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Flat knot 6.1812

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,1,1,2,0,1,1,1,-1,1,1,0,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.1812']
Arrow polynomial of the knot is: 4K13 - 12K1*2 - 4K1K2 - K1 + 6K2 + K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1345', '6.1510', '6.1565', '6.1691', '6.1812']
Outer characteristic polynomial of the knot is: t^7+31t^5+75t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1812']
2-strand cable arrow polynomial of the knot is: -320K14K2*2 + 736K1*4K2 - 2880K14 + 416K1*3K2K3 - 224K1*3K3 - 384K12K2*4 + 1760K1*2K2*3 + 64K1*2K2*2K4 - 7424K12K2*2 - 480K1*2K2K4 + 8016K1*2K2 - 192K12K3*2 - 3068K1*2 + 576K1K23K3 - 1056K1K2*2K3 - 192K1K2*2K5 + 5360K1K2K3 + 304K1K3K4 + 16K1K4K5 - 32K2*6 + 64K2*4K4 - 1728K24 - 32K2*3K6 - 240K22K3*2 - 16K2*2K4*2 + 1248K2*2K4 - 2054K22 + 112K2K3K5 + 16K2K4K6 - 888K3*2 - 160K4*2 - 12K5*2 - 2K6**2 + 2694
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1812']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4681', 'vk6.4980', 'vk6.6157', 'vk6.6634', 'vk6.8154', 'vk6.8566', 'vk6.9540', 'vk6.9887', 'vk6.20694', 'vk6.22134', 'vk6.28219', 'vk6.29644', 'vk6.39679', 'vk6.41920', 'vk6.46259', 'vk6.47866', 'vk6.48721', 'vk6.48940', 'vk6.49507', 'vk6.49712', 'vk6.50743', 'vk6.50952', 'vk6.51226', 'vk6.51419', 'vk6.57621', 'vk6.58779', 'vk6.62297', 'vk6.63230', 'vk6.67091', 'vk6.67955', 'vk6.69695', 'vk6.70378']
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: [-2,-1,0,0,1,2,0,1,1,1,2,0,1,1,1,-1,1,1,0,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.1812']

Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 4*K1*K2 - K1 + 6*K2 + K3 + 7

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1345', '6.1510', '6.1565', '6.1691', '6.1812']

Outer characteristic polynomial of the knot is: t^7+31t^5+75t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1812']

2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 736*K1**4*K2 - 2880*K1**4 + 416*K1**3*K2*K3 - 224*K1**3*K3 - 384*K1**2*K2**4 + 1760*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 7424*K1**2*K2**2 - 480*K1**2*K2*K4 + 8016*K1**2*K2 - 192*K1**2*K3**2 - 3068*K1**2 + 576*K1*K2**3*K3 - 1056*K1*K2**2*K3 - 192*K1*K2**2*K5 + 5360*K1*K2*K3 + 304*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1728*K2**4 - 32*K2**3*K6 - 240*K2**2*K3**2 - 16*K2**2*K4**2 + 1248*K2**2*K4 - 2054*K2**2 + 112*K2*K3*K5 + 16*K2*K4*K6 - 888*K3**2 - 160*K4**2 - 12*K5**2 - 2*K6**2 + 2694

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1812']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4681', 'vk6.4980', 'vk6.6157', 'vk6.6634', 'vk6.8154', 'vk6.8566', 'vk6.9540', 'vk6.9887', 'vk6.20694', 'vk6.22134', 'vk6.28219', 'vk6.29644', 'vk6.39679', 'vk6.41920', 'vk6.46259', 'vk6.47866', 'vk6.48721', 'vk6.48940', 'vk6.49507', 'vk6.49712', 'vk6.50743', 'vk6.50952', 'vk6.51226', 'vk6.51419', 'vk6.57621', 'vk6.58779', 'vk6.62297', 'vk6.63230', 'vk6.67091', 'vk6.67955', 'vk6.69695', 'vk6.70378']

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	O1O2O3U4U5O6O5U1U6O4U2U3
R3 orbit	{'O1O2O3U4U5O6O5U1U6O4U2U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3U1U2O4U5U3O6O5U6U4
Gauss code of K*	O1O2U3O4O5U1U4U5O3O6U2U6
Gauss code of -K*	O1O2U3O4O5U6U4O6O3U1U2U5
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 2 -1 1 0],[ 2 0 1 2 1 2 0],[ 0 -1 0 1 0 1 -1],[-2 -2 -1 0 -2 -1 -1],[ 1 -1 0 2 0 1 1],[-1 -2 -1 1 -1 0 0],[ 0 0 1 1 -1 0 0]]
Primitive based matrix	[[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 -1 -1 -2],[ 0 1 0 0 1 -1 0],[ 0 1 1 -1 0 0 -1],[ 1 2 1 1 0 0 -1],[ 2 2 2 0 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,0,1,2,1,1,1,2,2,0,1,1,2,-1,1,0,0,1,1]
Phi over symmetry	[-2,-1,0,0,1,2,0,1,1,1,2,0,1,1,1,-1,1,1,0,2,0]
Phi of -K	[-2,-1,0,0,1,2,0,1,2,1,2,1,0,1,1,1,0,1,1,1,0]
Phi of K*	[-2,-1,0,0,1,2,0,1,1,1,2,0,1,1,1,-1,1,1,0,2,0]
Phi of -K*	[-2,-1,0,0,1,2,1,0,1,2,2,1,0,1,2,1,0,1,1,1,1]
Symmetry type of based matrix	c
u-polynomial	0
Normalized Jones-Krushkal polynomial	3z^2+20z+29
Enhanced Jones-Krushkal polynomial	3w^3z^2+20w^2z+29w
Inner characteristic polynomial	t^6+21t^4+47t^2
Outer characteristic polynomial	t^7+31t^5+75t^3+4t
Flat arrow polynomial	4K13 - 12K1*2 - 4K1K2 - K1 + 6K2 + K3 + 7
2-strand cable arrow polynomial	-320K14K2*2 + 736K1*4K2 - 2880K14 + 416K1*3K2K3 - 224K1*3K3 - 384K12K2*4 + 1760K1*2K2*3 + 64K1*2K2*2K4 - 7424K12K2*2 - 480K1*2K2K4 + 8016K1*2K2 - 192K12K3*2 - 3068K1*2 + 576K1K23K3 - 1056K1K2*2K3 - 192K1K2*2K5 + 5360K1K2K3 + 304K1K3K4 + 16K1K4K5 - 32K2*6 + 64K2*4K4 - 1728K24 - 32K2*3K6 - 240K22K3*2 - 16K2*2K4*2 + 1248K2*2K4 - 2054K22 + 112K2K3K5 + 16K2K4K6 - 888K3*2 - 160K4*2 - 12K5*2 - 2K6**2 + 2694
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {2, 4}, {1}]]
If K is slice	False

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