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Flat knot 6.1211

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,1,1,1,3,1,1,1,3,0,0,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1211']
Arrow polynomial of the knot is: -10K12 - 8K1K2 + 4K1 - 4K22 + 5K2 + 4K3 + 2K4 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1211']
Outer characteristic polynomial of the knot is: t^7+45t^5+36t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1211']
2-strand cable arrow polynomial of the knot is: -128K16 - 192K1*4K2*2 + 2432K1*4K2 - 6384K14 + 608K1*3K2K3 + 224K1*3K3K4 - 2208K1*3K3 + 32K13K4K5 + 128K1*2K2*3 - 3840K1*2K2*2 - 992K1*2K2K4 + 11848K1*2K2 - 1872K12K3*2 - 480K1*2K3K5 - 640K1*2K4*2 - 64K1*2K4K6 - 96K1*2K5*2 - 7920K1*2 - 480K1K22K3 - 32K1K2*2K5 - 320K1K2K3K4 + 8384K1K2K3 + 4552K1K3K4 + 1416K1K4K5 + 120K1K5K6 - 136K24 - 192K2*2K3*2 - 96K2*2K4*2 + 928K2*2K4 - 5832K22 - 64K2K32K4 + 464K2K3K5 + 128K2K4K6 - 96K34 - 128K3*2K4*2 + 128K3*2K6 - 3760K32 + 128K3K4K7 - 16K44 + 32K4*2K8 - 2082K42 - 544K5*2 - 88K6*2 - 32K7*2 - 12K8**2 + 7172
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1211']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3664', 'vk6.3759', 'vk6.3954', 'vk6.4049', 'vk6.4498', 'vk6.4593', 'vk6.5880', 'vk6.6007', 'vk6.7155', 'vk6.7334', 'vk6.7425', 'vk6.7937', 'vk6.8056', 'vk6.9367', 'vk6.17909', 'vk6.18004', 'vk6.18754', 'vk6.24448', 'vk6.24873', 'vk6.25336', 'vk6.37493', 'vk6.43883', 'vk6.44220', 'vk6.44525', 'vk6.48288', 'vk6.48351', 'vk6.50077', 'vk6.50191', 'vk6.50582', 'vk6.50645', 'vk6.55852', 'vk6.60712']
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,-2,0,1,1,2,0,1,1,1,3,1,1,1,3,0,0,0,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1211']

Arrow polynomial of the knot is: -10*K1**2 - 8*K1*K2 + 4*K1 - 4*K2**2 + 5*K2 + 4*K3 + 2*K4 + 8

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1211']

Outer characteristic polynomial of the knot is: t^7+45t^5+36t^3+2t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1211']

2-strand cable arrow polynomial of the knot is: -128*K1**6 - 192*K1**4*K2**2 + 2432*K1**4*K2 - 6384*K1**4 + 608*K1**3*K2*K3 + 224*K1**3*K3*K4 - 2208*K1**3*K3 + 32*K1**3*K4*K5 + 128*K1**2*K2**3 - 3840*K1**2*K2**2 - 992*K1**2*K2*K4 + 11848*K1**2*K2 - 1872*K1**2*K3**2 - 480*K1**2*K3*K5 - 640*K1**2*K4**2 - 64*K1**2*K4*K6 - 96*K1**2*K5**2 - 7920*K1**2 - 480*K1*K2**2*K3 - 32*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 8384*K1*K2*K3 + 4552*K1*K3*K4 + 1416*K1*K4*K5 + 120*K1*K5*K6 - 136*K2**4 - 192*K2**2*K3**2 - 96*K2**2*K4**2 + 928*K2**2*K4 - 5832*K2**2 - 64*K2*K3**2*K4 + 464*K2*K3*K5 + 128*K2*K4*K6 - 96*K3**4 - 128*K3**2*K4**2 + 128*K3**2*K6 - 3760*K3**2 + 128*K3*K4*K7 - 16*K4**4 + 32*K4**2*K8 - 2082*K4**2 - 544*K5**2 - 88*K6**2 - 32*K7**2 - 12*K8**2 + 7172

