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Normalized braids

Canonical braids up to 10 crossings, lexicographically minimal among cyclic rotations and mirror images, with each generator appearing at least twice.

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length ?
int
Length of the braid 		braid ?
short_code · Mathematica
Braid word used to compute $F_K$ 		yakdb_id ?
string
YakDB ID 		positive ?
bool
Is the braid positive? 		homogeneous ?
bool
Is the braid homogeneous? 		is_minimum_braid ?
bool
Is the braid a minimum braid? 		minimum_braid ?
short_code · Mathematica
Shortest braid word for this knot
3 [1, 1, 1] -3_1 Yes Yes Yes [1, 1, 1]
4 [1, -2, -1, -2] 0_1 No No No []
4 [1, -2, -1, 2] 0_1 No No No []
4 [1, -2, 1, -2] 4_1 No No Yes [1, -2, 1, -2]
4 [1, 2, -1, -2] 0_1 No No No []
4 [1, 2, -1, 2] 0_1 No No No []
4 [1, 2, 1, -2] 0_1 No No No []
4 [1, 2, 1, 2] -3_1 Yes Yes No [1, 1, 1]
5 [1, 1, 1, 1, 1] -5_1 Yes Yes Yes [1, 1, 1, 1, 1]
6 [1, -2, -1, -1, -1, -2] 5_2 No No No [1, -2, -2, -1, -1, -2]
6 [1, -2, -1, -1, -1, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
6 [1, -2, -1, -1, -2, -2] 5_2 No No No [1, -2, -2, -1, -1, -2]
6 [1, -2, -1, -1, 2, 2] 4_1 No No No [1, -2, 1, -2]
6 [1, -2, -1, -2, -2, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
6 [1, -2, -1, 2, 2, 2] -3_1 No No No [1, 1, 1]
6 [1, -2, -2, -1, -1, -2] 5_2 No No Yes [1, -2, -2, -1, -1, -2]
6 [1, -2, -2, -1, -1, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
6 [1, -2, -2, -2, -1, -2] 3_1 No No Yes [1, -2, -2, -2, -1, -2]
6 [1, -2, -2, -2, -1, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
6 [1, -2, 1, -2, -2, -2] 6_2 No No Yes [1, -2, 1, -2, -2, -2]
6 [1, 1, -2, -1, -2, -2] 0_1 No No No []
6 [1, 1, -2, -1, 2, 2] -3_1 No No No [1, 1, 1]
6 [1, 1, -2, -2, -1, -2] 0_1 No No No []
6 [1, 1, -2, -2, -1, 2] 4_1 No No No [1, -2, 1, -2]
6 [1, 1, -2, -2, 1, -2] 6_3 No No Yes [1, 1, -2, -2, 1, -2]
6 [1, 1, -2, -2, 1, 2] 0_1 No No No []
6 [1, 1, -2, 1, -2, -2] 6_3 No No No [1, 1, -2, -2, 1, -2]
6 [1, 1, -2, 1, 2, 2] -5_2 No No Yes [1, 1, -2, 1, 2, 2]
6 [1, 1, 1, -2, -1, -2] 0_1 No No No []
6 [1, 1, 1, -2, -1, 2] -3_1 No No No [1, 1, 1]
6 [1, 1, 1, -2, -2, -2] 3_1 # -3_1 No No Yes [1, 1, 1, -2, -2, -2]
6 [1, 1, 1, -2, 1, -2] -6_2 No No Yes [1, 1, 1, -2, 1, -2]
6 [1, 1, 1, -2, 1, 2] -3_1 No No No [1, 1, 1]
6 [1, 1, 1, 2, -1, -2] -3_1 No No No [1, 1, 1]
6 [1, 1, 1, 2, -1, 2] -5_2 No No No [1, 1, -2, 1, 2, 2]
6 [1, 1, 1, 2, 1, -2] -3_1 No No No [1, 1, 1]
6 [1, 1, 1, 2, 1, 2] -5_1 Yes Yes No [1, 1, 1, 1, 1]
6 [1, 1, 1, 2, 2, 2] -3_1 # -3_1 Yes Yes Yes [1, 1, 1, 2, 2, 2]
6 [1, 1, 2, -1, -2, -2] 4_1 No No No [1, -2, 1, -2]
6 [1, 1, 2, -1, 2, 2] -5_2 No No No [1, 1, -2, 1, 2, 2]
6 [1, 1, 2, 1, -2, -2] 0_1 No No No []
6 [1, 1, 2, 1, 2, 2] -5_1 Yes Yes No [1, 1, 1, 1, 1]
6 [1, 1, 2, 2, -1, -2] -3_1 No No No [1, 1, 1]
6 [1, 1, 2, 2, -1, 2] -5_2 No No No [1, 1, -2, 1, 2, 2]
6 [1, 1, 2, 2, 1, -2] -5_2 No No No [1, 1, -2, 1, 2, 2]
