Compactified Kontsevich configuration spaces and faces

The critical points are isolated in the compact one-dimensional skeleton and absent from the tangle boundary, so there are finitely many critical points and critical values. On every open interval between consecutive critical values or boundary levels, the inverse image is a finite union of arcs on which $t$ is a local diffeomorphism. Thus the number of branches, and the finite set of allowed pairs at a level, are constant on each such interval.

At a regular level, each branch is a smooth graph $z=z_{\lambda }(t)$, so its evaluation extends smoothly to any regular face. Near a critical point, choose a local parameter $x$ with $t(0)=c$, $t^{\mathrm{\prime }}(0)=0$, and $t^{\mathrm{\prime }\mathrm{\prime }}(0)\mathrm{\ne }0$. The function $h(x)=(t(x)-c)\mathrm{/}{x}^2$ extends smoothly with $h(0)=t^{\mathrm{\prime }\mathrm{\prime }}(0)\mathrm{/}2\mathrm{\ne }0$; after shrinking the coordinate interval, $s=x\sqrt{\mathrm{\mid }h(x)\mathrm{\mid }}$ is a smooth coordinate with nonzero derivative and $t=c+\mathrm{sgn}(h(0))s^2$. Since $z=z(x(s))$ is smooth, the two local branches and their limiting common point extend to the critical face in this coordinate.

Fix a chamber and let $k_r$ be the number of its ordered levels in the $r$th interval between critical or boundary levels. The chamber is a product of open simplices, one for each interval, with $k_r$ ordered coordinates. Its closure is the product of the corresponding closed simplices, hence compact. The facets of a closed simplex are exactly adjacent-coordinate equalities or a coordinate reaching one endpoint of its interval; across the finite family of chambers those endpoints are the knot extrema, tangle boundary levels, or critical values. Thus before resolving endpoint collisions the only faces are level collisions, outer endpoints, and critical-level events.

In a regular-level collision, endpoints on distinct branches remain distinct because $K$ is embedded; endpoints on one branch coalesce along a smooth arc. In the latter case, use local parameters $s_1,\dots ,s_k$ for the colliding endpoints, their center $p=k^{-1}\sum s_j$, the radius $r=({\sum }_j\mathrm{\mid }{s}_j-p{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, and the normalized shape $(s_j-p)\mathrm{/}r$. Replacing the diagonal $r=0$ by this shape data is the real oriented blow-up of that collision diagonal; its local coordinates are $r\ge 0$ and smooth shape coordinates. At a critical point, step 1.2 supplies the same form with the Morse coordinate $s$.

For a cluster of several endpoints, first blow up the full collision diagonal and then each proper collision diagonal in its limiting shape space, from larger clusters to smaller ones. Collision subsets in any limiting configuration are either nested or disjoint, so these blow-ups introduce finitely many nonnegative radii $r_C$ and smooth shape coordinates. Each resulting chart is a product of half-lines $r_C\ge 0$ and an open smooth coordinate set; transition maps are smooth because they are changes of center, scale, and unit direction. Hence the resolved cell is a manifold with corners. Each added boundary hypersurface is a collision cluster, and its boundary orientation is determined by the outward vector $-{\mathrm{\partial }}_{r_C}$. The original simplex orientation and the local coordinates $s\ge 0$ at critical levels give the stated orientations on all remaining faces.

Each chamber closure is compact by step 2.1, and real oriented blow-up replaces a closed collision diagonal by a compact sphere of limiting directions without losing compactness. Only finitely many chambers, branch pairings, and endpoint subsets occur at fixed $m$, so their resolved union is compact. The boundary labels from steps 2.1 and 3.1 exhaust its boundary faces; no additional face can occur because every boundary coordinate is either a level-gap, a distance to a critical/global endpoint, or one of the finitely many collision radii. The branch evaluations extend by steps 1.2 and 2.2.