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Flat knot 6.1936

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,1,1,1,2,1,1,2,1,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1936']
Arrow polynomial of the knot is: 4K12K2 - 4K1K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']
Outer characteristic polynomial of the knot is: t^7+38t^5+39t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1936']
2-strand cable arrow polynomial of the knot is: -1152K14K2*2 + 4608K1*4K2 - 3712K14 - 896K1*3K3 + 2688K12K2*3 - 5568K1*2K2*2 - 608K1*2K2K4 + 4144K1*2K2 - 384K12K3K5 - 192K1*2K4*2 - 160K1*2K4K6 - 128K1*2K5*2 - 1760K1*2 + 192K1K22K3K4 - 1280K1K22K3 + 192K1K2*2K4K5 - 256K1K22K5 - 384K1K2K3K4 - 64K1K2K3K6 + 3360K1K2K3 - 320K1K2K4K5 - 64K1K2K4K7 - 128K1K2K5K6 + 1152K1K3K4 + 1152K1K4K5 + 432K1K5K6 + 48K1K6K7 - 32K2*4K4*2 + 128K2*4K4 - 1104K24 + 64K2*3K3K5 + 64K2*3K4K6 - 96K2*3K6 - 192K22K3*2 - 64K2*2K3K7 - 320K2*2K4*2 - 32K2*2K4K8 + 1312K2*2K4 - 128K22K5*2 - 16K2*2K6*2 - 1488K2*2 + 640K2K3K5 + 416K2K4K6 + 144K2K5K7 + 16K2K6K8 - 800K3*2 - 784K4*2 - 544K5*2 - 192K6*2 - 32K7*2 - 2K8**2 + 2128
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1936']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4892', 'vk6.5235', 'vk6.6491', 'vk6.6901', 'vk6.8453', 'vk6.8869', 'vk6.9787', 'vk6.10078', 'vk6.20823', 'vk6.22221', 'vk6.29784', 'vk6.39879', 'vk6.46432', 'vk6.47986', 'vk6.48845', 'vk6.49115', 'vk6.51369', 'vk6.51579', 'vk6.63282', 'vk6.67119']
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,-2,1,1,1,1,0,1,1,1,2,1,1,2,1,0,0,0,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1936']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']

Outer characteristic polynomial of the knot is: t^7+38t^5+39t^3+2t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1936']

2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 4608*K1**4*K2 - 3712*K1**4 - 896*K1**3*K3 + 2688*K1**2*K2**3 - 5568*K1**2*K2**2 - 608*K1**2*K2*K4 + 4144*K1**2*K2 - 384*K1**2*K3*K5 - 192*K1**2*K4**2 - 160*K1**2*K4*K6 - 128*K1**2*K5**2 - 1760*K1**2 + 192*K1*K2**2*K3*K4 - 1280*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 - 384*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 3360*K1*K2*K3 - 320*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 128*K1*K2*K5*K6 + 1152*K1*K3*K4 + 1152*K1*K4*K5 + 432*K1*K5*K6 + 48*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1104*K2**4 + 64*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 192*K2**2*K3**2 - 64*K2**2*K3*K7 - 320*K2**2*K4**2 - 32*K2**2*K4*K8 + 1312*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 1488*K2**2 + 640*K2*K3*K5 + 416*K2*K4*K6 + 144*K2*K5*K7 + 16*K2*K6*K8 - 800*K3**2 - 784*K4**2 - 544*K5**2 - 192*K6**2 - 32*K7**2 - 2*K8**2 + 2128

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1936']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4892', 'vk6.5235', 'vk6.6491', 'vk6.6901', 'vk6.8453', 'vk6.8869', 'vk6.9787', 'vk6.10078', 'vk6.20823', 'vk6.22221', 'vk6.29784', 'vk6.39879', 'vk6.46432', 'vk6.47986', 'vk6.48845', 'vk6.49115', 'vk6.51369', 'vk6.51579', 'vk6.63282', 'vk6.67119']

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	O1O2O3U1U4U5O6O5O4U6U3U2
R3 orbit	{'O1O2O3U1U4U5O6O5O4U6U3U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3U2U1U4O5O6O4U6U5U3
Gauss code of K*	O1O2O3U4U3U2O4O5O6U1U6U5
Gauss code of -K*	O1O2O3U4U5U3O5O4O6U2U1U6
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 1 1 1 1 -2],[ 2 0 2 1 2 2 0],[-1 -2 0 0 0 0 -2],[-1 -1 0 0 0 0 -2],[-1 -2 0 0 0 0 -1],[-1 -2 0 0 0 0 -2],[ 2 0 2 2 1 2 0]]
Primitive based matrix	[[ 0 1 1 1 1 -2 -2],[-1 0 0 0 0 -1 -2],[-1 0 0 0 0 -2 -1],[-1 0 0 0 0 -2 -2],[-1 0 0 0 0 -2 -2],[ 2 1 2 2 2 0 0],[ 2 2 1 2 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,2,2,0,0,0,1,2,0,0,2,1,0,2,2,2,2,0]
Phi over symmetry	[-2,-2,1,1,1,1,0,1,1,1,2,1,1,2,1,0,0,0,0,0,0]
Phi of -K	[-2,-2,1,1,1,1,0,1,1,1,2,1,1,2,1,0,0,0,0,0,0]
Phi of K*	[-1,-1,-1,-1,2,2,0,0,0,1,1,0,0,1,1,0,1,2,2,1,0]
Phi of -K*	[-2,-2,1,1,1,1,0,1,2,2,2,2,1,2,2,0,0,0,0,0,0]
Symmetry type of based matrix	c
u-polynomial	2t^2-4t
Normalized Jones-Krushkal polynomial	9z^2+30z+25
Enhanced Jones-Krushkal polynomial	9w^3z^2+30w^2z+25w
Inner characteristic polynomial	t^6+26t^4+25t^2
Outer characteristic polynomial	t^7+38t^5+39t^3+2t
Flat arrow polynomial	4K12K2 - 4K1K3 + K4
2-strand cable arrow polynomial	-1152K14K2*2 + 4608K1*4K2 - 3712K14 - 896K1*3K3 + 2688K12K2*3 - 5568K1*2K2*2 - 608K1*2K2K4 + 4144K1*2K2 - 384K12K3K5 - 192K1*2K4*2 - 160K1*2K4K6 - 128K1*2K5*2 - 1760K1*2 + 192K1K22K3K4 - 1280K1K22K3 + 192K1K2*2K4K5 - 256K1K22K5 - 384K1K2K3K4 - 64K1K2K3K6 + 3360K1K2K3 - 320K1K2K4K5 - 64K1K2K4K7 - 128K1K2K5K6 + 1152K1K3K4 + 1152K1K4K5 + 432K1K5K6 + 48K1K6K7 - 32K2*4K4*2 + 128K2*4K4 - 1104K24 + 64K2*3K3K5 + 64K2*3K4K6 - 96K2*3K6 - 192K22K3*2 - 64K2*2K3K7 - 320K2*2K4*2 - 32K2*2K4K8 + 1312K2*2K4 - 128K22K5*2 - 16K2*2K6*2 - 1488K2*2 + 640K2K3K5 + 416K2K4K6 + 144K2K5K7 + 16K2K6K8 - 800K3*2 - 784K4*2 - 544K5*2 - 192K6*2 - 32K7*2 - 2K8**2 + 2128
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {5}, {1, 4}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice	False

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