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Flat knot 6.398

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,3,3,3,0,1,2,1,-1,-1,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.398']
Arrow polynomial of the knot is: -2K1K2 + K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['4.1', '4.3', '6.59', '6.66', '6.112', '6.215', '6.297', '6.306', '6.346', '6.351', '6.352', '6.353', '6.368', '6.393', '6.398', '6.402', '6.420', '6.422', '6.524', '6.529', '6.630', '6.632', '6.633', '6.642', '6.684', '6.707', '6.708', '6.717', '6.719', '6.721', '6.722', '6.737', '6.793', '6.835', '6.837', '6.847', '6.849', '6.857', '6.858', '6.883', '6.902', '6.913', '6.1084', '6.1092', '6.1097', '6.1136', '6.1146', '6.1155', '6.1159', '6.1374', '7.349', '7.365', '7.690', '7.2260', '7.2269', '7.2612', '7.2624', '7.2972', '7.2975', '7.4214', '7.4542', '7.4546', '7.9686', '7.9695', '7.9947', '7.10639', '7.10643', '7.10829', '7.10833', '7.13433', '7.15124', '7.15128', '7.15638', '7.15647', '7.15703', '7.15845', '7.16115', '7.16120', '7.16150', '7.19418', '7.19470', '7.19474', '7.19871', '7.20310', '7.20362', '7.20421', '7.20424', '7.23942', '7.24011', '7.24100', '7.24114', '7.24116', '7.24445', '7.26258', '7.26318', '7.26811', '7.26827', '7.27967', '7.28040', '7.28124', '7.28138', '7.29092', '7.29107', '7.29452', '7.29853', '7.30091', '7.30098', '7.30140', '7.30193', '7.30339', '7.30350', '7.30354']
Outer characteristic polynomial of the knot is: t^7+62t^5+52t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.398']
2-strand cable arrow polynomial of the knot is: 640K14K2 - 2192K14 - 128K1*3K2K3K4 + 480K13K2K3 + 192K1*3K3K4 - 960K1*3K3 + 224K12K2*2K4 - 2624K12K2*2 + 256K1*2K2K32 + 352K1*2K2K42 - 1184K1*2K2K4 + 7112K1*2K2 - 880K12K3*2 - 592K1*2K4*2 - 5824K1*2 + 64K1K23K3 + 224K1K2*2K3K4 - 672K1K22K3 - 960K1K2K3K4 + 6192K1K2K3 - 192K1K2K4K5 + 2512K1K3K4 + 488K1K4K5 - 64K2*4 - 320K2*2K3*2 - 392K2*2K4*2 + 1528K2*2K4 - 4742K22 + 408K2K3K5 + 184K2K4K6 - 2500K3*2 - 1248K4*2 - 116K5*2 - 10K6**2 + 4742
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.398']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11501', 'vk6.11816', 'vk6.12831', 'vk6.13156', 'vk6.17064', 'vk6.17305', 'vk6.20907', 'vk6.21060', 'vk6.22319', 'vk6.22486', 'vk6.23784', 'vk6.28383', 'vk6.31260', 'vk6.31621', 'vk6.32831', 'vk6.35580', 'vk6.36033', 'vk6.40029', 'vk6.40299', 'vk6.42083', 'vk6.43275', 'vk6.46559', 'vk6.46762', 'vk6.48017', 'vk6.52256', 'vk6.53411', 'vk6.57713', 'vk6.57718', 'vk6.58899', 'vk6.59947', 'vk6.64425', 'vk6.69753']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,3,3,3,0,1,2,1,-1,-1,0,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.398']

Arrow polynomial of the knot is: -2*K1*K2 + K1 + K3 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['4.1', '4.3', '6.59', '6.66', '6.112', '6.215', '6.297', '6.306', '6.346', '6.351', '6.352', '6.353', '6.368', '6.393', '6.398', '6.402', '6.420', '6.422', '6.524', '6.529', '6.630', '6.632', '6.633', '6.642', '6.684', '6.707', '6.708', '6.717', '6.719', '6.721', '6.722', '6.737', '6.793', '6.835', '6.837', '6.847', '6.849', '6.857', '6.858', '6.883', '6.902', '6.913', '6.1084', '6.1092', '6.1097', '6.1136', '6.1146', '6.1155', '6.1159', '6.1374', '7.349', '7.365', '7.690', '7.2260', '7.2269', '7.2612', '7.2624', '7.2972', '7.2975', '7.4214', '7.4542', '7.4546', '7.9686', '7.9695', '7.9947', '7.10639', '7.10643', '7.10829', '7.10833', '7.13433', '7.15124', '7.15128', '7.15638', '7.15647', '7.15703', '7.15845', '7.16115', '7.16120', '7.16150', '7.19418', '7.19470', '7.19474', '7.19871', '7.20310', '7.20362', '7.20421', '7.20424', '7.23942', '7.24011', '7.24100', '7.24114', '7.24116', '7.24445', '7.26258', '7.26318', '7.26811', '7.26827', '7.27967', '7.28040', '7.28124', '7.28138', '7.29092', '7.29107', '7.29452', '7.29853', '7.30091', '7.30098', '7.30140', '7.30193', '7.30339', '7.30350', '7.30354']

