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Flat knot 6.1080

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,2,0,0,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1080']
Arrow polynomial of the knot is: 4K13 - 6K1*2 - 8K1K2 + K1 + 3K2 + 3*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1080', '6.1837', '6.1841', '6.1865']
Outer characteristic polynomial of the knot is: t^7+24t^5+27t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1080']
2-strand cable arrow polynomial of the knot is: -64K16 + 288K1*4K2 - 960K14 + 32K1*3K2K3 - 544K1*3K3 + 192K12K2*3 - 1296K1*2K2*2 + 160K1*2K2K32 - 128K1*2K2K4 + 3880K1*2K2 - 512K12K3*2 - 64K1*2K3K5 - 48K1*2K4*2 - 3068K1*2 + 96K1K23K3 - 832K1K2*2K3 - 32K1K2*2K5 + 32K1K2K33 - 96K1K2K3K4 - 32K1K2K3K6 + 3304K1K2K3 + 992K1K3K4 + 80K1K4K5 + 8K1K5K6 - 32K2*6 + 64K2*4K4 - 312K24 - 32K2*3K6 - 400K22K3*2 - 64K2*2K4*2 + 584K2*2K4 - 2154K22 - 32K2K32K4 + 272K2K3K5 + 56K2K4K6 - 32K34 + 40K3*2K6 - 1188K32 - 354K4*2 - 48K5*2 - 14K6**2 + 2256
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1080']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4207', 'vk6.4286', 'vk6.5464', 'vk6.5575', 'vk6.7574', 'vk6.7664', 'vk6.9076', 'vk6.9155', 'vk6.11183', 'vk6.12267', 'vk6.12374', 'vk6.19385', 'vk6.19678', 'vk6.19773', 'vk6.26169', 'vk6.26206', 'vk6.26585', 'vk6.26651', 'vk6.30773', 'vk6.31320', 'vk6.31715', 'vk6.31974', 'vk6.32474', 'vk6.32889', 'vk6.38169', 'vk6.38186', 'vk6.39082', 'vk6.41338', 'vk6.44830', 'vk6.44927', 'vk6.45834', 'vk6.48517', 'vk6.49317', 'vk6.52312', 'vk6.53152', 'vk6.58430', 'vk6.62950', 'vk6.63601', 'vk6.66321', 'vk6.66334']
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,-1,0,1,1,1,0,1,1,1,2,1,1,1,2,0,0,0,-1,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1080']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1080', '6.1837', '6.1841', '6.1865']

Outer characteristic polynomial of the knot is: t^7+24t^5+27t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1080']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 288*K1**4*K2 - 960*K1**4 + 32*K1**3*K2*K3 - 544*K1**3*K3 + 192*K1**2*K2**3 - 1296*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 128*K1**2*K2*K4 + 3880*K1**2*K2 - 512*K1**2*K3**2 - 64*K1**2*K3*K5 - 48*K1**2*K4**2 - 3068*K1**2 + 96*K1*K2**3*K3 - 832*K1*K2**2*K3 - 32*K1*K2**2*K5 + 32*K1*K2*K3**3 - 96*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3304*K1*K2*K3 + 992*K1*K3*K4 + 80*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 312*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 64*K2**2*K4**2 + 584*K2**2*K4 - 2154*K2**2 - 32*K2*K3**2*K4 + 272*K2*K3*K5 + 56*K2*K4*K6 - 32*K3**4 + 40*K3**2*K6 - 1188*K3**2 - 354*K4**2 - 48*K5**2 - 14*K6**2 + 2256

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1080']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4207', 'vk6.4286', 'vk6.5464', 'vk6.5575', 'vk6.7574', 'vk6.7664', 'vk6.9076', 'vk6.9155', 'vk6.11183', 'vk6.12267', 'vk6.12374', 'vk6.19385', 'vk6.19678', 'vk6.19773', 'vk6.26169', 'vk6.26206', 'vk6.26585', 'vk6.26651', 'vk6.30773', 'vk6.31320', 'vk6.31715', 'vk6.31974', 'vk6.32474', 'vk6.32889', 'vk6.38169', 'vk6.38186', 'vk6.39082', 'vk6.41338', 'vk6.44830', 'vk6.44927', 'vk6.45834', 'vk6.48517', 'vk6.49317', 'vk6.52312', 'vk6.53152', 'vk6.58430', 'vk6.62950', 'vk6.63601', 'vk6.66321', 'vk6.66334']

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	O1O2O3O4U5U4O6U3O5U1U6U2
R3 orbit	{'O1O2O3O4U5U4O6U3O5U1U6U2', 'O1O2O3O4U5U4U2O6O5U1U3U6'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4U3U5U4O6U2O5U1U6
Gauss code of K*	O1O2O3U1U3U4U5O6O5U2O4U6
Gauss code of -K*	O1O2O3U4O5U2O6O4U6U5U1U3
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 1 0 1 -1 1],[ 2 0 2 1 1 0 1],[-1 -2 0 0 1 -2 0],[ 0 -1 0 0 0 -1 0],[-1 -1 -1 0 0 -1 -1],[ 1 0 2 1 1 0 1],[-1 -1 0 0 1 -1 0]]
Primitive based matrix	[[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -1],[-1 -1 0 -1 0 -1 -1],[-1 0 1 0 0 -2 -2],[ 0 0 0 0 0 -1 -1],[ 1 1 1 2 1 0 0],[ 2 1 1 2 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,1,2,-1,0,0,1,1,1,0,1,1,0,2,2,1,1,0]
Phi over symmetry	[-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,2,0,0,0,-1,-1,0]
Phi of -K	[-2,-1,0,1,1,1,1,1,1,2,2,0,0,1,1,1,1,1,-1,0,1]
Phi of K*	[-1,-1,-1,0,1,2,-1,-1,1,1,2,0,1,0,1,1,1,2,0,1,1]
Phi of -K*	[-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,2,0,0,0,-1,-1,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	z^2+14z+25
Enhanced Jones-Krushkal polynomial	w^3z^2+14w^2z+25w
Inner characteristic polynomial	t^6+16t^4+6t^2
Outer characteristic polynomial	t^7+24t^5+27t^3+3t
Flat arrow polynomial	4K13 - 6K1*2 - 8K1K2 + K1 + 3K2 + 3*K3 + 4
2-strand cable arrow polynomial	-64K16 + 288K1*4K2 - 960K14 + 32K1*3K2K3 - 544K1*3K3 + 192K12K2*3 - 1296K1*2K2*2 + 160K1*2K2K32 - 128K1*2K2K4 + 3880K1*2K2 - 512K12K3*2 - 64K1*2K3K5 - 48K1*2K4*2 - 3068K1*2 + 96K1K23K3 - 832K1K2*2K3 - 32K1K2*2K5 + 32K1K2K33 - 96K1K2K3K4 - 32K1K2K3K6 + 3304K1K2K3 + 992K1K3K4 + 80K1K4K5 + 8K1K5K6 - 32K2*6 + 64K2*4K4 - 312K24 - 32K2*3K6 - 400K22K3*2 - 64K2*2K4*2 + 584K2*2K4 - 2154K22 - 32K2K32K4 + 272K2K3K5 + 56K2K4K6 - 32K34 + 40K3*2K6 - 1188K32 - 354K4*2 - 48K5*2 - 14K6**2 + 2256
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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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