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Flat knot 6.1787

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,1,0,0,1,1,0,0,1,0,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1787', '7.45992']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923']
Outer characteristic polynomial of the knot is: t^7+16t^5+29t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1787', '6.1818']
2-strand cable arrow polynomial of the knot is: -1280*K1**6 - 2496*K1**4*K2**2 + 5408*K1**4*K2 - 8080*K1**4 + 2080*K1**3*K2*K3 - 1184*K1**3*K3 - 1472*K1**2*K2**4 + 5216*K1**2*K2**3 + 480*K1**2*K2**2*K4 - 14128*K1**2*K2**2 - 1376*K1**2*K2*K4 + 11864*K1**2*K2 - 656*K1**2*K3**2 - 112*K1**2*K4**2 - 496*K1**2 + 1888*K1*K2**3*K3 - 2272*K1*K2**2*K3 - 448*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7760*K1*K2*K3 + 616*K1*K3*K4 + 104*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2928*K2**4 - 64*K2**3*K6 - 528*K2**2*K3**2 - 128*K2**2*K4**2 + 1976*K2**2*K4 - 1436*K2**2 + 264*K2*K3*K5 + 48*K2*K4*K6 - 796*K3**2 - 220*K4**2 - 36*K5**2 - 4*K6**2 + 2626
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1787']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.505', 'vk6.596', 'vk6.623', 'vk6.1005', 'vk6.1102', 'vk6.1127', 'vk6.1665', 'vk6.1840', 'vk6.2179', 'vk6.2180', 'vk6.2286', 'vk6.2308', 'vk6.2784', 'vk6.2885', 'vk6.3067', 'vk6.3193', 'vk6.5261', 'vk6.6516', 'vk6.8890', 'vk6.9805', 'vk6.20816', 'vk6.21050', 'vk6.22211', 'vk6.22473', 'vk6.28495', 'vk6.29777', 'vk6.39876', 'vk6.40275', 'vk6.46430', 'vk6.46924', 'vk6.49149', 'vk6.58826']
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 O1O2O3U4U5O6O4U6U3O5U1U2
R3 orbit {'O1O2O3U4U5O6O4U6U3O5U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U3O4U1U5O6O5U4U6
Gauss code of K* O1O2U3O4O5U4U5U2O6O3U1U6
Gauss code of -K* O1O2U3O4O5U6U5O3O6U4U1U2
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 -1 1 1 0 0 -1],[ 1 0 1 1 1 1 -1],[-1 -1 0 1 -1 -1 -1],[-1 -1 -1 0 0 0 -1],[ 0 -1 1 0 0 0 -1],[ 0 -1 1 0 0 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 -1 -1 -1 -1],[-1 -1 0 0 0 -1 -1],[ 0 1 0 0 0 -1 -1],[ 0 1 0 0 0 -1 -1],[ 1 1 1 1 1 0 1],[ 1 1 1 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,1,1,1,1,0,0,1,1,0,1,1,1,1,-1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,1,0,0,1,1,0,0,1,0,1,-1]
Phi of -K [-1,-1,0,0,1,1,-1,0,0,1,1,0,0,1,1,0,0,1,0,1,-1]
Phi of K* [-1,-1,0,0,1,1,-1,1,1,1,1,0,0,1,1,0,0,0,0,0,-1]
Phi of -K* [-1,-1,0,0,1,1,-1,1,1,1,1,1,1,1,1,0,0,1,0,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+12t^4+19t^2
Outer characteristic polynomial t^7+16t^5+29t^3+2t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -1280*K1**6 - 2496*K1**4*K2**2 + 5408*K1**4*K2 - 8080*K1**4 + 2080*K1**3*K2*K3 - 1184*K1**3*K3 - 1472*K1**2*K2**4 + 5216*K1**2*K2**3 + 480*K1**2*K2**2*K4 - 14128*K1**2*K2**2 - 1376*K1**2*K2*K4 + 11864*K1**2*K2 - 656*K1**2*K3**2 - 112*K1**2*K4**2 - 496*K1**2 + 1888*K1*K2**3*K3 - 2272*K1*K2**2*K3 - 448*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7760*K1*K2*K3 + 616*K1*K3*K4 + 104*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2928*K2**4 - 64*K2**3*K6 - 528*K2**2*K3**2 - 128*K2**2*K4**2 + 1976*K2**2*K4 - 1436*K2**2 + 264*K2*K3*K5 + 48*K2*K4*K6 - 796*K3**2 - 220*K4**2 - 36*K5**2 - 4*K6**2 + 2626
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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