Why teafork works

The graphical Reidemeister moves (from Jiang, Jin and Deng, "Determining the component number of links corresponding to triangular and honeycomb lattices," 2012.)

A Tait graph of a knot projection can be transformed to a Tait graph of any other knot projection by a sequence of unsigned graphical Reidemeister moves (see Noble and Welsh, below, and Jiang, Jin and Deng above.)

The graphical Reidemeister moves (from Noble and Welsh, Knot Graphs, 2000.)

For example, using the graphical Reidemeister moves, we can demonstrate that, in a 3-regular plane graph, vertex truncation, t, does indeed preserve cursality (figure below.) We will use the Roman numerals of Noble and Welsh to designate the moves.

From left to right, a demonstration that truncate, t, preserves cursality in a 3-regular plane graph: a vertex in the graph; same vertex after inverse-IV is applied to all edges in the graph; same neighborhood after V is applied.

Dually, we can demonstrate that kis, k, preserves cursality when applied to a 4-regular plane graph (figure below.)

From left to right, a demonstration that kis, k, preserves cursality in a face-3-regular plane graph: a face in the graph; same face after inverse-III is applied to all edges in the graph; same neighborhood after V is applied.

The final figures seem to leave some stray vertices and parallel edges laying about, but these are destined to be "used up" as the same operations are applied to neighboring vertices or faces.

Similarly, we can demonstrate that cursality is preserved in a 4-regular plane graph when k4t is applied (figure below.)

A demonstration that k4t preserves cursality in a 4-regular plane graph: a) a vertex in the graph; b) same vertex after inverse-IV is applied to all edges in the graph; c) same neighborhood after inverse-IV is applied to the central vertex; d) after V is applied to the two Y vertices; e) after inverse-IV is applied to the central vertex; f) after V is applied to the outer vertices in "Orion's belt."

Dually, we can demonstrate that cursality is preserved in a face-4-regular graph when teafork, t4k, is applied (figure below.)

A demonstration that t4k preserves cursality in a face-4-regular plane graph: a) a face in the graph; b) same face after inverse-III is applied to all edges in the graph; c) same neighborhood after inverse-III is used to bisect the face; d) after V is applied to two triangles; e) after inverse-III is used to bisect central quad; f) after V is applied to the resulting two triangles.