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1211']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3664', 'vk6.3759', 'vk6.3954', 'vk6.4049', 'vk6.4498', 'vk6.4593', 'vk6.5880', 'vk6.6007', 'vk6.7155', 'vk6.7334', 'vk6.7425', 'vk6.7937', 'vk6.8056', 'vk6.9367', 'vk6.17909', 'vk6.18004', 'vk6.18754', 'vk6.24448', 'vk6.24873', 'vk6.25336', 'vk6.37493', 'vk6.43883', 'vk6.44220', 'vk6.44525', 'vk6.48288', 'vk6.48351', 'vk6.50077', 'vk6.50191', 'vk6.50582', 'vk6.50645', 'vk6.55852', 'vk6.60712']

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	O1O2O3O4U2U5U4O5O6U3U1U6
R3 orbit	{'O1O2O3O4U2U5U4O5O6U3U1U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U4U2O5O6U1U6U3
Gauss code of K*	O1O2O3U2U4O5O6O4U6U1U5U3
Gauss code of -K*	O1O2O3U1U4O5O4O6U5U3U6U2
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 -2 0 2 -1 2],[ 1 0 -2 1 2 0 2],[ 2 2 0 2 1 1 1],[ 0 -1 -2 0 1 -1 1],[-2 -2 -1 -1 0 -2 0],[ 1 0 -1 1 2 0 2],[-2 -2 -1 -1 0 -2 0]]
Primitive based matrix	[[ 0 2 2 0 -1 -1 -2],[-2 0 0 -1 -2 -2 -1],[-2 0 0 -1 -2 -2 -1],[ 0 1 1 0 -1 -1 -2],[ 1 2 2 1 0 0 -1],[ 1 2 2 1 0 0 -2],[ 2 1 1 2 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,1,1,2,0,1,2,2,1,1,2,2,1,1,1,2,0,1,2]
Phi over symmetry	[-2,-2,0,1,1,2,0,1,1,1,3,1,1,1,3,0,0,0,0,-1,0]
Phi of -K	[-2,-1,-1,0,2,2,-1,0,0,3,3,0,0,1,1,0,1,1,1,1,0]
Phi of K*	[-2,-2,0,1,1,2,0,1,1,1,3,1,1,1,3,0,0,0,0,-1,0]
Phi of -K*	[-2,-1,-1,0,2,2,1,2,2,1,1,0,1,2,2,1,2,2,1,1,0]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	21z+43
Enhanced Jones-Krushkal polynomial	21w^2z+43w
Inner characteristic polynomial	t^6+31t^4+19t^2
Outer characteristic polynomial	t^7+45t^5+36t^3+2t
Flat arrow polynomial	-10K12 - 8K1K2 + 4K1 - 4K22 + 5K2 + 4K3 + 2K4 + 8
2-strand cable arrow polynomial	-128K16 - 192K1*4K2*2 + 2432K1*4K2 - 6384K14 + 608K1*3K2K3 + 224K1*3K3K4 - 2208K1*3K3 + 32K13K4K5 + 128K1*2K2*3 - 3840K1*2K2*2 - 992K1*2K2K4 + 11848K1*2K2 - 1872K12K3*2 - 480K1*2K3K5 - 640K1*2K4*2 - 64K1*2K4K6 - 96K1*2K5*2 - 7920K1*2 - 480K1K22K3 - 32K1K2*2K5 - 320K1K2K3K4 + 8384K1K2K3 + 4552K1K3K4 + 1416K1K4K5 + 120K1K5K6 - 136K24 - 192K2*2K3*2 - 96K2*2K4*2 + 928K2*2K4 - 5832K22 - 64K2K32K4 + 464K2K3K5 + 128K2K4K6 - 96K34 - 128K3*2K4*2 + 128K3*2K6 - 3760K32 + 128K3K4K7 - 16K44 + 32K4*2K8 - 2082K42 - 544K5*2 - 88K6*2 - 32K7*2 - 12K8**2 + 7172
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}, {2, 5}, {4}, {3}, {1}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{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}, {1, 3}, {2}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}]]
If K is slice	False

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