Reading the figure
Paired bars show the two Arf counts in dimensions two through ten, indexed by half dimension one through five. The vertical scale is logarithmic; the table retains each exact count.
| Half dimension r | Arf 0 | Arf 1 |
|---|---|---|
| 1 | 3 | 1 |
| 2 | 10 | 6 |
| 3 | 36 | 28 |
| 4 | 136 | 120 |
| 5 | 528 | 496 |
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
1 2 3 4 5 Half dimension r 1 0 0 1 0 1 1 0 2 Number of refinements Arf 0 Arf 1
Original figure caption
Exact counts of refinements of a fixed nondegenerate alternating space.
The difference between two refinements is linear. Nondegeneracy identifies every linear form with \(b(v,\cdot)\), so all refinements are
The numbers of the two Arf types are
For \(r=1\) these are 3 and 1. For \(r=3\) they are 36 and 28. Orthogonal sums add Arf invariants and multiply normalized Gauss signs.
Reading the diagram
Equal dimension and equal alternating form do not force equal refinement. An embedding fixed in advance must preserve \(q\) pointwise.