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Flat knot 6.364

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,-1,1,2,3,2,1,2,2,1,0,-1,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.364']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 8K12 - 6K1K2 - 4K1K3 + 4K2 + 2*K3 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.364', '6.895', '6.907']
Outer characteristic polynomial of the knot is: t^7+52t^5+51t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.364']
2-strand cable arrow polynomial of the knot is: 256K14K2*3 - 640K1*4K2*2 + 1152K1*4K2 - 2080K14 - 512K1*3K2*2K3 + 992K13K2K3 + 64K1*3K3K4 - 896K1*3K3 - 384K12K2*4 + 2048K1*2K2*3 - 5568K1*2K2*2 + 352K1*2K2K32 - 1024K1*2K2K4 + 7064K1*2K2 - 864K12K3*2 - 128K1*2K3K5 - 64K1*2K4*2 - 5360K1*2 + 1216K1K23K3 - 1120K1K2*2K3 - 64K1K2*2K5 - 320K1K2K3K4 + 6552K1K2K3 - 32K1K32K5 - 32K1K3K4K6 + 1864K1K3K4 + 224K1K4K5 + 88K1K5K6 + 56K1K6K7 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 1392K24 + 128K2*3K3K5 + 64K2*3K4K6 - 1008K2*2K3*2 - 248K2*2K4*2 + 1312K2*2K4 - 112K22K5*2 - 48K2*2K6*2 - 3468K2*2 + 856K2K3K5 + 272K2K4K6 + 72K2K5K7 + 16K2K6K8 + 48K3*2K6 - 2396K32 + 48K3K4K7 - 1008K42 - 348K5*2 - 172K6*2 - 64K7*2 - 2K8**2 + 4576
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.364']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4421', 'vk6.4518', 'vk6.5803', 'vk6.5932', 'vk6.7866', 'vk6.7973', 'vk6.9282', 'vk6.9403', 'vk6.10159', 'vk6.10230', 'vk6.10373', 'vk6.17899', 'vk6.17962', 'vk6.18290', 'vk6.18625', 'vk6.24402', 'vk6.25176', 'vk6.30042', 'vk6.30103', 'vk6.36900', 'vk6.37358', 'vk6.43829', 'vk6.44117', 'vk6.44440', 'vk6.48614', 'vk6.50513', 'vk6.50594', 'vk6.51115', 'vk6.51668', 'vk6.55847', 'vk6.56088', 'vk6.65509']
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: [-3,-2,1,1,1,2,-1,1,2,3,2,1,2,2,1,0,-1,0,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.364']

Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 4*K1*K3 + 4*K2 + 2*K3 + K4 + 4

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.364', '6.895', '6.907']

Outer characteristic polynomial of the knot is: t^7+52t^5+51t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.364']

2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 640*K1**4*K2**2 + 1152*K1**4*K2 - 2080*K1**4 - 512*K1**3*K2**2*K3 + 992*K1**3*K2*K3 + 64*K1**3*K3*K4 - 896*K1**3*K3 - 384*K1**2*K2**4 + 2048*K1**2*K2**3 - 5568*K1**2*K2**2 + 352*K1**2*K2*K3**2 - 1024*K1**2*K2*K4 + 7064*K1**2*K2 - 864*K1**2*K3**2 - 128*K1**2*K3*K5 - 64*K1**2*K4**2 - 5360*K1**2 + 1216*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 64*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 6552*K1*K2*K3 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 1864*K1*K3*K4 + 224*K1*K4*K5 + 88*K1*K5*K6 + 56*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1392*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 1008*K2**2*K3**2 - 248*K2**2*K4**2 + 1312*K2**2*K4 - 112*K2**2*K5**2 - 48*K2**2*K6**2 - 3468*K2**2 + 856*K2*K3*K5 + 272*K2*K4*K6 + 72*K2*K5*K7 + 16*K2*K6*K8 + 48*K3**2*K6 - 2396*K3**2 + 48*K3*K4*K7 - 1008*K4**2 - 348*K5**2 - 172*K6**2 - 64*K7**2 - 2*K8**2 + 4576

