Manuscript VII-F@1 · VII-F.tex · section · 2d81c03f

Transition words

Adapted affine charts and exact holonomy › Transition words · lines 1083–1105

VII-F@1#eq:cocycle

For an oriented crossing from a face chart $f$ to a face chart $g$ , define the seam exponent

n_{g f} = ⎩ ⎨ ⎧ 1, - 1, 0, upper seam to lower seam, lower seam to upper seam, a crossing not using the marked seam .

If a chart overlap has more than one connected component, the component is part of the crossing data.

The affine transition is

g_{g \leftarrow f} = A_{g} J^{n_{g f}} A_{f}^{- 1} .

This specifies every adjacency transition: ordinary adjacencies have $n = 0$ , and the marked-seam adjacencies have $n = \pm 1$ .

Displayed formulas in this passage

VII-F@1#eq:cocycle · line 1099
$g_{g \leftarrow f} = A_{g} J^{n_{g f}} A_{f}^{- 1} .$
unlabeled · line 1087
$n_{g f} = ⎩ ⎨ ⎧ 1, - 1, 0, upper seam to lower seam, lower seam to upper seam, a crossing not using the marked seam .$