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Flat knot 6.1013

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,3,1,1,1,2,1,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1013']
Arrow polynomial of the knot is: -6K12 - 4K1K2 + 2K1 + 3K2 + 2K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866']
Outer characteristic polynomial of the knot is: t^7+43t^5+25t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1013']
2-strand cable arrow polynomial of the knot is: -112K14 + 32K1*3K2K3 - 128K1*3K3 - 1664K12K2*2 + 32K1*2K2K32 - 352K1*2K2K4 + 2704K1*2K2 - 112K12K3*2 - 80K1*2K4*2 - 2712K1*2 + 224K1K23K3 - 320K1K2*2K3 - 192K1K2*2K5 - 96K1K2K3K4 + 3128K1K2K3 + 704K1K3K4 + 136K1K4K5 - 392K2*4 - 256K2*2K3*2 - 48K2*2K4*2 + 784K2*2K4 - 1964K22 + 248K2K3K5 + 32K2K4K6 - 1116K3*2 - 438K4*2 - 68K5*2 - 4K6**2 + 2028
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1013']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71458', 'vk6.71490', 'vk6.71493', 'vk6.71509', 'vk6.71528', 'vk6.71549', 'vk6.71552', 'vk6.71984', 'vk6.72002', 'vk6.72039', 'vk6.72055', 'vk6.72523', 'vk6.72543', 'vk6.72641', 'vk6.72663', 'vk6.72914', 'vk6.72954', 'vk6.73117', 'vk6.73141', 'vk6.77079', 'vk6.77113', 'vk6.77117', 'vk6.77127', 'vk6.77150', 'vk6.77168', 'vk6.77171', 'vk6.77463', 'vk6.77466', 'vk6.81290', 'vk6.81430', 'vk6.81533', 'vk6.81541', 'vk6.85470', 'vk6.85473', 'vk6.86886', 'vk6.86909', 'vk6.87254', 'vk6.87723', 'vk6.89349', 'vk6.89506']
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: [-2,-2,0,1,1,2,-1,1,1,2,3,1,1,1,2,1,1,2,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1013']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866']

Outer characteristic polynomial of the knot is: t^7+43t^5+25t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1013']

2-strand cable arrow polynomial of the knot is: -112*K1**4 + 32*K1**3*K2*K3 - 128*K1**3*K3 - 1664*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 2704*K1**2*K2 - 112*K1**2*K3**2 - 80*K1**2*K4**2 - 2712*K1**2 + 224*K1*K2**3*K3 - 320*K1*K2**2*K3 - 192*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 3128*K1*K2*K3 + 704*K1*K3*K4 + 136*K1*K4*K5 - 392*K2**4 - 256*K2**2*K3**2 - 48*K2**2*K4**2 + 784*K2**2*K4 - 1964*K2**2 + 248*K2*K3*K5 + 32*K2*K4*K6 - 1116*K3**2 - 438*K4**2 - 68*K5**2 - 4*K6**2 + 2028

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1013']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71458', 'vk6.71490', 'vk6.71493', 'vk6.71509', 'vk6.71528', 'vk6.71549', 'vk6.71552', 'vk6.71984', 'vk6.72002', 'vk6.72039', 'vk6.72055', 'vk6.72523', 'vk6.72543', 'vk6.72641', 'vk6.72663', 'vk6.72914', 'vk6.72954', 'vk6.73117', 'vk6.73141', 'vk6.77079', 'vk6.77113', 'vk6.77117', 'vk6.77127', 'vk6.77150', 'vk6.77168', 'vk6.77171', 'vk6.77463', 'vk6.77466', 'vk6.81290', 'vk6.81430', 'vk6.81533', 'vk6.81541', 'vk6.85470', 'vk6.85473', 'vk6.86886', 'vk6.86909', 'vk6.87254', 'vk6.87723', 'vk6.89349', 'vk6.89506']

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	O1O2O3O4U2U5O6U4O5U1U6U3
R3 orbit	{'O1O2O3O4U2U5O6U4O5U1U6U3', 'O1O2O3O4U2U5U3O6O5U1U4U6'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4U2U5U4O6U1O5U6U3
Gauss code of K*	O1O2O3U1U4U3U5O4O6U2O5U6
Gauss code of -K*	O1O2O3U4O5U2O4O6U5U1U6U3
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 -2 -2 2 1 0 1],[ 2 0 -1 3 2 1 1],[ 2 1 0 2 1 1 1],[-2 -3 -2 0 0 -2 0],[-1 -2 -1 0 0 -1 0],[ 0 -1 -1 2 1 0 1],[-1 -1 -1 0 0 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -2 -2],[-2 0 0 0 -2 -2 -3],[-1 0 0 0 -1 -1 -1],[-1 0 0 0 -1 -1 -2],[ 0 2 1 1 0 -1 -1],[ 2 2 1 1 1 0 1],[ 2 3 1 2 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,2,2,0,0,2,2,3,0,1,1,1,1,1,2,1,1,-1]
Phi over symmetry	[-2,-2,0,1,1,2,-1,1,1,2,3,1,1,1,2,1,1,2,0,0,0]
Phi of -K	[-2,-2,0,1,1,2,-1,1,2,2,2,1,1,2,1,0,0,0,0,1,1]
Phi of K*	[-2,-1,-1,0,2,2,1,1,0,1,2,0,0,1,2,0,2,2,1,1,-1]
Phi of -K*	[-2,-2,0,1,1,2,-1,1,1,2,3,1,1,1,2,1,1,2,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	4w^3z^2+17w^2z+19w
Inner characteristic polynomial	t^6+29t^4+8t^2
Outer characteristic polynomial	t^7+43t^5+25t^3+3t
Flat arrow polynomial	-6K12 - 4K1K2 + 2K1 + 3K2 + 2K3 + 4
2-strand cable arrow polynomial	-112K14 + 32K1*3K2K3 - 128K1*3K3 - 1664K12K2*2 + 32K1*2K2K32 - 352K1*2K2K4 + 2704K1*2K2 - 112K12K3*2 - 80K1*2K4*2 - 2712K1*2 + 224K1K23K3 - 320K1K2*2K3 - 192K1K2*2K5 - 96K1K2K3K4 + 3128K1K2K3 + 704K1K3K4 + 136K1K4K5 - 392K2*4 - 256K2*2K3*2 - 48K2*2K4*2 + 784K2*2K4 - 1964K22 + 248K2K3K5 + 32K2K4K6 - 1116K3*2 - 438K4*2 - 68K5*2 - 4K6**2 + 2028
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}, {1, 5}, {4}, {3}, {2}], [{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}, {4}, {3}, {1, 2}]]
If K is slice	False

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