Manuscript VII-E-ST@1 · VII-E-ST.tex · section · d28d9c5d

Compact transverse collar

A four-face certificate and its supported 3+1 lift › Compact transverse collar · lines 843–881

VII-E-ST@1#eq:transverse-cap

Let

f_{s} (x, y) = min {F_{s} (x, y), 2 + s} .

The nonhorizontal region is

R_{s} = [- X_{s}, X_{s}] \times [- Y_{s}, Y_{s}], X_{s} = \frac{10 + s}{3}, Y_{s} = \frac{26 + s}{5} .

Take

K = [- 14/3, 14/3] \times [- 6, 6] .

The collar margins are

\frac{14}{3} - X_{s} = \frac{4 - s}{3}, 6 - Y_{s} = \frac{4 - s}{5} .

They are strictly positive throughout the parameter window.

The northeast resting collar polygon has spatial vertices

(X_{s}, 0), (14/3, 0), (14/3, 6), (0, 6), (0, Y_{s}), (X_{s}, Y_{s}) .

Reflections give the other three. They partition $K ∖ R_{s}$ and lie at $t = 2 + s$ . The collar chart is

A_{0} (t, x, y) = (t - 2, x, y),

and the core charts are $A_{i} = C_{i}$ . Thus all transitions are $A_{i} A_{j}^{- 1}$ , including their translations. Closed mixed words telescope in the full affine group.

Displayed formulas in this passage

VII-E-ST@1#eq:transverse-cap · line 846
$f_{s} (x, y) = min {F_{s} (x, y), 2 + s} .$
unlabeled · line 851
$R_{s} = [- X_{s}, X_{s}] \times [- Y_{s}, Y_{s}], X_{s} = \frac{10 + s}{3}, Y_{s} = \frac{26 + s}{5} .$
unlabeled · line 856
$K = [- 14/3, 14/3] \times [- 6, 6] .$
unlabeled · line 860
$\frac{14}{3} - X_{s} = \frac{4 - s}{3}, 6 - Y_{s} = \frac{4 - s}{5} .$
unlabeled · line 867
$(X_{s}, 0), (14/3, 0), (14/3, 6), (0, 6), (0, Y_{s}), (X_{s}, Y_{s}) .$
unlabeled · line 874
$A_{0} (t, x, y) = (t - 2, x, y),$

Nearby in this section

proposition: Proposition: Complete transverse noncentral data