Exact number of chord diagrams and an estimation of the number of spine diagrams of ordern |
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Authors: | Banghe Li Hongwei Sun |
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Institution: | LI Banghe and SUN Hongwei
Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, China |
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Abstract: | The main results are as follows:
( i ) For the number of chord diagrams of order n, an exact formula is given.
( ii ) For the number of spine diagrams of order n, the upper and lower bounds are obtained. These bounds show that the estimation is asymptotically the best.
As a byproduct, an upper bound is obtained, for the dimension of Vassiliev knot invariants of order n, that is, 1/2 ( n -1)! for any n≥3, and 1/2( n - 1)! - 1/2( n - 2)! for bigger n . Our upper bound is based on the work of Chmutov and Duzhin and is an improvement of their bound ( n - 1)! . For n = 3, and 4,1/2( n - 1)! is already the best. |
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Keywords: | knot Vassiliev invariant chord diagram spine diagram combinatorics |
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