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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
10 [1, -2, -1, 2, -4, 3, 3, 3, 4, -3] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 3, 3, 4, 3] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 3, 3, 4, 4] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 3, 4, -3, -4] 4_1 No No No [1, -2, 1, -2]
10 [1, -2, -1, 2, -4, 3, 3, 4, -3, 4] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 3, 4, 3, -4] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 3, 4, 3, 4] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 3, 4, 4, -3] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 3, 4, 4, 3] -5_2 No No No [1, 1, -2, 1, 2, 2]
10 [1, -2, -1, 2, -4, 3, 4, -3, -2, -2] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, -3, -2, -3] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, -3, -2, 3] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, -3, -3, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
10 [1, -2, -1, 2, -4, 3, 4, -3, -3, -4] 3_1 No No No [1, -2, -2, -2, -1, -2]
10 [1, -2, -1, 2, -4, 3, 4, -3, -3, 4] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, -3, -4, -4] 3_1 No No No [1, -2, -2, -2, -1, -2]
10 [1, -2, -1, 2, -4, 3, 4, -3, 2, -3] 4_1 No No No [1, -2, 1, -2]
10 [1, -2, -1, 2, -4, 3, 4, -3, 2, 2] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 4, -3, 2, 3] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, -3, 4, 4] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, 3, -2, -2] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, 3, -2, -3] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, 3, -2, 3] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, 3, -4, -4] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, 3, 2, -3] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, 3, 2, 2] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 4, 3, 2, 3] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 4, 3, 3, -4] 0_1 No No No []
10 [1, -2, -1, 2, -4, 3, 4, 3, 3, 3] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 4, 3, 3, 4] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 4, 3, 4, 4] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 4, 4, -3, -3] 4_1 No No No [1, -2, 1, -2]
10 [1, -2, -1, 2, -4, 3, 4, 4, 3, 3] -5_2 No No No [1, 1, -2, 1, 2, 2]
10 [1, -2, -1, 2, -4, 3, 4, 4, 4, -3] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, -4, 3, 4, 4, 4, 3] -5_2 No No No [1, 1, -2, 1, 2, 2]
10 [1, -2, -1, 2, 1, -2, -1, -2, -1, -2] 5_2 No No No [1, -2, -2, -1, -1, -2]
10 [1, -2, -1, 2, 1, -2, -1, -2, -1, 2] 3_1 No No No [1, -2, -2, -2, -1, -2]
10 [1, -2, -1, 2, 1, -2, -1, -2, -2, -2] 5_2 No No No [1, -2, -2, -1, -1, -2]
10 [1, -2, -1, 2, 1, -2, -1, 2, -1, -2] 8_20 No No No [1, -2, -2, -1, 2, 2, -1, -2]
10 [1, -2, -1, 2, 1, -2, -1, 2, -1, 2] 6_3 No No No [1, 1, -2, -2, 1, -2]
10 [1, -2, -1, 2, 1, -2, -1, 2, 2, 2] -6_2 No No No [1, 1, 1, -2, 1, -2]
10 [1, -2, -1, 2, 1, -2, -2, -1, -2, -2] 5_2 No No No [1, -2, -2, -1, -1, -2]
10 [1, -2, -1, 2, 1, -2, -2, -1, 2, 2] 6_3 No No No [1, 1, -2, -2, 1, -2]
10 [1, -2, -1, 2, 1, -2, -2, -2, -1, -2] 3_1 No No No [1, -2, -2, -2, -1, -2]
10 [1, -2, -1, 2, 1, -2, -2, -2, -1, 2] 6_2 No No No [1, -2, 1, -2, -2, -2]
10 [1, -2, -1, 2, 1, -2, -2, 1, -2, -2] -8_10 No No No [1, 1, -2, -2, -2, 1, -2, -2]
10 [1, -2, -1, 2, 2, -1, -1, -1, -1, -2] 10_126 No No No [1, -2, -2, -2, -2, -1, 2, 2, -1, -2]
10 [1, -2, -1, 2, 2, -1, -1, -1, -1, 2] -3_1 # 5_1 No No No [1, 1, 1, -2, -2, -2, -2, -2]
10 [1, -2, -1, 2, 2, -1, -1, -1, -2, -2] 10_143 No No No [1, -2, -2, -2, -1, -1, 2, 2, -1, -2]
