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Flat knot 6.1371

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,0,3,1,2,3,1,0,1,0,1,0,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1371']
Arrow polynomial of the knot is: 4K13 - 8K1*2 - 6K1K2 + 4K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.211', '6.557', '6.676', '6.685', '6.750', '6.751', '6.856', '6.919', '6.1093', '6.1371']
Outer characteristic polynomial of the knot is: t^7+40t^5+76t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1371']
2-strand cable arrow polynomial of the knot is: -384K14K2*2 + 1024K1*4K2 - 4672K14 + 640K1*3K2K3 - 192K1*3K3 - 256K12K2*4 + 1088K1*2K2*3 + 256K1*2K2*2K4 - 7264K12K2*2 - 416K1*2K2K4 + 10992K1*2K2 - 704K12K3*2 - 128K1*2K3K5 - 64K1*2K4*2 - 64K1*2K4K6 - 5224K1*2 + 448K1K23K3 - 1920K1K2*2K3 - 192K1K2*2K5 - 640K1K2K3K4 + 7536K1K2K3 + 1856K1K3K4 + 624K1K4K5 + 80K1K5K6 - 32K26 + 96K2*4K4 - 1104K24 - 32K2*3K6 - 288K22K3*2 - 88K2*2K4*2 + 2080K2*2K4 - 5100K22 + 656K2K3K5 + 112K2K4K6 - 2400K3*2 - 1228K4*2 - 392K5*2 - 60K6**2 + 5354
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1371']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4852', 'vk6.5197', 'vk6.6423', 'vk6.6853', 'vk6.8392', 'vk6.8815', 'vk6.9751', 'vk6.10048', 'vk6.20779', 'vk6.22181', 'vk6.29746', 'vk6.39820', 'vk6.46384', 'vk6.47959', 'vk6.49091', 'vk6.49923', 'vk6.51343', 'vk6.51557', 'vk6.58808', 'vk6.63274']
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: [-3,0,0,1,1,1,0,3,1,2,3,1,0,1,0,1,0,1,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1371']

Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.211', '6.557', '6.676', '6.685', '6.750', '6.751', '6.856', '6.919', '6.1093', '6.1371']

Outer characteristic polynomial of the knot is: t^7+40t^5+76t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1371']

2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 1024*K1**4*K2 - 4672*K1**4 + 640*K1**3*K2*K3 - 192*K1**3*K3 - 256*K1**2*K2**4 + 1088*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 7264*K1**2*K2**2 - 416*K1**2*K2*K4 + 10992*K1**2*K2 - 704*K1**2*K3**2 - 128*K1**2*K3*K5 - 64*K1**2*K4**2 - 64*K1**2*K4*K6 - 5224*K1**2 + 448*K1*K2**3*K3 - 1920*K1*K2**2*K3 - 192*K1*K2**2*K5 - 640*K1*K2*K3*K4 + 7536*K1*K2*K3 + 1856*K1*K3*K4 + 624*K1*K4*K5 + 80*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 1104*K2**4 - 32*K2**3*K6 - 288*K2**2*K3**2 - 88*K2**2*K4**2 + 2080*K2**2*K4 - 5100*K2**2 + 656*K2*K3*K5 + 112*K2*K4*K6 - 2400*K3**2 - 1228*K4**2 - 392*K5**2 - 60*K6**2 + 5354

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1371']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4852', 'vk6.5197', 'vk6.6423', 'vk6.6853', 'vk6.8392', 'vk6.8815', 'vk6.9751', 'vk6.10048', 'vk6.20779', 'vk6.22181', 'vk6.29746', 'vk6.39820', 'vk6.46384', 'vk6.47959', 'vk6.49091', 'vk6.49923', 'vk6.51343', 'vk6.51557', 'vk6.58808', 'vk6.63274']

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	O1O2O3U4O5O4U1U5U3O6U2U6
R3 orbit	{'O1O2O3U4O5O4U1U5U3O6U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U2O4U1U5U3O6O5U6
Gauss code of K*	Same
Gauss code of -K*	O1O2O3U4U2O4U1U5U3O6O5U6
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 -3 0 1 1 0 1],[ 3 0 3 2 3 0 1],[ 0 -3 0 0 1 -1 1],[-1 -2 0 0 0 -1 0],[-1 -3 -1 0 0 0 1],[ 0 0 1 1 0 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix	[[ 0 1 1 1 0 0 -3],[-1 0 1 0 0 -1 -3],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 -1 0 -2],[ 0 0 0 1 0 1 0],[ 0 1 1 0 -1 0 -3],[ 3 3 1 2 0 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,0,3,-1,0,0,1,3,0,0,1,1,1,0,2,-1,0,3]
Phi over symmetry	[-3,0,0,1,1,1,0,3,1,2,3,1,0,1,0,1,0,1,0,-1,0]
Phi of -K	[-3,0,0,1,1,1,0,3,1,2,3,1,0,1,0,1,0,1,0,-1,0]
Phi of K*	[-1,-1,-1,0,0,3,-1,0,0,1,3,0,0,1,1,1,0,2,-1,0,3]
Phi of -K*	[-3,0,0,1,1,1,0,3,1,2,3,1,0,1,0,1,0,1,0,-1,0]
Symmetry type of based matrix	+
u-polynomial	t^3-3t
Normalized Jones-Krushkal polynomial	2z^2+22z+37
Enhanced Jones-Krushkal polynomial	2w^3z^2+22w^2z+37w
Inner characteristic polynomial	t^6+28t^4+38t^2+1
Outer characteristic polynomial	t^7+40t^5+76t^3+7t
Flat arrow polynomial	4K13 - 8K1*2 - 6K1K2 + 4K2 + 2*K3 + 5
2-strand cable arrow polynomial	-384K14K2*2 + 1024K1*4K2 - 4672K14 + 640K1*3K2K3 - 192K1*3K3 - 256K12K2*4 + 1088K1*2K2*3 + 256K1*2K2*2K4 - 7264K12K2*2 - 416K1*2K2K4 + 10992K1*2K2 - 704K12K3*2 - 128K1*2K3K5 - 64K1*2K4*2 - 64K1*2K4K6 - 5224K1*2 + 448K1K23K3 - 1920K1K2*2K3 - 192K1K2*2K5 - 640K1K2K3K4 + 7536K1K2K3 + 1856K1K3K4 + 624K1K4K5 + 80K1K5K6 - 32K26 + 96K2*4K4 - 1104K24 - 32K2*3K6 - 288K22K3*2 - 88K2*2K4*2 + 2080K2*2K4 - 5100K22 + 656K2K3K5 + 112K2K4K6 - 2400K3*2 - 1228K4*2 - 392K5*2 - 60K6**2 + 5354
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}]]
If K is slice	False

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