FK Edge-Heap Lab | Minho Song

Random FK word

Mode

Vertices

Min word length

Max word length

Word length

R_n star mode (K_1,n initial words) The initial word uses only edges incident to the highest-numbered vertex. FK triangle moves may later create edges between outer vertices.

Quick rules

×²

Repeat dies

${x_{i j}}^{2} = 0$ . Adjacent identical generators make the monomial vanish.

↔

Disjoint swap

If the two edges are disjoint, use $x_{i j} x_{k l} = x_{k l} x_{i j}$ .

△

Triangle split

For $i < j < k$ , use $x_{i j} x_{j k} = x_{j k} x_{i k} + x_{i k} x_{i j}$ . Hover over △ to highlight the selected arrows.

Automatic bookkeeping

Like words merge, opposite coefficients cancel, and dead terms fade automatically.

Goal: use the defining relations to eliminate every term. Wrong answers, passes, and nonminimal reductions open a solution popup.

∂ reverse-derivative probe

Original word:

This is not an equality move. Repeated derivatives reaching a nonzero scalar certify that the original word is nonzero.

Current derivative expression:

A global Sₙ relabeling preserves zero/nonzero but usually does not simplify the word, so it is not included as a player move.

Reduction board

Each row is the current linear combination. Like words merge and opposite coefficients cancel automatically.

Live terms remain

R_n experiment

Word extending lab

Extend a star word while its original-order repeated derivative remains nonzero. Long surviving examples are intended to reveal useful patterns. The height-path panel and folder-style nonzero archive are there to help you compare those patterns visually. Open the folder for a near-fullscreen multi-column view, then select several lengths such as 5, 6, and 7 at once. Archive entries remain only in the current browser on the current computer; they are not uploaded or synchronized. The archive window can export the currently filtered collection as one large PNG image, with SVG fallback for very large collections.

R_n star mode Checked: use letters a,b,c,…. Unchecked: enter arbitrary generators without x_.

Live word expression

Compute power

Current length 0

Longest nonzero so far 0

Original-order derivative Ready

Exact algebra zero test Waiting

Height-path view Available in R_n star mode

In star mode, a word such as ababc is plotted by placing vertex i at (i,h_i), where a,b,c,d,e,... correspond to heights 1,2,3,4,5,....

Expanded word degree 0

Exact repeated-derivative result

No word yet.