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Flat knot 6.516

Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,0,0,2,2,3,1,3,3,3,1,0,-1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.516']
Arrow polynomial of the knot is: -4K12 - 4K1K2 + 2K1 + 2K2 + 2K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932']
Outer characteristic polynomial of the knot is: t^7+88t^5+161t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.516']
2-strand cable arrow polynomial of the knot is: 160K14K2 - 496K14 - 448K1*3K3 + 96K12K2*2K4 - 2032K12K2*2 - 512K1*2K2K4 + 5160K1*2K2 - 80K12K3*2 - 96K1*2K4*2 - 5336K1*2 + 192K1K23K3 - 1088K1K2*2K3 - 192K1K2*2K5 - 192K1K2K3K4 + 5568K1K2K3 - 32K1K2K4K5 + 1680K1K3K4 + 208K1K4K5 + 40K1K5K6 - 880K24 - 592K2*2K3*2 - 80K2*2K4*2 + 1960K2*2K4 - 4228K22 - 64K2K32K4 + 720K2K3K5 + 136K2K4K6 + 48K32K6 - 2468K32 - 1192K4*2 - 244K5*2 - 68K6**2 + 4350
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.516']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73744', 'vk6.73862', 'vk6.74200', 'vk6.74820', 'vk6.75661', 'vk6.75865', 'vk6.76379', 'vk6.76879', 'vk6.78676', 'vk6.78865', 'vk6.79237', 'vk6.79716', 'vk6.80290', 'vk6.80419', 'vk6.80727', 'vk6.81078', 'vk6.81621', 'vk6.81807', 'vk6.82002', 'vk6.82320', 'vk6.82365', 'vk6.82729', 'vk6.83224', 'vk6.84233', 'vk6.84316', 'vk6.84404', 'vk6.84489', 'vk6.85660', 'vk6.86552', 'vk6.87561', 'vk6.88261', 'vk6.89416']
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,-3,1,1,2,2,0,0,2,2,3,1,3,3,3,1,0,-1,0,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.516']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932']

Outer characteristic polynomial of the knot is: t^7+88t^5+161t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.516']

2-strand cable arrow polynomial of the knot is: 160*K1**4*K2 - 496*K1**4 - 448*K1**3*K3 + 96*K1**2*K2**2*K4 - 2032*K1**2*K2**2 - 512*K1**2*K2*K4 + 5160*K1**2*K2 - 80*K1**2*K3**2 - 96*K1**2*K4**2 - 5336*K1**2 + 192*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 192*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5568*K1*K2*K3 - 32*K1*K2*K4*K5 + 1680*K1*K3*K4 + 208*K1*K4*K5 + 40*K1*K5*K6 - 880*K2**4 - 592*K2**2*K3**2 - 80*K2**2*K4**2 + 1960*K2**2*K4 - 4228*K2**2 - 64*K2*K3**2*K4 + 720*K2*K3*K5 + 136*K2*K4*K6 + 48*K3**2*K6 - 2468*K3**2 - 1192*K4**2 - 244*K5**2 - 68*K6**2 + 4350

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.516']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73744', 'vk6.73862', 'vk6.74200', 'vk6.74820', 'vk6.75661', 'vk6.75865', 'vk6.76379', 'vk6.76879', 'vk6.78676', 'vk6.78865', 'vk6.79237', 'vk6.79716', 'vk6.80290', 'vk6.80419', 'vk6.80727', 'vk6.81078', 'vk6.81621', 'vk6.81807', 'vk6.82002', 'vk6.82320', 'vk6.82365', 'vk6.82729', 'vk6.83224', 'vk6.84233', 'vk6.84316', 'vk6.84404', 'vk6.84489', 'vk6.85660', 'vk6.86552', 'vk6.87561', 'vk6.88261', 'vk6.89416']

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	O1O2O3O4O5U2U1U4O6U5U3U6
R3 orbit	{'O1O2O3O4O5U2U1U4O6U5U3U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U3U1O6U2U5U4
Gauss code of K*	O1O2O3U4O5O6O4U2U1U6U3U5
Gauss code of -K*	O1O2O3U1O4O5O6U3U4U2U6U5
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 -3 1 1 2 2],[ 3 0 0 4 2 3 2],[ 3 0 0 3 1 2 2],[-1 -4 -3 0 -1 1 2],[-1 -2 -1 1 0 1 1],[-2 -3 -2 -1 -1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix	[[ 0 2 2 1 1 -3 -3],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -1 -2 -2 -2],[-1 1 1 0 1 -1 -2],[-1 1 2 -1 0 -3 -4],[ 3 2 2 1 3 0 0],[ 3 3 2 2 4 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,-1,3,3,-1,1,1,2,3,1,2,2,2,-1,1,2,3,4,0]
Phi over symmetry	[-3,-3,1,1,2,2,0,0,2,2,3,1,3,3,3,1,0,-1,0,0,-1]
Phi of -K	[-3,-3,1,1,2,2,0,0,2,2,3,1,3,3,3,1,0,-1,0,0,-1]
Phi of K*	[-2,-2,-1,-1,3,3,-1,-1,0,3,3,0,0,2,3,-1,0,1,2,3,0]
Phi of -K*	[-3,-3,1,1,2,2,0,1,3,2,2,2,4,2,3,1,1,1,2,1,-1]
Symmetry type of based matrix	c
u-polynomial	2t^3-2t^2-2t
Normalized Jones-Krushkal polynomial	5z^2+22z+25
Enhanced Jones-Krushkal polynomial	-2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w
Inner characteristic polynomial	t^6+60t^4+27t^2+1
Outer characteristic polynomial	t^7+88t^5+161t^3+9t
Flat arrow polynomial	-4K12 - 4K1K2 + 2K1 + 2K2 + 2K3 + 3
2-strand cable arrow polynomial	160K14K2 - 496K14 - 448K1*3K3 + 96K12K2*2K4 - 2032K12K2*2 - 512K1*2K2K4 + 5160K1*2K2 - 80K12K3*2 - 96K1*2K4*2 - 5336K1*2 + 192K1K23K3 - 1088K1K2*2K3 - 192K1K2*2K5 - 192K1K2K3K4 + 5568K1K2K3 - 32K1K2K4K5 + 1680K1K3K4 + 208K1K4K5 + 40K1K5K6 - 880K24 - 592K2*2K3*2 - 80K2*2K4*2 + 1960K2*2K4 - 4228K22 - 64K2K32K4 + 720K2K3K5 + 136K2K4K6 + 48K32K6 - 2468K32 - 1192K4*2 - 244K5*2 - 68K6**2 + 4350
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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}]]
If K is slice	False

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