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Flat knot 6.1351

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-1,1,2,2,3,0,1,1,2,-1,0,-1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1351']
Arrow polynomial of the knot is: 4K13 - 12K1*2 - 8K1K2 + K1 + 6K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070']
Outer characteristic polynomial of the knot is: t^7+40t^5+39t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1351']
2-strand cable arrow polynomial of the knot is: -384K14K2*2 + 1216K1*4K2 - 3968K14 + 640K1*3K2K3 - 512K1*3K3 - 256K12K2*4 + 1280K1*2K2*3 + 256K1*2K2*2K4 - 6784K12K2*2 - 608K1*2K2K4 + 8960K1*2K2 - 512K12K3*2 - 64K1*2K4*2 - 3744K1*2 + 512K1K23K3 - 1344K1K2*2K3 - 256K1K2*2K5 - 384K1K2K3K4 + 6368K1K2K3 + 1040K1K3K4 + 192K1K4K5 - 32K2*6 + 96K2*4K4 - 1312K24 - 32K2*3K6 - 320K22K3*2 - 128K2*2K4*2 + 1624K2*2K4 - 3434K22 + 384K2K3K5 + 104K2K4K6 - 1600K3*2 - 604K4*2 - 112K5*2 - 22K6**2 + 3738
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1351']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4862', 'vk6.5206', 'vk6.6435', 'vk6.6858', 'vk6.8396', 'vk6.8818', 'vk6.9754', 'vk6.10050', 'vk6.20788', 'vk6.22189', 'vk6.29751', 'vk6.39830', 'vk6.46389', 'vk6.47965', 'vk6.49088', 'vk6.49921', 'vk6.51334', 'vk6.51552', 'vk6.58804', 'vk6.63267']
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: [-2,-2,1,1,1,1,-1,1,2,2,3,0,1,1,2,-1,0,-1,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1351']

Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070']

Outer characteristic polynomial of the knot is: t^7+40t^5+39t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1351']

2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 1216*K1**4*K2 - 3968*K1**4 + 640*K1**3*K2*K3 - 512*K1**3*K3 - 256*K1**2*K2**4 + 1280*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 6784*K1**2*K2**2 - 608*K1**2*K2*K4 + 8960*K1**2*K2 - 512*K1**2*K3**2 - 64*K1**2*K4**2 - 3744*K1**2 + 512*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 256*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 6368*K1*K2*K3 + 1040*K1*K3*K4 + 192*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1312*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 1624*K2**2*K4 - 3434*K2**2 + 384*K2*K3*K5 + 104*K2*K4*K6 - 1600*K3**2 - 604*K4**2 - 112*K5**2 - 22*K6**2 + 3738

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1351']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4862', 'vk6.5206', 'vk6.6435', 'vk6.6858', 'vk6.8396', 'vk6.8818', 'vk6.9754', 'vk6.10050', 'vk6.20788', 'vk6.22189', 'vk6.29751', 'vk6.39830', 'vk6.46389', 'vk6.47965', 'vk6.49088', 'vk6.49921', 'vk6.51334', 'vk6.51552', 'vk6.58804', 'vk6.63267']

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	O1O2O3U1O4O5U6U5U3O6U4U2
R3 orbit	{'O1O2O3U1O4O5U6U5U3O6U4U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3U2U4O5U1U6U5O6O4U3
Gauss code of K*	Same
Gauss code of -K*	O1O2O3U2U4O5U1U6U5O6O4U3
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 -2 1 1 1 1 -2],[ 2 0 2 1 1 0 1],[-1 -2 0 0 1 1 -3],[-1 -1 0 0 0 0 -2],[-1 -1 -1 0 0 1 -2],[-1 0 -1 0 -1 0 -1],[ 2 -1 3 2 2 1 0]]
Primitive based matrix	[[ 0 1 1 1 1 -2 -2],[-1 0 1 1 0 -2 -3],[-1 -1 0 1 0 -1 -2],[-1 -1 -1 0 0 0 -1],[-1 0 0 0 0 -1 -2],[ 2 2 1 0 1 0 1],[ 2 3 2 1 2 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,2,2,-1,-1,0,2,3,-1,0,1,2,0,0,1,1,2,-1]
Phi over symmetry	[-2,-2,1,1,1,1,-1,1,2,2,3,0,1,1,2,-1,0,-1,0,-1,0]
Phi of -K	[-2,-2,1,1,1,1,-1,1,2,2,3,0,1,1,2,-1,0,-1,0,-1,0]
Phi of K*	[-1,-1,-1,-1,2,2,-1,-1,0,2,3,-1,0,1,2,0,0,1,1,2,-1]
Phi of -K*	[-2,-2,1,1,1,1,-1,1,2,2,3,0,1,1,2,-1,0,-1,0,-1,0]
Symmetry type of based matrix	+
u-polynomial	2t^2-4t
Normalized Jones-Krushkal polynomial	2z^2+19z+31
Enhanced Jones-Krushkal polynomial	2w^3z^2+19w^2z+31w
Inner characteristic polynomial	t^6+28t^4+23t^2
Outer characteristic polynomial	t^7+40t^5+39t^3+4t
Flat arrow polynomial	4K13 - 12K1*2 - 8K1K2 + K1 + 6K2 + 3*K3 + 7
2-strand cable arrow polynomial	-384K14K2*2 + 1216K1*4K2 - 3968K14 + 640K1*3K2K3 - 512K1*3K3 - 256K12K2*4 + 1280K1*2K2*3 + 256K1*2K2*2K4 - 6784K12K2*2 - 608K1*2K2K4 + 8960K1*2K2 - 512K12K3*2 - 64K1*2K4*2 - 3744K1*2 + 512K1K23K3 - 1344K1K2*2K3 - 256K1K2*2K5 - 384K1K2K3K4 + 6368K1K2K3 + 1040K1K3K4 + 192K1K4K5 - 32K2*6 + 96K2*4K4 - 1312K24 - 32K2*3K6 - 320K22K3*2 - 128K2*2K4*2 + 1624K2*2K4 - 3434K22 + 384K2K3K5 + 104K2K4K6 - 1600K3*2 - 604K4*2 - 112K5*2 - 22K6**2 + 3738
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {3, 4}, {2}]]
If K is slice	False

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