Links and cubic 3-polytopes |
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Authors: | Weiling Yang Fuji Zhang |
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Institution: | School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, People's Republic of China ; School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, People's Republic of China |
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Abstract: | It is well known that a prime link diagram corresponds to a signed plane graph without cut vertices (Kauffman, 1989). In this paper, we present a new relation between prime links and cubic 3-polytopes. Let be the set of links such that each has a diagram whose corresponding signed plane graph is the graph of a cubic 3-polytope. We show that all nontrivial prime links, except -torus links and -pretzel links, can be obtained from by using some operation of untwining. Furthermore, we define the generalized cubic 3-polytope chains and then show that any nontrivial link can be obtained from by some untwining operations, where is the set of links corresponding to generalized cubic 3-polytope chains. These results are used to simplify the computation of the Kauffman brackets of links so that the computing can be done in a unified way for many infinite families of links. |
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Keywords: | Link diagram signed plane graph cubic 3-polytope generalized cubic 3-polytope chain untwining chain polynomial Kauffman bracket polynomial |
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