Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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The Tait graphs of a connected projection are connected
Statement
For a connected regular link projection on with at least one crossing and either checkerboard coloring, both Tait graphs are connected.
Statement references
Facts & Assumptions
Given: A connected regular link projection and a checkerboard coloring of its complementary regions.
A connected projection is a connected finite 4-valent graph, with crossingless components represented by embedded circles (Regular oriented link diagrams).
Its complementary regions admit a checkerboard coloring (Connected four-valent link projections admit checkerboard colorings).
Each Tait graph has one vertex per region of its color and one edge per crossing joining the opposite sectors (Checkerboard coloring and Tait graphs).
Proof
Associate to each crossing the edge of a chosen Tait graph. Along any projection edge joining crossings, the region of the chosen color on one side is the same at both ends, so the two corresponding Tait edges share that region vertex. Since the projection graph is connected, a path between any two crossings gives a sequence of Tait edges in which consecutive edges share a vertex. Every region vertex is incident to an edge because every face boundary of a connected projection with crossings contains a crossing. Therefore the Tait graph is connected. The same argument applies to the other color.
The only excluded connected projection is a crossingless circle, which is not covered by this statement; it will be treated directly in the bracket-span proof. This closes the claimed case.
Depends on
Used by
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Louis H. Kauffman, New Invariants in the Theory of Knots, Astérisque 163–164 (1988), §III, pp. 160–163 (standard reference, not scraped)
- Danny Calegari, Introduction to Knot Theory, §5.2, printed pp. 44–45 (standard reference, not scraped)