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Flat knot 6.833

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,1,3,3,-1,1,1,1,1,0,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.833']
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+52t^5+93t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.833']
2-strand cable arrow polynomial of the knot is: -128K16 + 736K1*4K2 - 3632K14 + 128K1*3K2K3 - 736K1*3K3 + 160K12K2*3 - 3472K1*2K2*2 + 64K1*2K2K32 - 352K1*2K2K4 + 7672K1*2K2 - 944K12K3*2 - 48K1*2K4*2 - 3696K1*2 - 320K1K22K3 - 32K1K2*2K5 - 128K1K2K3K4 + 4912K1K2K3 + 1136K1K3K4 + 56K1K4K5 - 184K2*4 - 80K2*2K3*2 - 8K2*2K4*2 + 464K2*2K4 - 3238K22 + 120K2K3K5 + 8K2K4K6 - 1392K3*2 - 366K4*2 - 24K5*2 - 2K6**2 + 3324
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.833']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4820', 'vk6.5163', 'vk6.6380', 'vk6.6811', 'vk6.8341', 'vk6.8775', 'vk6.9713', 'vk6.10016', 'vk6.11617', 'vk6.11968', 'vk6.12959', 'vk6.20474', 'vk6.20744', 'vk6.21828', 'vk6.27861', 'vk6.29370', 'vk6.31420', 'vk6.32594', 'vk6.39289', 'vk6.39776', 'vk6.41467', 'vk6.46340', 'vk6.47591', 'vk6.47915', 'vk6.49057', 'vk6.49885', 'vk6.51311', 'vk6.51528', 'vk6.53224', 'vk6.57333', 'vk6.62022', 'vk6.64305']
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,-1,0,1,1,2,1,1,1,3,3,-1,1,1,1,1,0,1,-1,-1,1]

Flat knots (up to 7 crossings) with same phi are :['6.833']

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+52t^5+93t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.833']

2-strand cable arrow polynomial of the knot is: -128*K1**6 + 736*K1**4*K2 - 3632*K1**4 + 128*K1**3*K2*K3 - 736*K1**3*K3 + 160*K1**2*K2**3 - 3472*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 7672*K1**2*K2 - 944*K1**2*K3**2 - 48*K1**2*K4**2 - 3696*K1**2 - 320*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 4912*K1*K2*K3 + 1136*K1*K3*K4 + 56*K1*K4*K5 - 184*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 464*K2**2*K4 - 3238*K2**2 + 120*K2*K3*K5 + 8*K2*K4*K6 - 1392*K3**2 - 366*K4**2 - 24*K5**2 - 2*K6**2 + 3324

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.833']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4820', 'vk6.5163', 'vk6.6380', 'vk6.6811', 'vk6.8341', 'vk6.8775', 'vk6.9713', 'vk6.10016', 'vk6.11617', 'vk6.11968', 'vk6.12959', 'vk6.20474', 'vk6.20744', 'vk6.21828', 'vk6.27861', 'vk6.29370', 'vk6.31420', 'vk6.32594', 'vk6.39289', 'vk6.39776', 'vk6.41467', 'vk6.46340', 'vk6.47591', 'vk6.47915', 'vk6.49057', 'vk6.49885', 'vk6.51311', 'vk6.51528', 'vk6.53224', 'vk6.57333', 'vk6.62022', 'vk6.64305']

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	O1O2O3O4U3O5U2U4U6U1O6U5
R3 orbit	{'O1O2O3O4U3O5U2U4U6U1O6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5O6U4U6U1U3O5U2
Gauss code of K*	O1O2O3O4U3O5U4U1U6U2O6U5
Gauss code of -K*	O1O2O3O4U5O6U3U6U4U1O5U2
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 0 -2 -1 1 3 -1],[ 0 0 -1 -1 2 2 0],[ 2 1 0 0 2 2 2],[ 1 1 0 0 1 1 1],[-1 -2 -2 -1 0 1 -1],[-3 -2 -2 -1 -1 0 -3],[ 1 0 -2 -1 1 3 0]]
Primitive based matrix	[[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -1 -3 -2],[-1 1 0 -2 -1 -1 -2],[ 0 2 2 0 -1 0 -1],[ 1 1 1 1 0 1 0],[ 1 3 1 0 -1 0 -2],[ 2 2 2 1 0 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,0,1,1,2,1,2,1,3,2,2,1,1,2,1,0,1,-1,0,2]
Phi over symmetry	[-3,-1,0,1,1,2,1,1,1,3,3,-1,1,1,1,1,0,1,-1,-1,1]
Phi of -K	[-2,-1,-1,0,1,3,-1,1,1,1,3,1,1,1,1,0,1,3,-1,1,1]
Phi of K*	[-3,-1,0,1,1,2,1,1,1,3,3,-1,1,1,1,1,0,1,-1,-1,1]
Phi of -K*	[-2,-1,-1,0,1,3,0,2,1,2,2,1,1,1,1,0,1,3,2,2,1]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	3z^2+22z+33
Enhanced Jones-Krushkal polynomial	3w^3z^2+22w^2z+33w
Inner characteristic polynomial	t^6+36t^4+38t^2+1
Outer characteristic polynomial	t^7+52t^5+93t^3+6t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-128K16 + 736K1*4K2 - 3632K14 + 128K1*3K2K3 - 736K1*3K3 + 160K12K2*3 - 3472K1*2K2*2 + 64K1*2K2K32 - 352K1*2K2K4 + 7672K1*2K2 - 944K12K3*2 - 48K1*2K4*2 - 3696K1*2 - 320K1K22K3 - 32K1K2*2K5 - 128K1K2K3K4 + 4912K1K2K3 + 1136K1K3K4 + 56K1K4K5 - 184K2*4 - 80K2*2K3*2 - 8K2*2K4*2 + 464K2*2K4 - 3238K22 + 120K2K3K5 + 8K2K4K6 - 1392K3*2 - 366K4*2 - 24K5*2 - 2K6**2 + 3324
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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

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