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Flat knot 6.1050

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,0,1,3,-1,1,0,-1,0,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1050']
Arrow polynomial of the knot is: 8K13 - 8K1*2 - 6K1K2 - 3K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']
Outer characteristic polynomial of the knot is: t^7+52t^5+217t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1050']
2-strand cable arrow polynomial of the knot is: -256K16 + 1984K1*4K2 - 5248K14 + 288K1*3K2K3 + 32K1*3K3K4 - 1504K1*3K3 - 192K12K2*4 + 1504K1*2K2*3 + 256K1*2K2*2K4 - 7168K12K2*2 + 128K1*2K2K32 - 736K1*2K2K4 + 12816K1*2K2 - 1088K12K3*2 - 64K1*2K3K5 - 112K1*2K4*2 - 7240K1*2 + 256K1K23K3 + 32K1K2*2K3K4 - 1824K1K22K3 - 192K1K2*2K5 - 288K1K2K3K4 + 9480K1K2K3 - 32K1K2K4K5 + 1936K1K3K4 + 168K1K4K5 + 8K1K5K6 - 64K26 + 96K2*4K4 - 1472K24 - 32K2*3K6 - 384K22K3*2 - 40K2*2K4*2 + 1856K2*2K4 - 5722K22 + 400K2K3K5 + 32K2K4K6 - 2864K3*2 - 828K4*2 - 96K5*2 - 6K6**2 + 6178
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1050']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4815', 'vk6.5160', 'vk6.6377', 'vk6.6810', 'vk6.8342', 'vk6.8778', 'vk6.9716', 'vk6.10021', 'vk6.11622', 'vk6.11975', 'vk6.12968', 'vk6.20464', 'vk6.20741', 'vk6.21818', 'vk6.27851', 'vk6.29360', 'vk6.31425', 'vk6.32603', 'vk6.39284', 'vk6.39781', 'vk6.41464', 'vk6.46341', 'vk6.47589', 'vk6.47918', 'vk6.49054', 'vk6.49884', 'vk6.51314', 'vk6.51533', 'vk6.53217', 'vk6.57335', 'vk6.62024', 'vk6.64298']
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: [-3,0,0,1,1,1,1,2,0,1,3,-1,1,0,-1,0,0,1,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1050']

Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']

Outer characteristic polynomial of the knot is: t^7+52t^5+217t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1050']

2-strand cable arrow polynomial of the knot is: -256*K1**6 + 1984*K1**4*K2 - 5248*K1**4 + 288*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1504*K1**3*K3 - 192*K1**2*K2**4 + 1504*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 7168*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 736*K1**2*K2*K4 + 12816*K1**2*K2 - 1088*K1**2*K3**2 - 64*K1**2*K3*K5 - 112*K1**2*K4**2 - 7240*K1**2 + 256*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 192*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 9480*K1*K2*K3 - 32*K1*K2*K4*K5 + 1936*K1*K3*K4 + 168*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1472*K2**4 - 32*K2**3*K6 - 384*K2**2*K3**2 - 40*K2**2*K4**2 + 1856*K2**2*K4 - 5722*K2**2 + 400*K2*K3*K5 + 32*K2*K4*K6 - 2864*K3**2 - 828*K4**2 - 96*K5**2 - 6*K6**2 + 6178

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1050']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4815', 'vk6.5160', 'vk6.6377', 'vk6.6810', 'vk6.8342', 'vk6.8778', 'vk6.9716', 'vk6.10021', 'vk6.11622', 'vk6.11975', 'vk6.12968', 'vk6.20464', 'vk6.20741', 'vk6.21818', 'vk6.27851', 'vk6.29360', 'vk6.31425', 'vk6.32603', 'vk6.39284', 'vk6.39781', 'vk6.41464', 'vk6.46341', 'vk6.47589', 'vk6.47918', 'vk6.49054', 'vk6.49884', 'vk6.51314', 'vk6.51533', 'vk6.53217', 'vk6.57335', 'vk6.62024', 'vk6.64298']

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	O1O2O3O4U5U6O5U2O6U3U1U4
R3 orbit	{'O1O2O3O4U5U6O5U2O6U3U1U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U1U4U2O5U3O6U5U6
Gauss code of K*	O1O2O3U2U4U1U3O5O6U5O4U6
Gauss code of -K*	O1O2O3U4O5U6O4O6U1U3U5U2
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 -1 0 3 -1 0],[ 1 0 0 1 3 0 1],[ 1 0 0 0 1 1 2],[ 0 -1 0 0 1 -1 1],[-3 -3 -1 -1 0 -4 -2],[ 1 0 -1 1 4 0 0],[ 0 -1 -2 -1 2 0 0]]
Primitive based matrix	[[ 0 3 0 0 -1 -1 -1],[-3 0 -1 -2 -1 -3 -4],[ 0 1 0 1 0 -1 -1],[ 0 2 -1 0 -2 -1 0],[ 1 1 0 2 0 0 1],[ 1 3 1 1 0 0 0],[ 1 4 1 0 -1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,0,0,1,1,1,1,2,1,3,4,-1,0,1,1,2,1,0,0,-1,0]
Phi over symmetry	[-3,0,0,1,1,1,1,2,0,1,3,-1,1,0,-1,0,0,1,0,-1,0]
Phi of -K	[-1,-1,-1,0,0,3,-1,0,-1,1,3,0,1,0,0,0,0,1,1,1,2]
Phi of K*	[-3,0,0,1,1,1,1,2,0,1,3,-1,1,0,-1,0,0,1,0,-1,0]
Phi of -K*	[-1,-1,-1,0,0,3,-1,0,0,1,4,0,2,0,1,1,1,3,-1,2,1]
Symmetry type of based matrix	c
u-polynomial	-t^3+3t
Normalized Jones-Krushkal polynomial	3z^2+24z+37
Enhanced Jones-Krushkal polynomial	3w^3z^2+24w^2z+37w
Inner characteristic polynomial	t^6+40t^4+161t^2+1
Outer characteristic polynomial	t^7+52t^5+217t^3+8t
Flat arrow polynomial	8K13 - 8K1*2 - 6K1K2 - 3K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial	-256K16 + 1984K1*4K2 - 5248K14 + 288K1*3K2K3 + 32K1*3K3K4 - 1504K1*3K3 - 192K12K2*4 + 1504K1*2K2*3 + 256K1*2K2*2K4 - 7168K12K2*2 + 128K1*2K2K32 - 736K1*2K2K4 + 12816K1*2K2 - 1088K12K3*2 - 64K1*2K3K5 - 112K1*2K4*2 - 7240K1*2 + 256K1K23K3 + 32K1K2*2K3K4 - 1824K1K22K3 - 192K1K2*2K5 - 288K1K2K3K4 + 9480K1K2K3 - 32K1K2K4K5 + 1936K1K3K4 + 168K1K4K5 + 8K1K5K6 - 64K26 + 96K2*4K4 - 1472K24 - 32K2*3K6 - 384K22K3*2 - 40K2*2K4*2 + 1856K2*2K4 - 5722K22 + 400K2K3K5 + 32K2K4K6 - 2864K3*2 - 828K4*2 - 96K5*2 - 6K6**2 + 6178
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}], [{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}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

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