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Flat knot 6.725

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,0,3,3,3,-1,1,2,1,0,0,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.725']
Arrow polynomial of the knot is: -6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']
Outer characteristic polynomial of the knot is: t^7+44t^5+68t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.725', '6.1355']
2-strand cable arrow polynomial of the knot is: -320K16 + 416K1*4K2 - 3424K14 + 64K1*3K2K3 + 32K1*3K3K4 - 256K1*3K3 + 224K12K2*3 - 4080K1*2K2*2 - 288K1*2K2K4 + 7504K1*2K2 - 992K12K3*2 - 32K1*2K3K5 - 128K1*2K4*2 - 4012K1*2 - 416K1K22K3 - 128K1K2K3K4 + 5800K1K2K3 + 1928K1K3K4 + 256K1K4K5 + 24K1K5K6 - 280K24 - 48K2*2K3*2 - 8K2*2K4*2 + 672K2*2K4 - 3590K22 + 176K2K3K5 + 16K2K4K6 - 2152K3*2 - 874K4*2 - 148K5*2 - 18K6**2 + 4080
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.725']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4211', 'vk6.4291', 'vk6.5473', 'vk6.5586', 'vk6.7578', 'vk6.7671', 'vk6.9083', 'vk6.9163', 'vk6.11168', 'vk6.12250', 'vk6.12359', 'vk6.19364', 'vk6.19659', 'vk6.19785', 'vk6.26148', 'vk6.26218', 'vk6.26566', 'vk6.26663', 'vk6.30758', 'vk6.31959', 'vk6.38148', 'vk6.38206', 'vk6.44809', 'vk6.44947', 'vk6.48533', 'vk6.49231', 'vk6.49344', 'vk6.50320', 'vk6.52752', 'vk6.63590', 'vk6.66316', 'vk6.66346']
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,-1,0,1,1,2,1,0,3,3,3,-1,1,2,1,0,0,1,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.725']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']

Outer characteristic polynomial of the knot is: t^7+44t^5+68t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.725', '6.1355']

2-strand cable arrow polynomial of the knot is: -320*K1**6 + 416*K1**4*K2 - 3424*K1**4 + 64*K1**3*K2*K3 + 32*K1**3*K3*K4 - 256*K1**3*K3 + 224*K1**2*K2**3 - 4080*K1**2*K2**2 - 288*K1**2*K2*K4 + 7504*K1**2*K2 - 992*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 4012*K1**2 - 416*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 5800*K1*K2*K3 + 1928*K1*K3*K4 + 256*K1*K4*K5 + 24*K1*K5*K6 - 280*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 672*K2**2*K4 - 3590*K2**2 + 176*K2*K3*K5 + 16*K2*K4*K6 - 2152*K3**2 - 874*K4**2 - 148*K5**2 - 18*K6**2 + 4080

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.725']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4211', 'vk6.4291', 'vk6.5473', 'vk6.5586', 'vk6.7578', 'vk6.7671', 'vk6.9083', 'vk6.9163', 'vk6.11168', 'vk6.12250', 'vk6.12359', 'vk6.19364', 'vk6.19659', 'vk6.19785', 'vk6.26148', 'vk6.26218', 'vk6.26566', 'vk6.26663', 'vk6.30758', 'vk6.31959', 'vk6.38148', 'vk6.38206', 'vk6.44809', 'vk6.44947', 'vk6.48533', 'vk6.49231', 'vk6.49344', 'vk6.50320', 'vk6.52752', 'vk6.63590', 'vk6.66316', 'vk6.66346']

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	O1O2O3O4U1O5U3U5U4O6U2U6
R3 orbit	{'O1O2O3O4U1O5U3U5U4O6U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U3O5U1U6U2O6U4
Gauss code of K*	O1O2O3U4O5O4U6U5U1U3O6U2
Gauss code of -K*	O1O2O3U2O4U1U3U5U4O6O5U6
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 0 -1 2 1 1],[ 3 0 3 1 2 1 1],[ 0 -3 0 -2 1 1 1],[ 1 -1 2 0 2 1 0],[-2 -2 -1 -2 0 0 0],[-1 -1 -1 -1 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -1 -2 -2],[-1 0 0 0 -1 0 -1],[-1 0 0 0 -1 -1 -1],[ 0 1 1 1 0 -2 -3],[ 1 2 0 1 2 0 -1],[ 3 2 1 1 3 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,0,0,1,2,2,0,1,0,1,1,1,1,2,3,1]
Phi over symmetry	[-3,-1,0,1,1,2,1,0,3,3,3,-1,1,2,1,0,0,1,0,1,1]
Phi of -K	[-3,-1,0,1,1,2,1,0,3,3,3,-1,1,2,1,0,0,1,0,1,1]
Phi of K*	[-2,-1,-1,0,1,3,1,1,1,1,3,0,0,1,3,0,2,3,-1,0,1]
Phi of -K*	[-3,-1,0,1,1,2,1,3,1,1,2,2,0,1,2,1,1,1,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	3z^2+24z+37
Enhanced Jones-Krushkal polynomial	3w^3z^2+24w^2z+37w
Inner characteristic polynomial	t^6+28t^4+23t^2+1
Outer characteristic polynomial	t^7+44t^5+68t^3+7t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-320K16 + 416K1*4K2 - 3424K14 + 64K1*3K2K3 + 32K1*3K3K4 - 256K1*3K3 + 224K12K2*3 - 4080K1*2K2*2 - 288K1*2K2K4 + 7504K1*2K2 - 992K12K3*2 - 32K1*2K3K5 - 128K1*2K4*2 - 4012K1*2 - 416K1K22K3 - 128K1K2K3K4 + 5800K1K2K3 + 1928K1K3K4 + 256K1K4K5 + 24K1K5K6 - 280K24 - 48K2*2K3*2 - 8K2*2K4*2 + 672K2*2K4 - 3590K22 + 176K2K3K5 + 16K2K4K6 - 2152K3*2 - 874K4*2 - 148K5*2 - 18K6**2 + 4080
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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}, {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}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {2, 3}, {1}]]
If K is slice	False

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