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Flat knot 6.1810

Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,-1,1,1,2,0,0,1,1,1,1,0,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1810', '7.42854', '7.44841', '7.45941']
Arrow polynomial of the knot is: 8K13 - 8K1*2 - 8K1K2 - 2K1 + 4K2 + 2K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923']
Outer characteristic polynomial of the knot is: t^7+19t^5+35t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1810']
2-strand cable arrow polynomial of the knot is: -1024K16 - 2048K1*4K2*2 + 3840K1*4K2 - 4384K14 + 1728K1*3K2K3 - 608K1*3K3 - 1344K12K2*4 + 3456K1*2K2*3 + 512K1*2K2*2K4 - 8768K12K2*2 - 832K1*2K2K4 + 6448K1*2K2 - 448K12K3*2 - 96K1*2K4*2 + 8K1*2 + 1472K1K23K3 - 1632K1K2*2K3 - 352K1K2*2K5 - 192K1K2K3K4 + 4512K1K2K3 + 368K1K3K4 + 80K1K4K5 - 192K2*6 + 320K2*4K4 - 2080K24 - 64K2*3K6 - 448K22K3*2 - 128K2*2K4*2 + 1328K2*2K4 - 444K22 + 176K2K3K5 + 48K2K4K6 - 404K3*2 - 120K4*2 - 20K5*2 - 4K6**2 + 1334
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1810']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73', 'vk6.77', 'vk6.130', 'vk6.138', 'vk6.225', 'vk6.233', 'vk6.272', 'vk6.276', 'vk6.390', 'vk6.391', 'vk6.810', 'vk6.811', 'vk6.1282', 'vk6.1290', 'vk6.1371', 'vk6.1379', 'vk6.1416', 'vk6.1420', 'vk6.1551', 'vk6.1552', 'vk6.2695', 'vk6.2696', 'vk6.2945', 'vk6.2953', 'vk6.14838', 'vk6.14846', 'vk6.15994', 'vk6.16002', 'vk6.25959', 'vk6.25960', 'vk6.33347', 'vk6.33360']
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: [-1,-1,-1,1,1,1,-1,-1,1,1,2,0,0,1,1,1,1,0,-1,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1810', '7.42854', '7.44841', '7.45941']

Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923']

Outer characteristic polynomial of the knot is: t^7+19t^5+35t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1810']

2-strand cable arrow polynomial of the knot is: -1024*K1**6 - 2048*K1**4*K2**2 + 3840*K1**4*K2 - 4384*K1**4 + 1728*K1**3*K2*K3 - 608*K1**3*K3 - 1344*K1**2*K2**4 + 3456*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 8768*K1**2*K2**2 - 832*K1**2*K2*K4 + 6448*K1**2*K2 - 448*K1**2*K3**2 - 96*K1**2*K4**2 + 8*K1**2 + 1472*K1*K2**3*K3 - 1632*K1*K2**2*K3 - 352*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4512*K1*K2*K3 + 368*K1*K3*K4 + 80*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2080*K2**4 - 64*K2**3*K6 - 448*K2**2*K3**2 - 128*K2**2*K4**2 + 1328*K2**2*K4 - 444*K2**2 + 176*K2*K3*K5 + 48*K2*K4*K6 - 404*K3**2 - 120*K4**2 - 20*K5**2 - 4*K6**2 + 1334

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1810']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73', 'vk6.77', 'vk6.130', 'vk6.138', 'vk6.225', 'vk6.233', 'vk6.272', 'vk6.276', 'vk6.390', 'vk6.391', 'vk6.810', 'vk6.811', 'vk6.1282', 'vk6.1290', 'vk6.1371', 'vk6.1379', 'vk6.1416', 'vk6.1420', 'vk6.1551', 'vk6.1552', 'vk6.2695', 'vk6.2696', 'vk6.2945', 'vk6.2953', 'vk6.14838', 'vk6.14846', 'vk6.15994', 'vk6.16002', 'vk6.25959', 'vk6.25960', 'vk6.33347', 'vk6.33360']

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	O1O2O3U4U3O5O6U5U6O4U1U2
R3 orbit	{'O1O2O3U4U3O5O6U5U6O4U1U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3U2U3O4U5U6O5O6U1U4
Gauss code of K*	O1O2U3O4O5U4U5U6O3O6U1U2
Gauss code of -K*	O1O2U3O4O5U4U5O6O3U6U1U2
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 -1 1 1 -1 -1 1],[ 1 0 1 1 0 -1 1],[-1 -1 0 1 -2 -1 1],[-1 -1 -1 0 -1 0 0],[ 1 0 2 1 0 0 0],[ 1 1 1 0 0 0 1],[-1 -1 -1 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 -1 -1 -1],[-1 0 1 1 -1 -1 -2],[-1 -1 0 0 0 -1 -1],[-1 -1 0 0 -1 -1 0],[ 1 1 0 1 0 1 0],[ 1 1 1 1 -1 0 0],[ 1 2 1 0 0 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,1,1,1,-1,-1,1,1,2,0,0,1,1,1,1,0,-1,0,0]
Phi over symmetry	[-1,-1,-1,1,1,1,-1,-1,1,1,2,0,0,1,1,1,1,0,-1,0,0]
Phi of -K	[-1,-1,-1,1,1,1,-1,0,1,1,2,0,1,1,1,0,2,1,-1,-1,0]
Phi of K*	[-1,-1,-1,1,1,1,-1,0,1,1,2,1,0,1,1,2,1,1,0,0,-1]
Phi of -K*	[-1,-1,-1,1,1,1,-1,0,1,1,1,0,0,1,1,1,0,2,0,-1,-1]
Symmetry type of based matrix	c
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+13t^4+21t^2+1
Outer characteristic polynomial	t^7+19t^5+35t^3+5t
Flat arrow polynomial	8K13 - 8K1*2 - 8K1K2 - 2K1 + 4K2 + 2K3 + 5
2-strand cable arrow polynomial	-1024K16 - 2048K1*4K2*2 + 3840K1*4K2 - 4384K14 + 1728K1*3K2K3 - 608K1*3K3 - 1344K12K2*4 + 3456K1*2K2*3 + 512K1*2K2*2K4 - 8768K12K2*2 - 832K1*2K2K4 + 6448K1*2K2 - 448K12K3*2 - 96K1*2K4*2 + 8K1*2 + 1472K1K23K3 - 1632K1K2*2K3 - 352K1K2*2K5 - 192K1K2K3K4 + 4512K1K2K3 + 368K1K3K4 + 80K1K4K5 - 192K2*6 + 320K2*4K4 - 2080K24 - 64K2*3K6 - 448K22K3*2 - 128K2*2K4*2 + 1328K2*2K4 - 444K22 + 176K2K3K5 + 48K2K4K6 - 404K3*2 - 120K4*2 - 20K5*2 - 4K6**2 + 1334
Genus of based matrix	0
Fillings of based matrix	[[{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice	True

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