10 [1, -2, -1, 2, 2, -1, -1, -1, 2, 2] 8_9 No No No [1, 1, 1, -2, -2, -2, 1, -2]
10 [1, -2, -1, 2, 2, -1, -1, -2, -2, -2] 10_143 No No No [1, -2, -2, -2, -1, -1, 2, 2, -1, -2]
10 [1, -2, -1, 2, 2, -1, -1, 2, 2, 2] 8_10 No No No [1, 1, 1, -2, -2, 1, 1, -2]
10 [1, -2, -1, 2, 2, -1, -2, -1, -1, -2] 5_2 No No No [1, -2, -2, -1, -1, -2]
10 [1, -2, -1, 2, 2, -1, -2, -1, -1, 2] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -2, -2, -1, -2] 5_2 No No No [1, -2, -2, -1, -1, -2]
10 [1, -2, -1, 2, 2, -1, -2, -2, -1, 2] 8_20 No No No [1, -2, -2, -1, 2, 2, -1, -2]
10 [1, -2, -1, 2, 2, -1, -2, -2, -2, -2] 10_126 No No No [1, -2, -2, -2, -2, -1, 2, 2, -1, -2]
10 [1, -2, -1, 2, 2, -1, -3, -4, -3, -4] 3_1 No No No [1, -2, -2, -2, -1, -2]
10 [1, -2, -1, 2, 2, -1, -3, -4, -3, 4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -3, -4, 3, -4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -3, -4, 3, 4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -3, 4, -3, -4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -3, 4, -3, 4] 4_1 No No No [1, -2, 1, -2]
10 [1, -2, -1, 2, 2, -1, -3, 4, 3, -4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -3, 4, 3, 4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -4, -3, -4, -3] 3_1 No No No [1, -2, -2, -2, -1, -2]
10 [1, -2, -1, 2, 2, -1, -4, -3, -4, 3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -4, -3, 4, -3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -4, -3, 4, 3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -4, 3, -4, -3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -4, 3, -4, 3] 4_1 No No No [1, -2, 1, -2]
10 [1, -2, -1, 2, 2, -1, -4, 3, 4, -3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, -4, 3, 4, 3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 2, -1, -1, -2] 10_141 No No No [1, -2, -2, -1, -1, 2, 2, 2, -1, -2]
10 [1, -2, -1, 2, 2, -1, 2, -1, -1, 2] 8_9 No No No [1, 1, 1, -2, -2, -2, 1, -2]
10 [1, -2, -1, 2, 2, -1, 2, 2, -1, -2] 10_125 No No No [1, -2, -2, -1, 2, 2, 2, 2, -1, -2]
10 [1, -2, -1, 2, 2, -1, 2, 2, -1, 2] 8_10 No No No [1, 1, 1, -2, -2, 1, 1, -2]
10 [1, -2, -1, 2, 2, -1, 2, 2, 2, 2] 8_5 No No No [1, 1, 1, -2, 1, 1, 1, -2]
10 [1, -2, -1, 2, 2, -1, 3, -4, -3, -4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 3, -4, -3, 4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 3, -4, 3, -4] 4_1 No No No [1, -2, 1, -2]
10 [1, -2, -1, 2, 2, -1, 3, -4, 3, 4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 3, 4, -3, -4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 3, 4, -3, 4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 3, 4, 3, -4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 3, 4, 3, 4] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, 2, -1, 4, -3, -4, -3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 4, -3, -4, 3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 4, -3, 4, -3] 4_1 No No No [1, -2, 1, -2]
10 [1, -2, -1, 2, 2, -1, 4, -3, 4, 3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 4, 3, -4, -3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 4, 3, -4, 3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 4, 3, 4, -3] 0_1 No No No []
10 [1, -2, -1, 2, 2, -1, 4, 3, 4, 3] -3_1 No No No [1, 1, 1]
10 [1, -2, -1, 2, 2, -3, -2, -4, -3, -4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -3, -2, -4, -3, 4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -3, -2, -4, 3, -4] 4_1 No No No [1, -2, 1, -2]
10 [1, -2, -1, 2, 2, -3, -2, -4, 3, 4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -3, -2, 4, -3, -4] 0_1 No No No []
10 [1, -2, -1, 2, 2, -3, -2, 4, -3, 4] 0_1 No No No []
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