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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
9 [1, -2, -2, -2, -1, 3, -2, 1, 3] 0_1 No No No []
9 [1, -2, -2, -2, -1, 3, -2, 3, -2] 3_1 # 4_1 No No No [1, -2, 1, -3, -3, -3, -2]
9 [1, -2, -2, -2, -1, 3, -2, 3, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -1, 3, -2, 3, 3] 3_1 # -3_1 No No No [1, 1, 1, -2, -2, -2]
9 [1, -2, -2, -2, -1, 3, 2, -1, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -1, 3, 2, -1, 3] 0_1 No No No []
9 [1, -2, -2, -2, -1, 3, 2, -3, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -1, 3, 2, -3, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -1, 3, 2, -3, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -1, 3, 2, 1, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -1, 3, 2, 1, 3] 6_3 No No No [1, 1, -2, -2, 1, -2]
9 [1, -2, -2, -2, -1, 3, 2, 3, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -1, 3, 2, 3, 2] 3_1 # -3_1 No No No [1, 1, 1, -2, -2, -2]
9 [1, -2, -2, -2, -1, 3, 2, 3, 3] 3_1 # -3_1 No No No [1, 1, 1, -2, -2, -2]
9 [1, -2, -2, -2, -1, 3, 3, -2, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -1, 3, 3, -2, 3] 3_1 # -3_1 No No No [1, 1, 1, -2, -2, -2]
9 [1, -2, -2, -2, -1, 3, 3, 2, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -1, 3, 3, 2, 3] 3_1 # -3_1 No No No [1, 1, 1, -2, -2, -2]
9 [1, -2, -2, -2, -1, 3, 3, 3, -2] 3_1 # -3_1 No No No [1, 1, 1, -2, -2, -2]
9 [1, -2, -2, -2, -1, 3, 3, 3, 2] 3_1 # -3_1 No No No [1, 1, 1, -2, -2, -2]
9 [1, -2, -2, -2, -3, -2, -1, -1, -3] 5_1 No No No [1, -2, -2, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, -1, -1, 3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, -1, -2, -3] 5_1 No No No [1, -2, -2, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, -1, -2, 3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, -1, -3, -2] 5_1 No No No [1, -2, -2, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, -1, -3, 2] 5_1 No No No [1, -2, -2, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, -1, 2, -3] 5_1 No No No [1, -2, -2, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, -1, 2, 3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, -1, 3, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, -1, 3, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, 1, -2, -3] -9_43 No No No [1, -3, -3, -2, 1, -3, -3, -3, -2]
9 [1, -2, -2, -2, -3, -2, 1, -2, 3] 3_1 # 4_1 No No No [1, -2, 1, -3, -3, -3, -2]
9 [1, -2, -2, -2, -3, -2, 1, -3, -2] 8_2 No No No [1, -2, 1, -2, -2, -2, -2, -2]
9 [1, -2, -2, -2, -3, -2, 1, -3, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, -2, 1, 3, -2] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, -3, -2, 1, 3, 2] 0_1 No No No []
9 [1, -2, -2, -2, -3, 2, -1, -1, -3] 5_2 No No No [1, -2, -2, -1, -1, -2]
9 [1, -2, -2, -2, -3, 2, -1, -1, 3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, 2, -1, -2, -3] 5_2 No No No [1, -2, -2, -1, -1, -2]
9 [1, -2, -2, -2, -3, 2, -1, -2, 3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, 2, -1, -3, -2] 5_2 No No No [1, -2, -2, -1, -1, -2]
9 [1, -2, -2, -2, -3, 2, -1, -3, 2] 5_2 No No No [1, -2, -2, -1, -1, -2]
9 [1, -2, -2, -2, -3, 2, -1, 2, -3] 5_2 No No No [1, -2, -2, -1, -1, -2]
9 [1, -2, -2, -2, -3, 2, -1, 2, 3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, 2, -1, 3, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, 2, -1, 3, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, 2, 1, -2, -3] 7_6 No No No [1, -3, -3, -2, -1, -1, 2, -3, 2]
9 [1, -2, -2, -2, -3, 2, 1, -2, 3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, -3, 2, 1, -3, -2] 8_11 No No No [1, -2, -2, -3, -3, -2, 1, 3, -2]
9 [1, -2, -2, -2, -3, 2, 1, -3, 2] 0_1 No No No []
9 [1, -2, -2, -2, -3, 2, 1, 3, -2] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, -3, 2, 1, 3, 2] 0_1 No No No []
