Convergence and needle estimates for Kontsevich integrals

In a Morse band the colliding pair has $\mathrm{\mid }{z}_{+}(s)-z_{-}(s)\mathrm{\mid }\ge as$, while its derivative is bounded, so its short-chord form is bounded in absolute value by $Cds\mathrm{/}s$. Every other pair has separation at least $d$ and bounded derivative, so its logarithmic form is bounded by $Cds$; in a regular band the bound is $Cdt$. These constants are uniform under the stated family hypotheses. For a single fixed tangle the same bounds hold with tangle-dependent constants: the Morse expansions are $z_{\pm }(s)=z_c\pm vs+O(s^2)$ with $v\mathrm{\ne }0$, and all other branch pairs have nonzero limiting separation; shrink finitely many disjoint critical bands and use compactness on the remaining regular bands. Minima are treated by reversing height.

Discard every pairing diagram killed by 1T before taking absolute values. In a remaining pairing, let $0<{\delta }_k<\cdots <{\delta }_1<S$ be all $k\ge 1$ short-chord parameters in one whole Morse band, including parameters outside a smaller cap. If no endpoint of another chord lies on the oriented critical arc between the ends of the innermost short chord at ${\delta }_k$, these ends are consecutive and 1T kills the diagram. Any such intervening endpoint must belong to a long chord in the same band, since no short chord lies closer to the critical point. Thus every surviving pairing has a long witness with parameter $0<r<{\delta }_k$. Retain that witness factor and drop other ordering constraints only when enlarging a nonnegative absolute majorant.

Integrating the retained witness first gives at most $C{\delta }_k$. The remaining short factors have ordered integral ${\int }_{0<{\delta }_k<\cdots <{\delta }_1<S}{\delta }_k{\prod }_{j=1}^k\frac{d{\delta }_j}{{\delta }_j}=S\mathrm{.}$ Indeed, the innermost integration gives ${\delta }_{k-1}$ and repeating gives $S$. Every additional long factor is integrated once over a bounded interval. Distinct critical bands use distinct level variables, and a witness chord in one band cannot be a witness in another band; their parameter intervals are disjoint. Apply this estimate in each band, and bounded-factor estimates in regular bands. There are finitely many branch chambers and pairing types at degree $m$, and every surviving term is absolutely integrable. The quotient norm is bounded by the sum of these absolute majorants times the norms of the finitely many diagram classes. This proves the fixed-tangle convergence assertion, with uniform bounds when all the family controls hold.

For the cap band put $S=S_c$ and, by enlarging the cap when necessary, use the cutoff $h=A\epsilon <S$. Define $J_0(h,S)=h$ and, for $k\ge 1$, $J_k(h,S):={\int }_{0<{\delta }_k<\cdots <{\delta }_1<S}\min \{h,{\delta }_k\}{\prod }_{j=1}^k\frac{d{\delta }_j}{{\delta }_j}\mathrm{.}$ For $L=\log (S\mathrm{/}h)\ge 0$, integrating the upper $k-1$ short parameters at fixed $u={\delta }_k$ gives $J_k(h,S)=\frac{1}{(k-1)!}({\int }_0^h\log (S\mathrm{/}u{)}^{k-1}du+h{\int }_h^S\log (S\mathrm{/}u{)}^{k-1}\frac{du}{u})=h{\sum }_{j=0}^k\frac{L^j}{j!}\mathrm{.}$ The last equality follows by integration by parts in the first integral and the substitution $v=\log (S\mathrm{/}u)$ in the second; the boundary term at $u=0$ is zero since $u\log (S\mathrm{/}u{)}^j\to 0$ for every fixed $j$. The same formula gives $J_0=h$ when the sum has only its $j=0$ term. In particular $J_k(S,S)=S$, consistently with step 3.1.

Suppose a selected cap-meeting chord is long and $k\ge 1$. If it is the witness retained in step 2.1, its parameter is restricted to $0<r<\min \{h,{\delta }_k\}$; integrating this one long factor supplies $C\min \{h,{\delta }_k\}$ and hence the majorant $C{J}_k(h,S)$. If it is distinct from the chosen witness, retain both factors: integrate the selected cap long chord over a parameter interval of length at most $h$, and the witness over $(0,{\delta }_k)$. Dropping their mutual order enlarges the absolute integral and supplies $Ch{\delta }_k$, so step 3.1 gives the bound $ChS$. This is at most a fixed constant times $J_k(h,S)$, since $J_k(h,S)\ge h$ and $S$ is fixed. Neither long factor has been omitted or used twice. If $k=0$, a cap-meeting long factor directly supplies $Ch=C{J}_0(h,S)$.

Suppose instead that a selected cap-meeting chord is short. Some short parameter is at most $h$, hence the innermost one has ${\delta }_k\le h$. The witness from step 2.1 lies closer to the critical point, so it also lies inside the cap cutoff. Its integration supplies $C{\delta }_k$, while all upper short parameters still range through the whole band up to $S$. Therefore the ordered bound is ${\int }_{\begin{array}{c}0<{\delta }_k<\cdots <{\delta }_1<S\\ {\delta }_k\le h\end{array}}{\delta }_k{\prod }_{j=1}^k\frac{d{\delta }_j}{{\delta }_j}=h{\sum }_{j=0}^{k-1}\frac{\log (S\mathrm{/}h{)}^j}{j!}\le J_k(h,S)\mathrm{.}$ This follows from the first integral in step 4.1. In particular no parameter of an outer short chord has been artificially restricted to $h$.

The event that a pairing meets $N_{\epsilon }$ is covered by the finite union over its at most $m$ selected chords and the long/short cases in steps 5.1 and 5.2. Keep the associated cap and witness factors in that band; in every other critical band keep its separate witness as in step 2.1. Integrating all remaining bounded factors once and summing over the finitely many diagrams gives $C_m{J}_k(A\epsilon ,S)$ for some $k\le m-1$, or $C_m\epsilon$ in the case of distinct retained long factors. The inequality $k\le m-1$ holds because any surviving short-chord contribution has at least one long witness. Formula 4.1 gives the stated $C_m\epsilon (1+\mathrm{\mid }\log \epsilon \mathrm{\mid }{)}^{m-1}$ bound. For fixed $m$ it tends to zero, since with $x=-\log \epsilon$ it is bounded by a constant times $e^{-x}(1+x{)}^{m-1}$. If $m=0$, there is no chord endpoint and the cap-meeting contribution is exactly zero.

The final special-critical-movement assertion is exactly the separately labelled external input [F4] under its stated disk-region and single-critical-point hypotheses. It makes no assertion that individual pairing integrals, or arbitrary degenerating families, obey a uniform linear bound. The local convergence and controlled-cap estimates above were established independently.