Outer characteristic polynomial of the knot is: t^7+62t^5+52t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.398']

2-strand cable arrow polynomial of the knot is: 640*K1**4*K2 - 2192*K1**4 - 128*K1**3*K2*K3*K4 + 480*K1**3*K2*K3 + 192*K1**3*K3*K4 - 960*K1**3*K3 + 224*K1**2*K2**2*K4 - 2624*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 352*K1**2*K2*K4**2 - 1184*K1**2*K2*K4 + 7112*K1**2*K2 - 880*K1**2*K3**2 - 592*K1**2*K4**2 - 5824*K1**2 + 64*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 960*K1*K2*K3*K4 + 6192*K1*K2*K3 - 192*K1*K2*K4*K5 + 2512*K1*K3*K4 + 488*K1*K4*K5 - 64*K2**4 - 320*K2**2*K3**2 - 392*K2**2*K4**2 + 1528*K2**2*K4 - 4742*K2**2 + 408*K2*K3*K5 + 184*K2*K4*K6 - 2500*K3**2 - 1248*K4**2 - 116*K5**2 - 10*K6**2 + 4742

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.398']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11501', 'vk6.11816', 'vk6.12831', 'vk6.13156', 'vk6.17064', 'vk6.17305', 'vk6.20907', 'vk6.21060', 'vk6.22319', 'vk6.22486', 'vk6.23784', 'vk6.28383', 'vk6.31260', 'vk6.31621', 'vk6.32831', 'vk6.35580', 'vk6.36033', 'vk6.40029', 'vk6.40299', 'vk6.42083', 'vk6.43275', 'vk6.46559', 'vk6.46762', 'vk6.48017', 'vk6.52256', 'vk6.53411', 'vk6.57713', 'vk6.57718', 'vk6.58899', 'vk6.59947', 'vk6.64425', 'vk6.69753']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is c.

The reverse -K is

The mirror image K* is

The reversed mirror image -K* is

The fillings (up to the first 10) associated to the algebraic genus:

Or click here to check the fillings

invariant	value
Gauss code	O1O2O3O4O5U4U3O6U2U5U1U6
R3 orbit	{'O1O2O3O4O5U4U3O6U2U5U1U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U5U1U4O6U3U2
Gauss code of K*	O1O2O3O4U3U1U5U6U2O6O5U4
Gauss code of -K*	O1O2O3O4U1O5O6U3U6U5U4U2
Diagrammatic symmetry type	c
Flat genus of the diagram	3
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -1 -2 -1 -1 2 3],[ 1 0 -1 -1 -1 3 3],[ 2 1 0 0 0 3 2],[ 1 1 0 0 0 2 1],[ 1 1 0 0 0 1 1],[-2 -3 -3 -2 -1 0 1],[-3 -3 -2 -1 -1 -1 0]]
Primitive based matrix	[[ 0 3 2 -1 -1 -1 -2],[-3 0 -1 -1 -1 -3 -2],[-2 1 0 -1 -2 -3 -3],[ 1 1 1 0 0 1 0],[ 1 1 2 0 0 1 0],[ 1 3 3 -1 -1 0 -1],[ 2 2 3 0 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,1,1,1,2,1,1,1,3,2,1,2,3,3,0,-1,0,-1,0,1]
Phi over symmetry	[-3,-2,1,1,1,2,0,1,3,3,3,0,1,2,1,-1,-1,0,0,1,1]
Phi of -K	[-2,-1,-1,-1,2,3,0,1,1,1,3,1,1,0,1,0,1,3,2,3,0]
Phi of K*	[-3,-2,1,1,1,2,0,1,3,3,3,0,1,2,1,-1,-1,0,0,1,1]
Phi of -K*	[-2,-1,-1,-1,2,3,0,0,1,3,2,0,1,1,1,1,2,1,3,3,1]
Symmetry type of based matrix	c
u-polynomial	-t^3+3t
Normalized Jones-Krushkal polynomial	7z^2+28z+29
Enhanced Jones-Krushkal polynomial	7w^3z^2+28w^2z+29w
Inner characteristic polynomial	t^6+42t^4+20t^2+1
Outer characteristic polynomial	t^7+62t^5+52t^3+8t
Flat arrow polynomial	-2K1K2 + K1 + K3 + 1
2-strand cable arrow polynomial	640K14K2 - 2192K14 - 128K1*3K2K3K4 + 480K13K2K3 + 192K1*3K3K4 - 960K1*3K3 + 224K12K2*2K4 - 2624K12K2*2 + 256K1*2K2K32 + 352K1*2K2K42 - 1184K1*2K2K4 + 7112K1*2K2 - 880K12K3*2 - 592K1*2K4*2 - 5824K1*2 + 64K1K23K3 + 224K1K2*2K3K4 - 672K1K22K3 - 960K1K2K3K4 + 6192K1K2K3 - 192K1K2K4K5 + 2512K1K3K4 + 488K1K4K5 - 64K2*4 - 320K2*2K3*2 - 392K2*2K4*2 + 1528K2*2K4 - 4742K22 + 408K2K3K5 + 184K2K4K6 - 2500K3*2 - 1248K4*2 - 116K5*2 - 10K6**2 + 4742
Genus of based matrix	2
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{2, 6}, {5}, {4}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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