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Flat knot 6.1348

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,0,1,0,2,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1348']
Arrow polynomial of the knot is: 8K13 - 6K1K2 - 3K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.67', '6.535', '6.1347', '6.1348', '6.1368']
Outer characteristic polynomial of the knot is: t^7+49t^5+90t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1348', '6.1374', '6.1376']
2-strand cable arrow polynomial of the knot is: 1536K14K2 - 1984K14 + 512K1*3K2K3 - 256K1*3K3 - 768K12K2*4 + 2688K1*2K2*3 + 256K1*2K2*2K4 - 9216K12K2*2 - 640K1*2K2K4 + 6464K1*2K2 - 320K12K3*2 - 2392K1*2 + 1344K1K23K3 - 2304K1K2*2K3 - 576K1K2*2K5 - 192K1K2K3K4 + 6608K1K2K3 + 672K1K3K4 + 96K1K4K5 - 704K2*6 + 768K2*4K4 - 3584K24 - 160K2*3K6 - 896K22K3*2 - 88K2*2K4*2 + 2984K2*2K4 - 1270K22 + 784K2K3K5 + 24K2K4K6 - 1240K3*2 - 476K4*2 - 176K5*2 - 2K6**2 + 2506
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1348']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74177', 'vk6.74181', 'vk6.74783', 'vk6.74787', 'vk6.76329', 'vk6.76848', 'vk6.76855', 'vk6.79210', 'vk6.79679', 'vk6.79685', 'vk6.81057', 'vk6.81060', 'vk6.82915', 'vk6.83431', 'vk6.85257', 'vk6.85315', 'vk6.85319', 'vk6.86568', 'vk6.87536', 'vk6.89246']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,0,1,0,2,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1348']

Arrow polynomial of the knot is: 8*K1**3 - 6*K1*K2 - 3*K1 + K3 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.67', '6.535', '6.1347', '6.1348', '6.1368']

Outer characteristic polynomial of the knot is: t^7+49t^5+90t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1348', '6.1374', '6.1376']

2-strand cable arrow polynomial of the knot is: 1536*K1**4*K2 - 1984*K1**4 + 512*K1**3*K2*K3 - 256*K1**3*K3 - 768*K1**2*K2**4 + 2688*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9216*K1**2*K2**2 - 640*K1**2*K2*K4 + 6464*K1**2*K2 - 320*K1**2*K3**2 - 2392*K1**2 + 1344*K1*K2**3*K3 - 2304*K1*K2**2*K3 - 576*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6608*K1*K2*K3 + 672*K1*K3*K4 + 96*K1*K4*K5 - 704*K2**6 + 768*K2**4*K4 - 3584*K2**4 - 160*K2**3*K6 - 896*K2**2*K3**2 - 88*K2**2*K4**2 + 2984*K2**2*K4 - 1270*K2**2 + 784*K2*K3*K5 + 24*K2*K4*K6 - 1240*K3**2 - 476*K4**2 - 176*K5**2 - 2*K6**2 + 2506

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1348']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.74177', 'vk6.74181', 'vk6.74783', 'vk6.74787', 'vk6.76329', 'vk6.76848', 'vk6.76855', 'vk6.79210', 'vk6.79679', 'vk6.79685', 'vk6.81057', 'vk6.81060', 'vk6.82915', 'vk6.83431', 'vk6.85257', 'vk6.85315', 'vk6.85319', 'vk6.86568', 'vk6.87536', 'vk6.89246']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is +.

The reverse -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	O1O2O3U1O4O5U6U3U2O6U4U5
R3 orbit	{'O1O2O3U1O4O5U6U3U2O6U4U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U5O6U2U1U6O4O5U3
Gauss code of K*	Same
Gauss code of -K*	O1O2O3U4U5O6U2U1U6O4O5U3
Diagrammatic symmetry type	+
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 0 0 1 3 -2],[ 2 0 2 1 2 2 0],[ 0 -2 0 0 1 2 -1],[ 0 -1 0 0 0 1 0],[-1 -2 -1 0 0 1 -1],[-3 -2 -2 -1 -1 0 -3],[ 2 0 1 0 1 3 0]]
Primitive based matrix	[[ 0 3 1 0 0 -2 -2],[-3 0 -1 -1 -2 -2 -3],[-1 1 0 0 -1 -2 -1],[ 0 1 0 0 0 -1 0],[ 0 2 1 0 0 -2 -1],[ 2 2 2 1 2 0 0],[ 2 3 1 0 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,0,1,0,2,1,0]
Phi over symmetry	[-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,0,1,0,2,1,0]
Phi of -K	[-2,-2,0,0,1,3,0,0,1,1,3,1,2,2,2,0,0,1,1,2,1]
Phi of K*	[-3,-1,0,0,2,2,1,1,2,2,3,0,1,2,1,0,1,0,2,1,0]
Phi of -K*	[-2,-2,0,0,1,3,0,0,1,1,3,1,2,2,2,0,0,1,1,2,1]
Symmetry type of based matrix	+
u-polynomial	-t^3+2t^2-t
Normalized Jones-Krushkal polynomial	8z^2+25z+19
Enhanced Jones-Krushkal polynomial	8w^3z^2+25w^2z+19w
Inner characteristic polynomial	t^6+31t^4+52t^2+4
Outer characteristic polynomial	t^7+49t^5+90t^3+8t
Flat arrow polynomial	8K13 - 6K1K2 - 3K1 + K3 + 1
2-strand cable arrow polynomial	1536K14K2 - 1984K14 + 512K1*3K2K3 - 256K1*3K3 - 768K12K2*4 + 2688K1*2K2*3 + 256K1*2K2*2K4 - 9216K12K2*2 - 640K1*2K2K4 + 6464K1*2K2 - 320K12K3*2 - 2392K1*2 + 1344K1K23K3 - 2304K1K2*2K3 - 576K1K2*2K5 - 192K1K2K3K4 + 6608K1K2K3 + 672K1K3K4 + 96K1K4K5 - 704K2*6 + 768K2*4K4 - 3584K24 - 160K2*3K6 - 896K22K3*2 - 88K2*2K4*2 + 2984K2*2K4 - 1270K22 + 784K2K3K5 + 24K2K4K6 - 1240K3*2 - 476K4*2 - 176K5*2 - 2K6**2 + 2506
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice	False

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