9 [1, -2, -2, -2, 1, -2, -3, -2, -3] -8_5 No No No [1, -2, -2, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -2, -3, -2, 3] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -2, -3, -3, -3] 3_1 # 6_2 No No No [1, -2, -2, -2, 1, -3, -3, -3, -2]
9 [1, -2, -2, -2, 1, -2, -3, 2, -3] 8_6 No No No [1, -2, -3, -3, -3, 2, 1, -3, -2]
9 [1, -2, -2, -2, 1, -2, -3, 2, 3] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -2, 3, -2, -3] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -2, 3, -2, 3] -9_22 No No Yes [1, -2, -2, -2, 1, -2, 3, -2, 3]
9 [1, -2, -2, -2, 1, -2, 3, 2, -3] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -2, 3, 2, 3] 0_1 No No No []
9 [1, -2, -2, -2, 1, -2, 3, 3, 3] -3_1 # 6_2 No No Yes [1, -2, -2, -2, 1, -2, 3, 3, 3]
9 [1, -2, -2, -2, 1, -3, -2, -1, -3] 5_1 No No No [1, -2, -2, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, 1, -3, -2, -1, 3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, 1, -3, -2, -3, -2] -8_5 No No No [1, -2, -2, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -3, -2, -3, -3] -8_5 No No No [1, -2, -2, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -3, -2, -3, 2] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -3, -2, 1, -3] -8_10 No No No [1, 1, -2, -2, -2, 1, -2, -2]
9 [1, -2, -2, -2, 1, -3, -2, 1, 3] 3_1 # -3_1 No No No [1, 1, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -3, -2, 3, -2] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -3, -2, 3, 2] 0_1 No No No []
9 [1, -2, -2, -2, 1, -3, -2, 3, 3] 0_1 No No No []
9 [1, -2, -2, -2, 1, -3, -3, -2, -3] -8_5 No No No [1, -2, -2, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -3, -3, -2, 3] 8_6 No No No [1, -2, -3, -3, -3, 2, 1, -3, -2]
9 [1, -2, -2, -2, 1, -3, -3, -3, -2] 3_1 # 6_2 No No Yes [1, -2, -2, -2, 1, -3, -3, -3, -2]
9 [1, -2, -2, -2, 1, -3, -3, -3, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, 1, -3, -3, 2, -3] 8_21 No No No [1, -2, -2, -1, -1, 2, -1, -2]
9 [1, -2, -2, -2, 1, -3, -3, 2, 3] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -3, 2, -1, -3] 5_2 No No No [1, -2, -2, -1, -1, -2]
9 [1, -2, -2, -2, 1, -3, 2, -1, 3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, 1, -3, 2, -3, -2] 8_6 No No No [1, -2, -3, -3, -3, 2, 1, -3, -2]
9 [1, -2, -2, -2, 1, -3, 2, -3, -3] 8_21 No No No [1, -2, -2, -1, -1, 2, -1, -2]
9 [1, -2, -2, -2, 1, -3, 2, -3, 2] -9_42 No No No [1, -3, -2, 1, -3, -3, 2, -3, 2]
9 [1, -2, -2, -2, 1, -3, 2, 1, -3] 8_20 No No No [1, -2, -2, -1, 2, 2, -1, -2]
9 [1, -2, -2, -2, 1, -3, 2, 1, 3] 3_1 # -3_1 No No No [1, 1, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -3, 2, 3, -2] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, -3, 2, 3, 2] 0_1 No No No []
9 [1, -2, -2, -2, 1, -3, 2, 3, 3] 6_3 No No No [1, 1, -2, -2, 1, -2]
9 [1, -2, -2, -2, 1, 3, -2, -1, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, 1, 3, -2, -1, 3] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, 3, -2, -3, -2] 6_2 No No No [1, -2, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, 3, -2, -3, -3] 8_6 No No No [1, -2, -3, -3, -3, 2, 1, -3, -2]
9 [1, -2, -2, -2, 1, 3, -2, -3, 2] 0_1 No No No []
9 [1, -2, -2, -2, 1, 3, -2, 1, -3] 3_1 # -3_1 No No No [1, 1, 1, -2, -2, -2]
9 [1, -2, -2, -2, 1, 3, -2, 1, 3] -9_24 No No Yes [1, -2, -2, -2, 1, 3, -2, 1, 3]
9 [1, -2, -2, -2, 1, 3, -2, 3, -2] -9_22 No No No [1, -2, -2, -2, 1, -2, 3, -2, 3]
9 [1, -2, -2, -2, 1, 3, -2, 3, 2] 6_3 No No No [1, 1, -2, -2, 1, -2]
9 [1, -2, -2, -2, 1, 3, -2, 3, 3] -9_24 No No No [1, -2, -2, -2, 1, 3, -2, 1, 3]
9 [1, -2, -2, -2, 1, 3, 2, -1, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
9 [1, -2, -2, -2, 1, 3, 2, -1, 3] 0_1 No No No []
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