on the oriented tangle skeleton. Its 4T relation is the same local alternating relation as above on the skeleton. The unframed tangle quotient also sets to zero any chord whose endpoints are consecutive in the order along the same skeleton component; for a circle the order is cyclic and for an arc it is linear. The completed degreewise tangle diagram space is denoted ${\widehat{\mathcal{A}}}_{\mathbb{C}}^{\mathrm{u}\mathrm{n}}(T)$.

The unframed configuration series of a strict Morse presentation $K$ is the formal degreewise series

$$Z(K):=\sum _{m\ge 0}\frac{1}{(2\pi i{)}^m}{\int }_{t_{\min }<t_m<\cdots <t_1<t_{\max }}\sum _P(-1{)}^{\downarrow (P)}[D_P{]}_{\mathrm{u}\mathrm{n}}\underset{j=1}{\overset{m}{\bigwedge }}\frac{d{z}_j-d{z}_j^{\mathrm{\prime }}}{z_j-z_j^{\mathrm{\prime }}}\in {\widehat{\mathcal{A}}}_{\mathbb{C}}^{\mathrm{u}\mathrm{n}}\mathrm{.}$$

The ordered simplex is divided into chambers by the critical values of $t{\mathrm{\mid }}_K$. In each chamber, $P$ ranges over choices, at each level $t_j$, of an unordered pair of distinct points $(z_j,t_j),(z_j^{\mathrm{\prime }},t_j)$ of $K$. The diagram $D_P$ joins the corresponding preimages on the oriented parameter circle. The integer $\downarrow (P)$ counts endpoints at which $t$ decreases along the orientation of $K$. The differential form is pulled back to the chamber by the selected local branches. The degree-zero term is the empty diagram with coefficient $1$; the displayed integrals are interpreted as improper integrals on each chamber.

For a compact strict Morse tangle $T$, $Z(T)$ is given by the same formula with $T$ in place of $K$, the interval $(a,b)$ in place of $(t_{\min },t_{\max })$, the chord diagrams on the tangle skeleton in place of circle diagrams, and the corresponding unframed tangle quotient in place of ${\widehat{\mathcal{A}}}_{\mathbb{C}}^{\mathrm{u}\mathrm{n}}$. A parenthesized braid is a special case with no interior critical points.

For a parenthesized braid path $\gamma$ in the configuration space of $q$ distinct complex points, define its Knizhnik–Zamolodchikov (KZ) series by the degreewise iterated-integral holonomy of ${\mathrm{\Omega }}_q=\frac{1}{2\pi i}{\sum }_{1\le i<j\le q}{t}_{ij}d\log (z_i-z_j)\mathrm{.}$ Explicitly, if $\gamma :[0,1]\to {\mathrm{Conf}}_q(\mathbb{C})$ is an ordinary path with noncolliding endpoints, its holonomy is $1+{\sum }_{m\ge 1}{\int }_{0<t_m<\cdots <t_1<1}{\gamma }^{\ast }{\mathrm{\Omega }}_q(t_1)\cdots {\gamma }^{\ast }{\mathrm{\Omega }}_q(t_m)\mathrm{.}$ A tangential endpoint is a limiting collision configuration together with a specified nonzero first-order direction of approach for each colliding cluster. When endpoints have such clusters prescribed by the parenthesizations, the holonomy is the constant term after removing the local logarithmic factors $\exp (t_{ij}\log \epsilon \mathrm{/}(2\pi i))$ at each tangential endpoint. A change between the two parenthesizations of three consecutive strands is encoded by the regularized transport ${\mathrm{\Phi }}_{\mathrm{K}\mathrm{Z}}$.

For parenthesized braids the KZ holonomy takes values in the framed strand algebra, where only tangled 4T is imposed; passing from a closed framed strand diagram to an unframed knot applies the quotient by isolated-chord 1T. Fix a standard hump $H$ by starting with the two-critical-point unknot and inserting one local birth of a maximum-minimum pair; thus $H$ has two local maxima. The normalized unframed Kontsevich series of $K$ is

$$I(K):=Z(K)Z(H{)}^{-r(K)}\in {\widehat{\mathcal{A}}}_{\mathbb{C}}^{\mathrm{u}\mathrm{n}},$$

The normalization uses one copy of $Z(H)$ for each local maximum; its degree-zero term is $1$, so its inverse is interpreted in the product completion. This definition makes no assertion that the unnormalized KZ coefficients are rational or that equal Kontsevich series separate knot types.

The following figure illustrates the positive Artin generator ${\sigma }_1$ and parenthesization conventions. Height increases upward, the right-bottom strand is the overstrand under the determinant crossing convention, and the labels at the ends record the ordered parenthesizations; no proof below relies on the drawing.

Singular-knot and finite-difference conventions

4T and 1T chord-diagram quotients

Positive Artin half-twist convention

Chmutov, Duzhin, and Mostovoy, §§8.2, 8.5–8.8, and 10.1, printed pp. 223–240 and 269–277

Bar-Natan and Stoimenow, §§2.3–2.5, PDF pp. 10–15

Le and Murakami, associator construction and regularization, pp. 6–11