Braid and knot isotopy invariance of the normalized integral

An orientation-preserving reparametrization of $K$ preserves the cyclic order of every chord endpoint, the set of points at each height, the down-endpoint sign, and the pulled-back differential form. Hence every degree coefficient of $Z(K)$ is unchanged, and the reparametrization preserves $r(K)$. A positive affine change $t\mapsto at+b$ of the height also preserves each coefficient: substitute $u_j=a{t}_j+b$ in the ordered level integrals; the branch pairings and logarithmic forms correspond term by term. Thus $I(K)$ is unchanged under both operations.

A horizontal deformation in the Statement is a smooth family of closed Morse knots: its height function is fixed and all its critical points remain nondegenerate. Apply [F16] directly to obtain $Z(K_0)=Z(K_1)$, even though regular strands in the critical slabs can move. The number of maxima is fixed, so $I(K_0)=I(K_1)$. The supplementary regular-tangle computation has a stronger boundary hypothesis; no assertion that entire critical slabs are fixed is needed.

By [F6], a standard hump birth multiplies the preliminary series by $Z(H)$ and increases $r(K)$ by one. Thus $I(K^{+})=Z(H)Z(K)Z(H{)}^{-r(K)-1}=I(K)$; deleting the hump is the reverse equality.

By definition of a critical-point move, the affected arc can be placed in a horizontal disk region meeting the knot in at most two points per disk, with exactly one nondegenerate critical point and no change outside that region. Thus [F15] applies and gives equality of the preliminary series. The number of maxima is unchanged during the move, so the normalized series is equal as well.

Between successive fold events of the generic movie [F7], every height function is Morse, including at a crossing of two critical values. The full Morse-class theorem [F16] applies on each compact subinterval, so $Z$ is constant there. The number of maxima cannot change without a degenerate critical point, so $r$ and $I$ are constant as well. In particular this handles critical-value crossings without assuming that the critical neighborhoods or the whole critical slabs remain fixed.

For the supplementary regular-tangle calculation, let $T_s$ have no critical points and fix its entire upper and lower boundary configurations. On each degree-$m$ branch chamber, form the finite signed diagram-valued sum ${\omega }_m$ of wedges of $d\log (z_i-z_j)$, taking full differentials in the levels and $s$. Embeddedness and compactness give nonzero uniformly separated branch differences, so these forms extend smoothly to the closed unblown chamber and are closed. Apply Stokes to its product with the deformation interval. On an outer endpoint face, the logarithmic factor at that level is zero because its two boundary points are fixed. At an equal-level face, if the two chords use the same two branches, their logarithmic 1-forms coincide and their wedge is zero. If they use four distinct branches, swapping the two chord choices gives the same diagram with the opposite wedge sign, so the terms cancel. If they use exactly three branches, the 4T relation [F5] groups the six choices into the common diagram coefficient times ${\alpha }_{12}\wedge {\alpha }_{23}+{\alpha }_{23}\wedge {\alpha }_{31}+{\alpha }_{31}\wedge {\alpha }_{12}$, where ${\alpha }_{ij}=d\log (z_i-z_j)$. Set $f=z_1-z_2$, $g=z_2-z_3$, and $h=z_3-z_1=-(f+g)$. Direct expansion gives ${\alpha }_{12}\wedge {\alpha }_{23}+{\alpha }_{23}\wedge {\alpha }_{31}+{\alpha }_{31}\wedge {\alpha }_{12}=(\frac{1}{fg}-\frac{1}{g(f+g)}-\frac{1}{f(f+g)})df\wedge dg=0$. These cases exhaust adjacent-level faces. The real blow-ups in [F2] over collision centers of codimension at least two add no Stokes term: on this no-critical tangle the branch form extends smoothly to the unblown closed chamber, and the blowdown restricted to a front face times the deformation interval has rank at most $m-1$, so its pullback $m$-form vanishes there. A codimension-one center is an adjacent-level face already handled above. Thus the only remaining boundary integrals are the two ends $s=0,1$, which are equal. Summing the finitely many chambers proves equality of the regular-tangle series. This proves the regular-tangle assertion only under its whole-boundary hypothesis. The closed-knot horizontal conclusion instead follows directly from [F16] in step 1.2.

At a fold, [F7] gives one local maximum-minimum birth or death. The affected arc is isolated in a small ball and has nonzero complex tangent at the degenerate point. Rotate the complex coordinate so $u=\mathrm{Re}z$ is a local parameter, then shrink the ball so $u$ is strictly monotone along the entire arc. Interpolating $\mathrm{Im}z$ to zero at fixed $(u,t)$ remains an embedding because distinct arc points have distinct $u$. On the resulting planar arc, move the two nondegenerate critical points one at a time to the maximum and minimum of the fixed hump, using critical-point moves; choose the hump scaled so its endpoint and critical heights agree. Each monotone subarc now has the same endpoint heights as the standard hump, so the unique height-matching reparametrization is orientation-preserving and smooth up to the critical collars in their Morse coordinates. The birth is therefore, up to the reparametrizations, horizontal deformations, and critical-point moves already handled, the fixed standard hump insertion. Step 1.3 gives equality across the fold. Since [F7] supplies finitely many folds and crossings, step 1.5 and the hump calculation give equality along the entire generic movie.

If $K_0,K_1$ are smoothly ambient isotopic, restrict the isotopy to the oriented circles, reparametrize its time by a smooth map constant near its endpoints, and apply [F7] to obtain a generic movie with the same endpoint embeddings. Steps 1.5 and 2.2 show that $I(K_0)=I(K_1)$. The only axiom used in this ambient-isotopy conclusion is ${\mathrm{A}\mathrm{C}}_{\omega }$ from [F7] and the explicitly stated genericity supplier.

Each Markov move preserves the oriented braid closure under ${\mathrm{A}\mathrm{C}}_{\omega }$ by [F13]; a finite Markov sequence therefore gives ambient isotopic closures, and step 3.1 gives equal normalized series. Each oriented Reidemeister move is likewise realized by ambient isotopy by [F10]; a finite sequence preserves $I$. Under [F9], any two regular diagrams of the same oriented link have such a finite move sequence.

For a fixed parenthesization, a braid path is a path in the configuration space of distinct strand points. Its KZ holonomy is unchanged under fixed-endpoint path homotopy by [F4]. This use is choice-free; changes of parenthesization and the associator coherence equations are handled by a separate result.