6 [1, 1, 2, 2, 1, 2] -5_1 Yes Yes No [1, 1, 1, 1, 1]
6 [1, 2, -1, -1, -1, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
6 [1, 2, -1, -1, -1, 2] 0_1 No No No []
6 [1, 2, -1, -1, -2, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
6 [1, 2, -1, -1, 2, 2] 0_1 No No No []
6 [1, 2, -1, -2, -2, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
6 [1, 2, -1, 2, 2, 2] -3_1 No No No [1, 1, 1]
6 [1, 2, 1, -2, -2, -2] 0_1 No No No []
6 [1, 2, 1, 2, 2, 2] -5_1 Yes Yes No [1, 1, 1, 1, 1]
6 [1, 2, 2, -1, -1, -2] 4_1 No No No [1, -2, 1, -2]
6 [1, 2, 2, -1, -1, 2] 0_1 No No No []
6 [1, 2, 2, 2, -1, -2] -3_1 No No No [1, 1, 1]
6 [1, 2, 2, 2, -1, 2] -3_1 No No No [1, 1, 1]
6 [1, 2, 2, 2, 1, -2] -5_2 No No No [1, 1, -2, 1, 2, 2]
7 [1, -2, -1, -1, -3, -2, -3] 5_2 No No No [1, -2, -2, -1, -1, -2]
7 [1, -2, -1, -1, -3, -2, 3] 0_1 No No No []
7 [1, -2, -1, -1, -3, 2, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -1, -3, 2, 3] 0_1 No No No []
7 [1, -2, -1, -1, 3, -2, -3] 0_1 No No No []
7 [1, -2, -1, -1, 3, -2, 3] 6_1 No No No [1, -2, -3, 2, 1, -3, -2]
7 [1, -2, -1, -1, 3, 2, -3] 0_1 No No No []
7 [1, -2, -1, -1, 3, 2, 3] 4_1 No No No [1, -2, 1, -2]
7 [1, -2, -1, -2, -3, -2, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -2, -3, -2, 3] 0_1 No No No []
7 [1, -2, -1, -2, -3, -3, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -2, -3, 2, -3] 0_1 No No No []
7 [1, -2, -1, -2, -3, 2, 3] 0_1 No No No []
7 [1, -2, -1, -2, 3, -2, -3] 0_1 No No No []
7 [1, -2, -1, -2, 3, -2, 3] 4_1 No No No [1, -2, 1, -2]
7 [1, -2, -1, -2, 3, 2, -3] 0_1 No No No []
7 [1, -2, -1, -2, 3, 2, 3] 0_1 No No No []
7 [1, -2, -1, -2, 3, 3, 3] -3_1 No No No [1, 1, 1]
7 [1, -2, -1, -3, -2, -1, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -3, -2, -1, 3] 0_1 No No No []
7 [1, -2, -1, -3, -2, -3, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -3, -2, -3, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -3, -2, -3, 2] 0_1 No No No []
7 [1, -2, -1, -3, -2, 1, -3] 0_1 No No No []
7 [1, -2, -1, -3, -2, 1, 3] 0_1 No No No []
7 [1, -2, -1, -3, -2, 3, -2] 0_1 No No No []
7 [1, -2, -1, -3, -2, 3, 2] 0_1 No No No []
7 [1, -2, -1, -3, -2, 3, 3] 0_1 No No No []
7 [1, -2, -1, -3, -3, -2, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -3, -3, -2, 3] 0_1 No No No []
7 [1, -2, -1, -3, -3, -3, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -3, -3, -3, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -3, -3, 2, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -3, -3, 2, 3] 0_1 No No No []
7 [1, -2, -1, -3, 2, -1, -3] 0_1 No No No []
7 [1, -2, -1, -3, 2, -1, 3] 0_1 No No No []
7 [1, -2, -1, -3, 2, -3, -2] 0_1 No No No []
7 [1, -2, -1, -3, 2, -3, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
7 [1, -2, -1, -3, 2, -3, 2] 4_1 No No No [1, -2, 1, -2]
7 [1, -2, -1, -3, 2, 1, -3] 4_1 No No No [1, -2, 1, -2]
7 [1, -2, -1, -3, 2, 1, 3] 0_1 No No No []
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