Knots Verses Graphs - Azram, Mohammad (2011)

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A theoretic and diagrammatic relationship between knots and planar graphs has enabled us to visualize, redefine and establish some variational, diagrammatic and illustrative results. It has been shown that the universes, LR-graphs and regions of reduced alternating knots (links) are path connected. Connected universe corresponding to reduced alternating knot (link) is unique. It has been shown that the regions, crossings and consequently the number of vertices, edges, and faces in the corresponding LR-graph are same and invariant. Establishment of new but pivotal moves such as R*-move, 2π-twist and π-twist enabled us to change connected knot (link) into a reduced form as well as to prove that the black regions can be changed into white regions via Reidemeister moves. It has been established that for reduced alternating knot (linked link), total regions are two more than the total crossings. In case the knot (linked link) is also achiral than total crossings = 2[total black(white) regions-1]. Finally, the equivalence of the companion graphs, necessary and sufficient conditions for achirality.