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Flat knot 6.614

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,3,1,2,3,2,1,1,3,1,2,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.614']
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+67t^5+192t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.614']
2-strand cable arrow polynomial of the knot is: 192K14K2 - 2944K14 + 192K1*3K2K3 + 64K1*3K3K4 - 352K1*3K3 + 128K12K2*3 - 2544K1*2K2*2 - 544K1*2K2K4 + 7472K1*2K2 - 1632K12K3*2 - 96K1*2K3K5 - 352K1*2K4*2 - 6116K1*2 - 576K1K22K3 - 32K1K2*2K5 - 64K1K2K3K4 + 6816K1K2K3 + 2976K1K3K4 + 480K1K4K5 - 120K2*4 - 32K2*2K3*2 - 8K2*2K4*2 + 672K2*2K4 - 4678K22 + 80K2K3K5 + 8K2K4K6 - 2972K3*2 - 1162K4*2 - 144K5*2 - 2K6**2 + 5288
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.614']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73289', 'vk6.73299', 'vk6.73432', 'vk6.73442', 'vk6.74090', 'vk6.74103', 'vk6.74661', 'vk6.74672', 'vk6.75432', 'vk6.75446', 'vk6.76128', 'vk6.76141', 'vk6.78162', 'vk6.78180', 'vk6.78394', 'vk6.78412', 'vk6.79100', 'vk6.79105', 'vk6.79983', 'vk6.80005', 'vk6.80136', 'vk6.80158', 'vk6.80608', 'vk6.80613', 'vk6.83803', 'vk6.83822', 'vk6.85112', 'vk6.85125', 'vk6.86594', 'vk6.86607', 'vk6.87387', 'vk6.87397']
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,-2,0,1,2,2,0,3,1,2,3,2,1,1,3,1,2,1,0,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.614']

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+67t^5+192t^3+16t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.614']

2-strand cable arrow polynomial of the knot is: 192*K1**4*K2 - 2944*K1**4 + 192*K1**3*K2*K3 + 64*K1**3*K3*K4 - 352*K1**3*K3 + 128*K1**2*K2**3 - 2544*K1**2*K2**2 - 544*K1**2*K2*K4 + 7472*K1**2*K2 - 1632*K1**2*K3**2 - 96*K1**2*K3*K5 - 352*K1**2*K4**2 - 6116*K1**2 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 6816*K1*K2*K3 + 2976*K1*K3*K4 + 480*K1*K4*K5 - 120*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 672*K2**2*K4 - 4678*K2**2 + 80*K2*K3*K5 + 8*K2*K4*K6 - 2972*K3**2 - 1162*K4**2 - 144*K5**2 - 2*K6**2 + 5288

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.614']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73289', 'vk6.73299', 'vk6.73432', 'vk6.73442', 'vk6.74090', 'vk6.74103', 'vk6.74661', 'vk6.74672', 'vk6.75432', 'vk6.75446', 'vk6.76128', 'vk6.76141', 'vk6.78162', 'vk6.78180', 'vk6.78394', 'vk6.78412', 'vk6.79100', 'vk6.79105', 'vk6.79983', 'vk6.80005', 'vk6.80136', 'vk6.80158', 'vk6.80608', 'vk6.80613', 'vk6.83803', 'vk6.83822', 'vk6.85112', 'vk6.85125', 'vk6.86594', 'vk6.86607', 'vk6.87387', 'vk6.87397']

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	O1O2O3O4U5O6U1O5U2U4U6U3
R3 orbit	{'O1O2O3O4U5O6U1O5U2U4U6U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U2U5U1U3O6U4O5U6
Gauss code of K*	O1O2O3O4U5U1U4U2O6U3O5U6
Gauss code of -K*	O1O2O3O4U5O6U2O5U3U1U4U6
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 -2 2 1 0 2],[ 3 0 0 3 1 3 2],[ 2 0 0 3 1 2 1],[-2 -3 -3 0 -1 -1 0],[-1 -1 -1 1 0 -1 0],[ 0 -3 -2 1 1 0 2],[-2 -2 -1 0 0 -2 0]]
Primitive based matrix	[[ 0 2 2 1 0 -2 -3],[-2 0 0 0 -2 -1 -2],[-2 0 0 -1 -1 -3 -3],[-1 0 1 0 -1 -1 -1],[ 0 2 1 1 0 -2 -3],[ 2 1 3 1 2 0 0],[ 3 2 3 1 3 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,2,3,0,0,2,1,2,1,1,3,3,1,1,1,2,3,0]
Phi over symmetry	[-3,-2,0,1,2,2,0,3,1,2,3,2,1,1,3,1,2,1,0,1,0]
Phi of -K	[-3,-2,0,1,2,2,1,0,3,2,3,0,2,1,3,0,1,0,0,1,0]
Phi of K*	[-2,-2,-1,0,2,3,0,0,1,1,2,1,0,3,3,0,2,3,0,0,1]
Phi of -K*	[-3,-2,0,1,2,2,0,3,1,2,3,2,1,1,3,1,2,1,0,1,0]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	z^2+22z+41
Enhanced Jones-Krushkal polynomial	w^3z^2+22w^2z+41w
Inner characteristic polynomial	t^6+45t^4+113t^2+9
Outer characteristic polynomial	t^7+67t^5+192t^3+16t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	192K14K2 - 2944K14 + 192K1*3K2K3 + 64K1*3K3K4 - 352K1*3K3 + 128K12K2*3 - 2544K1*2K2*2 - 544K1*2K2K4 + 7472K1*2K2 - 1632K12K3*2 - 96K1*2K3K5 - 352K1*2K4*2 - 6116K1*2 - 576K1K22K3 - 32K1K2*2K5 - 64K1K2K3K4 + 6816K1K2K3 + 2976K1K3K4 + 480K1K4K5 - 120K2*4 - 32K2*2K3*2 - 8K2*2K4*2 + 672K2*2K4 - 4678K22 + 80K2K3K5 + 8K2K4K6 - 2972K3*2 - 1162K4*2 - 144K5*2 - 2K6**2 + 5288
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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]]
If K is slice	False

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