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Flat knot 6.700

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,2,1,2,3,2,2,1,3,0,1,2,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.700']
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+64t^5+146t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.700']
2-strand cable arrow polynomial of the knot is: 352K14K2 - 1840K14 + 640K1*3K2K3 - 992K1*3K3 + 128K12K2*3 + 32K1*2K2*2K4 - 4592K12K2*2 + 256K1*2K2K32 - 896K1*2K2K4 + 8728K1*2K2 - 2064K12K3*2 - 80K1*2K4*2 - 7396K1*2 + 352K1K23K3 - 1152K1K2*2K3 - 352K1K2*2K5 - 320K1K2K3K4 + 9320K1K2K3 - 96K1K2K4K5 + 3048K1K3K4 + 288K1K4K5 + 24K1K5K6 - 504K24 - 480K2*2K3*2 - 48K2*2K4*2 + 1424K2*2K4 - 5460K22 + 536K2K3K5 + 88K2K4K6 - 3272K3*2 - 1226K4*2 - 172K5*2 - 20K6**2 + 5744
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.700']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81574', 'vk6.81577', 'vk6.81656', 'vk6.81658', 'vk6.81730', 'vk6.81732', 'vk6.81861', 'vk6.81866', 'vk6.82240', 'vk6.82244', 'vk6.82383', 'vk6.82387', 'vk6.82499', 'vk6.82503', 'vk6.82578', 'vk6.82584', 'vk6.83172', 'vk6.83174', 'vk6.83608', 'vk6.83610', 'vk6.84141', 'vk6.84147', 'vk6.84332', 'vk6.84336', 'vk6.84563', 'vk6.84570', 'vk6.86487', 'vk6.86491', 'vk6.88732', 'vk6.88736', 'vk6.88909', 'vk6.88921']
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,-2,0,1,1,3,0,2,1,2,3,2,2,1,3,0,1,2,0,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.700']

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+64t^5+146t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.700']

2-strand cable arrow polynomial of the knot is: 352*K1**4*K2 - 1840*K1**4 + 640*K1**3*K2*K3 - 992*K1**3*K3 + 128*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 4592*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 896*K1**2*K2*K4 + 8728*K1**2*K2 - 2064*K1**2*K3**2 - 80*K1**2*K4**2 - 7396*K1**2 + 352*K1*K2**3*K3 - 1152*K1*K2**2*K3 - 352*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 9320*K1*K2*K3 - 96*K1*K2*K4*K5 + 3048*K1*K3*K4 + 288*K1*K4*K5 + 24*K1*K5*K6 - 504*K2**4 - 480*K2**2*K3**2 - 48*K2**2*K4**2 + 1424*K2**2*K4 - 5460*K2**2 + 536*K2*K3*K5 + 88*K2*K4*K6 - 3272*K3**2 - 1226*K4**2 - 172*K5**2 - 20*K6**2 + 5744

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.700']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81574', 'vk6.81577', 'vk6.81656', 'vk6.81658', 'vk6.81730', 'vk6.81732', 'vk6.81861', 'vk6.81866', 'vk6.82240', 'vk6.82244', 'vk6.82383', 'vk6.82387', 'vk6.82499', 'vk6.82503', 'vk6.82578', 'vk6.82584', 'vk6.83172', 'vk6.83174', 'vk6.83608', 'vk6.83610', 'vk6.84141', 'vk6.84147', 'vk6.84332', 'vk6.84336', 'vk6.84563', 'vk6.84570', 'vk6.86487', 'vk6.86491', 'vk6.88732', 'vk6.88736', 'vk6.88909', 'vk6.88921']

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	O1O2O3O4U5O6U1U3O5U2U6U4
R3 orbit	{'O1O2O3O4U5O6U1U3O5U2U6U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U1U5U3O6U2U4O5U6
Gauss code of K*	O1O2O3U4U1U5U3O6U2O4O5U6
Gauss code of -K*	O1O2O3U4O5O6U2O4U1U5U3U6
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 -1 0 3 -1 2],[ 3 0 1 1 3 2 2],[ 1 -1 0 1 3 0 1],[ 0 -1 -1 0 1 0 0],[-3 -3 -3 -1 0 -2 -1],[ 1 -2 0 0 2 0 2],[-2 -2 -1 0 1 -2 0]]
Primitive based matrix	[[ 0 3 2 0 -1 -1 -3],[-3 0 -1 -1 -2 -3 -3],[-2 1 0 0 -2 -1 -2],[ 0 1 0 0 0 -1 -1],[ 1 2 2 0 0 0 -2],[ 1 3 1 1 0 0 -1],[ 3 3 2 1 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,0,1,1,3,1,1,2,3,3,0,2,1,2,0,1,1,0,2,1]
Phi over symmetry	[-3,-2,0,1,1,3,0,2,1,2,3,2,2,1,3,0,1,2,0,1,0]
Phi of -K	[-3,-1,-1,0,2,3,0,1,2,3,3,0,1,1,2,0,2,1,2,2,0]
Phi of K*	[-3,-2,0,1,1,3,0,2,1,2,3,2,2,1,3,0,1,2,0,1,0]
Phi of -K*	[-3,-1,-1,0,2,3,1,2,1,2,3,0,1,1,3,0,2,2,0,1,1]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	4z^2+25z+35
Enhanced Jones-Krushkal polynomial	4w^3z^2+25w^2z+35w
Inner characteristic polynomial	t^6+40t^4+75t^2
Outer characteristic polynomial	t^7+64t^5+146t^3+7t
Flat arrow polynomial	-6K12 - 4K1K2 + 2K1 + 3K2 + 2K3 + 4
2-strand cable arrow polynomial	352K14K2 - 1840K14 + 640K1*3K2K3 - 992K1*3K3 + 128K12K2*3 + 32K1*2K2*2K4 - 4592K12K2*2 + 256K1*2K2K32 - 896K1*2K2K4 + 8728K1*2K2 - 2064K12K3*2 - 80K1*2K4*2 - 7396K1*2 + 352K1K23K3 - 1152K1K2*2K3 - 352K1K2*2K5 - 320K1K2K3K4 + 9320K1K2K3 - 96K1K2K4K5 + 3048K1K3K4 + 288K1K4K5 + 24K1K5K6 - 504K24 - 480K2*2K3*2 - 48K2*2K4*2 + 1424K2*2K4 - 5460K22 + 536K2K3K5 + 88K2K4K6 - 3272K3*2 - 1226K4*2 - 172K5*2 - 20K6**2 + 5744
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}, {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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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