摘要
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.
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 greater than or equal to 3, and 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 hound (n -1)!. For n = 3, and 4, 1/2(n -1)! is already the best.
