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Flat knot 6.306

Min(phi) over symmetries of the knot is: [-3,-3,1,1,1,3,0,1,1,2,3,1,2,3,4,-1,-1,-1,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.306']
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+81t^5+141t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.306']
2-strand cable arrow polynomial of the knot is: -256K14 + 384K1*3K2K3 - 864K1*3K3 - 256K12K2*2K3*2 - 1984K1*2K2*2 + 128K1*2K2K32 + 32K1*2K2K3K5 - 192K12K2K4 + 3688K1*2K2 - 832K12K3*2 - 3464K1*2 + 928K1K23K3 + 224K1K2*2K3K4 - 320K1K22K3 - 32K1K2*2K5 - 512K1K2K3K4 + 4472K1K2K3 + 704K1K3K4 + 48K1K4K5 - 640K2*4 - 832K2*2K3*2 - 72K2*2K4*2 + 592K2*2K4 - 2046K22 + 432K2K3K5 + 8K2K4K6 - 1464K3*2 - 220K4*2 - 32K5*2 - 2K6**2 + 2378
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.306']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11503', 'vk6.11820', 'vk6.12835', 'vk6.13158', 'vk6.20278', 'vk6.21607', 'vk6.27552', 'vk6.29116', 'vk6.31264', 'vk6.31629', 'vk6.32400', 'vk6.32835', 'vk6.38953', 'vk6.41196', 'vk6.45732', 'vk6.47429', 'vk6.52266', 'vk6.52515', 'vk6.53089', 'vk6.53421', 'vk6.57115', 'vk6.58301', 'vk6.61707', 'vk6.62853', 'vk6.63789', 'vk6.63909', 'vk6.64223', 'vk6.64431', 'vk6.66742', 'vk6.67622', 'vk6.69400', 'vk6.70126']
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,-3,1,1,1,3,0,1,1,2,3,1,2,3,4,-1,-1,-1,1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.306']

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+81t^5+141t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.306']

2-strand cable arrow polynomial of the knot is: -256*K1**4 + 384*K1**3*K2*K3 - 864*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 1984*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 192*K1**2*K2*K4 + 3688*K1**2*K2 - 832*K1**2*K3**2 - 3464*K1**2 + 928*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 320*K1*K2**2*K3 - 32*K1*K2**2*K5 - 512*K1*K2*K3*K4 + 4472*K1*K2*K3 + 704*K1*K3*K4 + 48*K1*K4*K5 - 640*K2**4 - 832*K2**2*K3**2 - 72*K2**2*K4**2 + 592*K2**2*K4 - 2046*K2**2 + 432*K2*K3*K5 + 8*K2*K4*K6 - 1464*K3**2 - 220*K4**2 - 32*K5**2 - 2*K6**2 + 2378

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.306']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11503', 'vk6.11820', 'vk6.12835', 'vk6.13158', 'vk6.20278', 'vk6.21607', 'vk6.27552', 'vk6.29116', 'vk6.31264', 'vk6.31629', 'vk6.32400', 'vk6.32835', 'vk6.38953', 'vk6.41196', 'vk6.45732', 'vk6.47429', 'vk6.52266', 'vk6.52515', 'vk6.53089', 'vk6.53421', 'vk6.57115', 'vk6.58301', 'vk6.61707', 'vk6.62853', 'vk6.63789', 'vk6.63909', 'vk6.64223', 'vk6.64431', 'vk6.66742', 'vk6.67622', 'vk6.69400', 'vk6.70126']

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	O1O2O3O4O5U2U1O6U5U6U3U4
R3 orbit	{'O1O2O3O4O5U2U1O6U5U6U3U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U3U6U1O6U5U4
Gauss code of K*	O1O2O3O4U5U6U3U4U1O6O5U2
Gauss code of -K*	O1O2O3O4U3O5O6U4U1U2U6U5
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 -3 -3 1 3 1 1],[ 3 0 0 3 4 2 1],[ 3 0 0 2 3 1 1],[-1 -3 -2 0 1 -1 1],[-3 -4 -3 -1 0 -1 1],[-1 -2 -1 1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 3 1 1 1 -3 -3],[-3 0 1 -1 -1 -3 -4],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 1 -1 -2],[-1 1 1 -1 0 -2 -3],[ 3 3 1 1 2 0 0],[ 3 4 1 2 3 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,-1,3,3,-1,1,1,3,4,1,1,1,1,-1,1,2,2,3,0]
Phi over symmetry	[-3,-3,1,1,1,3,0,1,1,2,3,1,2,3,4,-1,-1,-1,1,1,1]
Phi of -K	[-3,-3,1,1,1,3,0,1,2,3,2,2,3,3,3,1,-1,1,-1,1,3]
Phi of K*	[-3,-1,-1,-1,3,3,1,1,3,2,3,-1,1,1,2,1,2,3,3,3,0]
Phi of -K*	[-3,-3,1,1,1,3,0,1,1,2,3,1,2,3,4,-1,-1,-1,1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^3-3t
Normalized Jones-Krushkal polynomial	6z^2+23z+23
Enhanced Jones-Krushkal polynomial	-2w^4z^2+8w^3z^2-2w^3z+25w^2z+23w
Inner characteristic polynomial	t^6+51t^4+25t^2+1
Outer characteristic polynomial	t^7+81t^5+141t^3+9t
Flat arrow polynomial	-2K1K2 + K1 + K3 + 1
2-strand cable arrow polynomial	-256K14 + 384K1*3K2K3 - 864K1*3K3 - 256K12K2*2K3*2 - 1984K1*2K2*2 + 128K1*2K2K32 + 32K1*2K2K3K5 - 192K12K2K4 + 3688K1*2K2 - 832K12K3*2 - 3464K1*2 + 928K1K23K3 + 224K1K2*2K3K4 - 320K1K22K3 - 32K1K2*2K5 - 512K1K2K3K4 + 4472K1K2K3 + 704K1K3K4 + 48K1K4K5 - 640K2*4 - 832K2*2K3*2 - 72K2*2K4*2 + 592K2*2K4 - 2046K22 + 432K2K3K5 + 8K2K4K6 - 1464K3*2 - 220K4*2 - 32K5*2 - 2K6**2 + 2378
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}], [{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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {2, 5}, {4}, {3}, {1}], [{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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