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.364']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4421', 'vk6.4518', 'vk6.5803', 'vk6.5932', 'vk6.7866', 'vk6.7973', 'vk6.9282', 'vk6.9403', 'vk6.10159', 'vk6.10230', 'vk6.10373', 'vk6.17899', 'vk6.17962', 'vk6.18290', 'vk6.18625', 'vk6.24402', 'vk6.25176', 'vk6.30042', 'vk6.30103', 'vk6.36900', 'vk6.37358', 'vk6.43829', 'vk6.44117', 'vk6.44440', 'vk6.48614', 'vk6.50513', 'vk6.50594', 'vk6.51115', 'vk6.51668', 'vk6.55847', 'vk6.56088', 'vk6.65509']

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	O1O2O3O4O5U3U5O6U1U4U6U2
R3 orbit	{'O1O2O3O4O5U3U5O6U1U4U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U6U2U5O6U1U3
Gauss code of K*	O1O2O3O4U1U4U5U2U6O5O6U3
Gauss code of -K*	O1O2O3O4U2O5O6U5U3U6U1U4
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 -3 1 -2 1 1 2],[ 3 0 3 -1 2 1 2],[-1 -3 0 -2 0 1 1],[ 2 1 2 0 2 1 1],[-1 -2 0 -2 0 0 1],[-1 -1 -1 -1 0 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 2 1 1 1 -2 -3],[-2 0 0 -1 -1 -1 -2],[-1 0 0 0 -1 -1 -1],[-1 1 0 0 0 -2 -2],[-1 1 1 0 0 -2 -3],[ 2 1 1 2 2 0 1],[ 3 2 1 2 3 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,-1,2,3,0,1,1,1,2,0,1,1,1,0,2,2,2,3,-1]
Phi over symmetry	[-3,-2,1,1,1,2,-1,1,2,3,2,1,2,2,1,0,-1,0,0,1,1]
Phi of -K	[-3,-2,1,1,1,2,2,1,2,3,3,1,1,2,3,0,-1,0,0,0,1]
Phi of K*	[-2,-1,-1,-1,2,3,0,0,1,3,3,0,0,1,2,1,1,1,2,3,2]
Phi of -K*	[-3,-2,1,1,1,2,-1,1,2,3,2,1,2,2,1,0,-1,0,0,1,1]
Symmetry type of based matrix	c
u-polynomial	t^3-3t
Normalized Jones-Krushkal polynomial	z^2+18z+33
Enhanced Jones-Krushkal polynomial	w^3z^2-4w^3z+22w^2z+33w
Inner characteristic polynomial	t^6+32t^4+23t^2
Outer characteristic polynomial	t^7+52t^5+51t^3+8t
Flat arrow polynomial	4K13 + 4K1*2K2 - 8K12 - 6K1K2 - 4K1K3 + 4K2 + 2*K3 + K4 + 4
2-strand cable arrow polynomial	256K14K2*3 - 640K1*4K2*2 + 1152K1*4K2 - 2080K14 - 512K1*3K2*2K3 + 992K13K2K3 + 64K1*3K3K4 - 896K1*3K3 - 384K12K2*4 + 2048K1*2K2*3 - 5568K1*2K2*2 + 352K1*2K2K32 - 1024K1*2K2K4 + 7064K1*2K2 - 864K12K3*2 - 128K1*2K3K5 - 64K1*2K4*2 - 5360K1*2 + 1216K1K23K3 - 1120K1K2*2K3 - 64K1K2*2K5 - 320K1K2K3K4 + 6552K1K2K3 - 32K1K32K5 - 32K1K3K4K6 + 1864K1K3K4 + 224K1K4K5 + 88K1K5K6 + 56K1K6K7 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 1392K24 + 128K2*3K3K5 + 64K2*3K4K6 - 1008K2*2K3*2 - 248K2*2K4*2 + 1312K2*2K4 - 112K22K5*2 - 48K2*2K6*2 - 3468K2*2 + 856K2K3K5 + 272K2K4K6 + 72K2K5K7 + 16K2K6K8 + 48K3*2K6 - 2396K32 + 48K3K4K7 - 1008K42 - 348K5*2 - 172K6*2 - 64K7*2 - 2K8**2 + 4576
Genus of based matrix	2
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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