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Flat knot 6.2083

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,1,1,1,0,1,1,1,-1,0,1,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.2083']
Arrow polynomial of the knot is: 4*K1**3 - 16*K1**2 - 8*K1*K2 + K1 + 8*K2 + 3*K3 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.2083']
Outer characteristic polynomial of the knot is: t^7+14t^5+30t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1996', '6.2083']
2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 3136*K1**4*K2 - 7872*K1**4 + 1408*K1**3*K2*K3 - 1344*K1**3*K3 - 256*K1**2*K2**4 + 1856*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 12992*K1**2*K2**2 - 1760*K1**2*K2*K4 + 16032*K1**2*K2 - 384*K1**2*K3**2 - 128*K1**2*K4**2 - 5776*K1**2 + 704*K1*K2**3*K3 - 1920*K1*K2**2*K3 - 512*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 11088*K1*K2*K3 + 1152*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 2032*K2**4 - 96*K2**3*K6 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 2456*K2**2*K4 - 5658*K2**2 + 336*K2*K3*K5 + 72*K2*K4*K6 - 2256*K3**2 - 664*K4**2 - 64*K5**2 - 6*K6**2 + 5862
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2083']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4856', 'vk6.5200', 'vk6.6427', 'vk6.6855', 'vk6.8388', 'vk6.8812', 'vk6.9748', 'vk6.10046', 'vk6.20786', 'vk6.22187', 'vk6.29753', 'vk6.39828', 'vk6.46391', 'vk6.47967', 'vk6.49094', 'vk6.49924', 'vk6.51340', 'vk6.51556', 'vk6.58802', 'vk6.63269']
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 O1O2U1U3O4O5U6U4O6O3U2U5
R3 orbit {'O1O2U1U3O4O5U6U4O6O3U2U5'}
R3 orbit length 1
Gauss code of -K O1O2U3U2O4O3U1U5O6O5U4U6
Gauss code of K* Same
Gauss code of -K* O1O2U3U2O4O3U1U5O6O5U4U6
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 -1 0 1 0 1 -1],[ 1 0 1 1 0 1 1],[ 0 -1 0 0 1 1 -1],[-1 -1 0 0 -1 0 -1],[ 0 0 -1 1 0 0 0],[-1 -1 -1 0 0 0 -1],[ 1 -1 1 1 0 1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 0 0 -1 -1 -1],[-1 0 0 -1 0 -1 -1],[ 0 0 1 0 1 -1 -1],[ 0 1 0 -1 0 0 0],[ 1 1 1 1 0 0 1],[ 1 1 1 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,0,0,1,1,1,1,0,1,1,-1,1,1,0,0,-1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,1,1,1,0,1,1,1,-1,0,1,1,0,0]
Phi of -K [-1,-1,0,0,1,1,-1,0,1,1,1,0,1,1,1,-1,0,1,1,0,0]
Phi of K* [-1,-1,0,0,1,1,0,0,1,1,1,1,0,1,1,-1,1,1,0,0,-1]
Phi of -K* [-1,-1,0,0,1,1,-1,0,1,1,1,0,1,1,1,-1,0,1,1,0,0]
Symmetry type of based matrix +
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+10t^4+18t^2+1
Outer characteristic polynomial t^7+14t^5+30t^3+5t
Flat arrow polynomial 4*K1**3 - 16*K1**2 - 8*K1*K2 + K1 + 8*K2 + 3*K3 + 9
2-strand cable arrow polynomial -512*K1**4*K2**2 + 3136*K1**4*K2 - 7872*K1**4 + 1408*K1**3*K2*K3 - 1344*K1**3*K3 - 256*K1**2*K2**4 + 1856*K1**2*K2**3 + 512*K1**2*K2**2*K4 - 12992*K1**2*K2**2 - 1760*K1**2*K2*K4 + 16032*K1**2*K2 - 384*K1**2*K3**2 - 128*K1**2*K4**2 - 5776*K1**2 + 704*K1*K2**3*K3 - 1920*K1*K2**2*K3 - 512*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 11088*K1*K2*K3 + 1152*K1*K3*K4 + 128*K1*K4*K5 - 32*K2**6 + 224*K2**4*K4 - 2032*K2**4 - 96*K2**3*K6 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 2456*K2**2*K4 - 5658*K2**2 + 336*K2*K3*K5 + 72*K2*K4*K6 - 2256*K3**2 - 664*K4**2 - 64*K5**2 - 6*K6**2 + 5862
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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