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Flat knot 6.1052

Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,2,2,4,3,-1,1,0,1,0,0,2,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1052']
Arrow polynomial of the knot is: 8K13 - 6K1*2 - 6K1K2 - 3K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.237', '6.602', '6.956', '6.986', '6.992', '6.1052', '6.1059']
Outer characteristic polynomial of the knot is: t^7+57t^5+211t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1052']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 448K1*4K2 - 1312K14 + 384K1*3K2K3 - 416K1*3K3 - 320K12K2*4 + 1280K1*2K2*3 - 192K1*2K2*2K3*2 + 64K1*2K2*2K4 - 5232K12K2*2 + 128K1*2K2K32 + 32K1*2K2K3K5 - 288K12K2K4 + 6880K1*2K2 - 320K12K3*2 - 4468K1*2 + 896K1K23K3 + 96K1K2*2K3K4 - 1088K1K22K3 - 128K1K2*2K5 - 96K1K2K3K4 + 5112K1K2K3 + 416K1K3K4 - 64K26 + 96K2*4K4 - 1320K24 - 608K2*2K3*2 - 72K2*2K4*2 + 968K2*2K4 - 2566K22 + 216K2K3K5 + 16K2K4K6 - 1216K3*2 - 186K4*2 - 12K5*2 - 2K6**2 + 3128
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1052']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4743', 'vk6.5072', 'vk6.6281', 'vk6.6722', 'vk6.8246', 'vk6.8697', 'vk6.9636', 'vk6.9953', 'vk6.20400', 'vk6.21751', 'vk6.27740', 'vk6.29276', 'vk6.39178', 'vk6.41412', 'vk6.45912', 'vk6.47547', 'vk6.48783', 'vk6.48996', 'vk6.49599', 'vk6.49804', 'vk6.50803', 'vk6.51020', 'vk6.51286', 'vk6.51483', 'vk6.57269', 'vk6.58496', 'vk6.61923', 'vk6.63022', 'vk6.66886', 'vk6.67770', 'vk6.69524', 'vk6.70232']
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: [-3,0,0,0,1,2,0,2,2,4,3,-1,1,0,1,0,0,2,1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.1052']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.237', '6.602', '6.956', '6.986', '6.992', '6.1052', '6.1059']

Outer characteristic polynomial of the knot is: t^7+57t^5+211t^3+13t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1052']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 1312*K1**4 + 384*K1**3*K2*K3 - 416*K1**3*K3 - 320*K1**2*K2**4 + 1280*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 5232*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 288*K1**2*K2*K4 + 6880*K1**2*K2 - 320*K1**2*K3**2 - 4468*K1**2 + 896*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 128*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5112*K1*K2*K3 + 416*K1*K3*K4 - 64*K2**6 + 96*K2**4*K4 - 1320*K2**4 - 608*K2**2*K3**2 - 72*K2**2*K4**2 + 968*K2**2*K4 - 2566*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 1216*K3**2 - 186*K4**2 - 12*K5**2 - 2*K6**2 + 3128

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1052']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4743', 'vk6.5072', 'vk6.6281', 'vk6.6722', 'vk6.8246', 'vk6.8697', 'vk6.9636', 'vk6.9953', 'vk6.20400', 'vk6.21751', 'vk6.27740', 'vk6.29276', 'vk6.39178', 'vk6.41412', 'vk6.45912', 'vk6.47547', 'vk6.48783', 'vk6.48996', 'vk6.49599', 'vk6.49804', 'vk6.50803', 'vk6.51020', 'vk6.51286', 'vk6.51483', 'vk6.57269', 'vk6.58496', 'vk6.61923', 'vk6.63022', 'vk6.66886', 'vk6.67770', 'vk6.69524', 'vk6.70232']

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	O1O2O3O4U5U6O5U3O6U1U2U4
R3 orbit	{'O1O2O3O4U5U6O5U3O6U1U2U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U1U3U4O5U2O6U5U6
Gauss code of K*	O1O2O3U1U2U4U3O5O6U5O4U6
Gauss code of -K*	O1O2O3U4O5U6O4O6U1U5U2U3
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 0 0 3 -1 0],[ 2 0 1 1 3 1 2],[ 0 -1 0 1 2 -1 0],[ 0 -1 -1 0 0 0 1],[-3 -3 -2 0 0 -4 -2],[ 1 -1 1 0 4 0 0],[ 0 -2 0 -1 2 0 0]]
Primitive based matrix	[[ 0 3 0 0 0 -1 -2],[-3 0 0 -2 -2 -4 -3],[ 0 0 0 1 -1 0 -1],[ 0 2 -1 0 0 0 -2],[ 0 2 1 0 0 -1 -1],[ 1 4 0 0 1 0 -1],[ 2 3 1 2 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,0,0,0,1,2,0,2,2,4,3,-1,1,0,1,0,0,2,1,1,1]
Phi over symmetry	[-3,0,0,0,1,2,0,2,2,4,3,-1,1,0,1,0,0,2,1,1,1]
Phi of -K	[-2,-1,0,0,0,3,0,0,1,1,2,1,0,1,0,0,1,1,-1,1,3]
Phi of K*	[-3,0,0,0,1,2,1,1,3,0,2,0,-1,1,0,1,0,1,1,1,0]
Phi of -K*	[-2,-1,0,0,0,3,1,1,1,2,3,0,1,0,4,-1,1,0,0,2,2]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	3z^2+20z+29
Enhanced Jones-Krushkal polynomial	3w^3z^2+20w^2z+29w
Inner characteristic polynomial	t^6+43t^4+154t^2+9
Outer characteristic polynomial	t^7+57t^5+211t^3+13t
Flat arrow polynomial	8K13 - 6K1*2 - 6K1K2 - 3K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial	-256K14K2*2 + 448K1*4K2 - 1312K14 + 384K1*3K2K3 - 416K1*3K3 - 320K12K2*4 + 1280K1*2K2*3 - 192K1*2K2*2K3*2 + 64K1*2K2*2K4 - 5232K12K2*2 + 128K1*2K2K32 + 32K1*2K2K3K5 - 288K12K2K4 + 6880K1*2K2 - 320K12K3*2 - 4468K1*2 + 896K1K23K3 + 96K1K2*2K3K4 - 1088K1K22K3 - 128K1K2*2K5 - 96K1K2K3K4 + 5112K1K2K3 + 416K1K3K4 - 64K26 + 96K2*4K4 - 1320K24 - 608K2*2K3*2 - 72K2*2K4*2 + 968K2*2K4 - 2566K22 + 216K2K3K5 + 16K2K4K6 - 1216K3*2 - 186K4*2 - 12K5*2 - 2K6**2 + 3128
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice	False

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