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Flat knot 6.1374

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,0,0,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.1374']
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+49t^5+90t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1348', '6.1374', '6.1376']
2-strand cable arrow polynomial of the knot is: 1536K14K2 - 3776K14 + 2048K1*3K2K3 - 1152K1*3K3 + 256K12K2*2K4 - 3520K12K2*2 - 1088K1*2K2K4 + 4688K1*2K2 - 2624K12K3*2 - 192K1*2K4*2 - 1176K1*2 + 64K1K23K3 - 448K1K2*2K3 - 192K1K2*2K5 - 256K1K2K3K4 + 4416K1K2K3 + 2288K1K3K4 + 224K1K4K5 - 160K2*4 - 64K2*2K3*2 - 8K2*2K4*2 + 552K2*2K4 - 1750K22 + 208K2K3K5 + 8K2K4K6 - 1160K3*2 - 572K4*2 - 80K5*2 - 2K6**2 + 1930
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1374']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.82820', 'vk6.82822', 'vk6.82934', 'vk6.82936', 'vk6.82937', 'vk6.82938', 'vk6.83257', 'vk6.83259', 'vk6.83261', 'vk6.83265', 'vk6.86076', 'vk6.86083', 'vk6.86798', 'vk6.86802', 'vk6.86804', 'vk6.86805', 'vk6.89663', 'vk6.89674', 'vk6.89999', 'vk6.90000']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,0,0,1,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.1374']

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+49t^5+90t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1348', '6.1374', '6.1376']

2-strand cable arrow polynomial of the knot is: 1536*K1**4*K2 - 3776*K1**4 + 2048*K1**3*K2*K3 - 1152*K1**3*K3 + 256*K1**2*K2**2*K4 - 3520*K1**2*K2**2 - 1088*K1**2*K2*K4 + 4688*K1**2*K2 - 2624*K1**2*K3**2 - 192*K1**2*K4**2 - 1176*K1**2 + 64*K1*K2**3*K3 - 448*K1*K2**2*K3 - 192*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 4416*K1*K2*K3 + 2288*K1*K3*K4 + 224*K1*K4*K5 - 160*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 552*K2**2*K4 - 1750*K2**2 + 208*K2*K3*K5 + 8*K2*K4*K6 - 1160*K3**2 - 572*K4**2 - 80*K5**2 - 2*K6**2 + 1930

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1374']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.82820', 'vk6.82822', 'vk6.82934', 'vk6.82936', 'vk6.82937', 'vk6.82938', 'vk6.83257', 'vk6.83259', 'vk6.83261', 'vk6.83265', 'vk6.86076', 'vk6.86083', 'vk6.86798', 'vk6.86802', 'vk6.86804', 'vk6.86805', 'vk6.89663', 'vk6.89674', 'vk6.89999', 'vk6.90000']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is +.

The reverse -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	O1O2O3U4O5O6U1U3U2O4U6U5
R3 orbit	{'O1O2O3U4O5O6U1U3U2O4U6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U5O6U2U1U3O5O4U6
Gauss code of K*	Same
Gauss code of -K*	O1O2O3U4U5O6U2U1U3O5O4U6
Diagrammatic symmetry type	+
Flat genus of the diagram	3
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -3 0 0 -1 2 2],[ 3 0 2 1 1 3 2],[ 0 -2 0 0 -1 2 1],[ 0 -1 0 0 0 1 0],[ 1 -1 1 0 0 1 2],[-2 -3 -2 -1 -1 0 0],[-2 -2 -1 0 -2 0 0]]
Primitive based matrix	[[ 0 2 2 0 0 -1 -3],[-2 0 0 0 -1 -2 -2],[-2 0 0 -1 -2 -1 -3],[ 0 0 1 0 0 0 -1],[ 0 1 2 0 0 -1 -2],[ 1 2 1 0 1 0 -1],[ 3 2 3 1 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,0,1,3,0,0,1,2,2,1,2,1,3,0,0,1,1,2,1]
Phi over symmetry	[-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,0,0,1,1,2,0]
Phi of -K	[-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,0,0,1,1,2,0]
Phi of K*	[-2,-2,0,0,1,3,0,0,1,2,2,1,2,1,3,0,0,1,1,2,1]
Phi of -K*	[-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,0,0,1,1,2,0]
Symmetry type of based matrix	+
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	6z^2+23z+23
Enhanced Jones-Krushkal polynomial	6w^3z^2+23w^2z+23w
Inner characteristic polynomial	t^6+31t^4+52t^2+4
Outer characteristic polynomial	t^7+49t^5+90t^3+8t
Flat arrow polynomial	-2K1K2 + K1 + K3 + 1
2-strand cable arrow polynomial	1536K14K2 - 3776K14 + 2048K1*3K2K3 - 1152K1*3K3 + 256K12K2*2K4 - 3520K12K2*2 - 1088K1*2K2K4 + 4688K1*2K2 - 2624K12K3*2 - 192K1*2K4*2 - 1176K1*2 + 64K1K23K3 - 448K1K2*2K3 - 192K1K2*2K5 - 256K1K2K3K4 + 4416K1K2K3 + 2288K1K3K4 + 224K1K4K5 - 160K2*4 - 64K2*2K3*2 - 8K2*2K4*2 + 552K2*2K4 - 1750K22 + 208K2K3K5 + 8K2K4K6 - 1160K3*2 - 572K4*2 - 80K5*2 - 2K6**2 + 1930
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice	False

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