Some applications of planar graph in knot theory

The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship between planar graph and almost planar Seifert surface is discussed. Using planar graph, we construct an alternating amphicheiral prime knot with crossing number n for any even number n ≥ 4. This gives an affirmative answer to problem 1.66(B) on Kirby's problem list.