The expansion of a chord diagram and the Tutte polynomial |
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Authors: | Tomoki Nakamigawa Tadashi Sakuma |
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Institution: | 1. Department of Information Science, Shonan Institute of Technology, 1-1-25 Tsujido-Nishikaigan, Fujisawa 251-8511, Japan;2. Systems Science and Information Studies, Faculty of Education, Art and Science, Yamagata University, 1-4-12 Kojirakawa, Yamagata 990-8560, Japan |
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Abstract: | A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram is called nonintersecting if contains no crossing. For a chord diagram having a crossing , the expansion of with respect to is to replace with or . For a chord diagram , let be the chord expansion number of , which is defined as the cardinality of the multiset of all nonintersecting chord diagrams generated from with a finite sequence of expansions.In this paper, it is shown that the chord expansion number equals the value of the Tutte polynomial at the point for the interlace graph corresponding to . The chord expansion number of a complete multipartite chord diagram is also studied. An extended abstract of the paper was published (Nakamigawa and Sakuma, 2017) 13]. |
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Keywords: | Chord diagram Complete multipartite graph Euler number Tutte polynomial |
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