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Flat knot 6.1261

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,0,1,2,2,0,0,1,1,1,2,2,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1261']
Arrow polynomial of the knot is: 4K12K2 - 8K12 - 4K1K2 - 4K1K3 + 2K1 + 4K2 + 2K3 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1261']
Outer characteristic polynomial of the knot is: t^7+34t^5+33t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1261']
2-strand cable arrow polynomial of the knot is: -224K14 + 384K1*3K2K3 - 704K1*3K3 + 128K12K2*3 - 128K1*2K2*2K3*2 - 3104K1*2K2*2 + 192K1*2K2K32 - 384K1*2K2K4 + 4912K1*2K2 - 1088K12K3*2 - 192K1*2K3K5 - 32K1*2K5*2 - 4888K1*2 + 1280K1K23K3 + 256K1K2*2K3K4 - 1216K1K22K3 + 64K1K2*2K4K5 - 128K1K22K5 + 64K1K2K33 - 448K1K2K3K4 - 192K1K2K3K6 + 6704K1K2K3 - 64K1K2K5K6 + 1440K1K3K4 + 304K1K4K5 + 112K1K5K6 + 16K1K6K7 - 32K24K4*2 + 128K2*4K4 - 1328K24 + 64K2*3K4K6 - 64K2*3K6 - 1376K22K3*2 - 272K2*2K4*2 + 1504K2*2K4 - 64K22K5*2 - 48K2*2K6*2 - 3212K2*2 + 912K2K3K5 + 240K2K4K6 + 80K2K5K7 + 16K2K6K8 - 2312K3*2 - 648K4*2 - 280K5*2 - 84K6*2 - 24K7*2 - 2K8**2 + 3736
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1261']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71645', 'vk6.71664', 'vk6.71824', 'vk6.71832', 'vk6.72242', 'vk6.72263', 'vk6.72372', 'vk6.77261', 'vk6.77359', 'vk6.77382', 'vk6.77605', 'vk6.77622', 'vk6.77699', 'vk6.77721', 'vk6.81410', 'vk6.81442', 'vk6.86957', 'vk6.87162', 'vk6.88000', 'vk6.89550']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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: [-2,-1,-1,1,1,2,0,0,1,2,2,0,0,1,1,1,2,2,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.1261']

Arrow polynomial of the knot is: 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 + 2*K1 + 4*K2 + 2*K3 + K4 + 4

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1261']

Outer characteristic polynomial of the knot is: t^7+34t^5+33t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1261']

2-strand cable arrow polynomial of the knot is: -224*K1**4 + 384*K1**3*K2*K3 - 704*K1**3*K3 + 128*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 3104*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 4912*K1**2*K2 - 1088*K1**2*K3**2 - 192*K1**2*K3*K5 - 32*K1**2*K5**2 - 4888*K1**2 + 1280*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1216*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 + 64*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 6704*K1*K2*K3 - 64*K1*K2*K5*K6 + 1440*K1*K3*K4 + 304*K1*K4*K5 + 112*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1328*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1376*K2**2*K3**2 - 272*K2**2*K4**2 + 1504*K2**2*K4 - 64*K2**2*K5**2 - 48*K2**2*K6**2 - 3212*K2**2 + 912*K2*K3*K5 + 240*K2*K4*K6 + 80*K2*K5*K7 + 16*K2*K6*K8 - 2312*K3**2 - 648*K4**2 - 280*K5**2 - 84*K6**2 - 24*K7**2 - 2*K8**2 + 3736

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1261']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71645', 'vk6.71664', 'vk6.71824', 'vk6.71832', 'vk6.72242', 'vk6.72263', 'vk6.72372', 'vk6.77261', 'vk6.77359', 'vk6.77382', 'vk6.77605', 'vk6.77622', 'vk6.77699', 'vk6.77721', 'vk6.81410', 'vk6.81442', 'vk6.86957', 'vk6.87162', 'vk6.88000', 'vk6.89550']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is r.

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	O1O2O3O4U5U2U4O6O5U1U3U6
R3 orbit	{'O1O2O3O4U5U2U4O6O5U1U3U6'}
R3 orbit length	1
Gauss code of -K	Same
Gauss code of K*	O1O2O3U4U1O5O6O4U5U2U6U3
Gauss code of -K*	O1O2O3U4U1O5O6O4U5U2U6U3
Diagrammatic symmetry type	r
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 -1 1 2 -1 1],[ 2 0 0 2 2 0 1],[ 1 0 0 1 1 0 0],[-1 -2 -1 0 1 -2 0],[-2 -2 -1 -1 0 -2 -1],[ 1 0 0 2 2 0 1],[-1 -1 0 0 1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 0 -1 -1],[-1 1 0 0 -1 -2 -2],[ 1 1 0 1 0 0 0],[ 1 2 1 2 0 0 0],[ 2 2 1 2 0 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,1,1,2,1,1,1,2,2,0,0,1,1,1,2,2,0,0,0]
Phi over symmetry	[-2,-1,-1,1,1,2,0,0,1,2,2,0,0,1,1,1,2,2,0,1,1]
Phi of -K	[-2,-1,-1,1,1,2,1,1,1,2,2,0,0,1,1,1,2,2,0,0,0]
Phi of K*	[-2,-1,-1,1,1,2,0,0,1,2,2,0,0,1,1,1,2,2,0,1,1]
Phi of -K*	[-2,-1,-1,1,1,2,0,0,1,2,2,0,0,1,1,1,2,2,0,1,1]
Symmetry type of based matrix	r
u-polynomial	0
Normalized Jones-Krushkal polynomial	4z^2+21z+27
Enhanced Jones-Krushkal polynomial	4w^3z^2+21w^2z+27w
Inner characteristic polynomial	t^6+22t^4+7t^2
Outer characteristic polynomial	t^7+34t^5+33t^3+4t
Flat arrow polynomial	4K12K2 - 8K12 - 4K1K2 - 4K1K3 + 2K1 + 4K2 + 2K3 + K4 + 4
2-strand cable arrow polynomial	-224K14 + 384K1*3K2K3 - 704K1*3K3 + 128K12K2*3 - 128K1*2K2*2K3*2 - 3104K1*2K2*2 + 192K1*2K2K32 - 384K1*2K2K4 + 4912K1*2K2 - 1088K12K3*2 - 192K1*2K3K5 - 32K1*2K5*2 - 4888K1*2 + 1280K1K23K3 + 256K1K2*2K3K4 - 1216K1K22K3 + 64K1K2*2K4K5 - 128K1K22K5 + 64K1K2K33 - 448K1K2K3K4 - 192K1K2K3K6 + 6704K1K2K3 - 64K1K2K5K6 + 1440K1K3K4 + 304K1K4K5 + 112K1K5K6 + 16K1K6K7 - 32K24K4*2 + 128K2*4K4 - 1328K24 + 64K2*3K4K6 - 64K2*3K6 - 1376K22K3*2 - 272K2*2K4*2 + 1504K2*2K4 - 64K22K5*2 - 48K2*2K6*2 - 3212K2*2 + 912K2K3K5 + 240K2K4K6 + 80K2K5K7 + 16K2K6K8 - 2312K3*2 - 648K4*2 - 280K5*2 - 84K6*2 - 24K7*2 - 2K8**2 + 3736
Genus of based matrix	1
Fillings of based matrix	[[{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice	False

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