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Flat knot 6.386

Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,-1,1,2,3,4,1,2,2,3,0,0,0,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.386']
Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.65', '6.137', '6.201', '6.203', '6.214', '6.310', '6.314', '6.332', '6.385', '6.386', '6.401', '6.516', '6.564', '6.571', '6.572', '6.578', '6.621', '6.626', '6.716', '6.773', '6.807', '6.814', '6.821', '6.940', '6.966', '6.1036', '6.1071', '6.1108', '6.1111', '6.1131', '6.1188', '6.1203', '6.1206', '6.1220', '6.1340', '6.1387', '6.1548', '6.1663', '6.1680', '6.1693', '6.1831', '6.1932']
Outer characteristic polynomial of the knot is: t^7+80t^5+48t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.386']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 128*K1**4*K2 - 1360*K1**4 + 64*K1**3*K2*K3 + 64*K1**3*K3*K4 - 592*K1**2*K2**2 + 1944*K1**2*K2 - 688*K1**2*K3**2 - 352*K1**2*K4**2 - 1420*K1**2 + 1904*K1*K2*K3 + 1152*K1*K3*K4 + 304*K1*K4*K5 + 16*K1*K5*K6 - 48*K2**4 - 48*K2**2*K3**2 - 16*K2**2*K4**2 + 232*K2**2*K4 - 1428*K2**2 + 128*K2*K3*K5 + 24*K2*K4*K6 + 8*K3**2*K6 - 1056*K3**2 - 520*K4**2 - 132*K5**2 - 20*K6**2 + 1774
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.386']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20009', 'vk6.20090', 'vk6.21279', 'vk6.21372', 'vk6.27056', 'vk6.27151', 'vk6.28759', 'vk6.28840', 'vk6.38457', 'vk6.38548', 'vk6.40644', 'vk6.40745', 'vk6.45337', 'vk6.45444', 'vk6.47104', 'vk6.47186', 'vk6.56808', 'vk6.56895', 'vk6.57940', 'vk6.58033', 'vk6.61322', 'vk6.61421', 'vk6.62496', 'vk6.62578', 'vk6.66528', 'vk6.66595', 'vk6.67315', 'vk6.67386', 'vk6.69170', 'vk6.69243', 'vk6.69919', 'vk6.69984']
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 O1O2O3O4O5U4U2O6U3U1U5U6
R3 orbit {'O1O2O3O4O5U4U2O6U3U1U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U1U5U3O6U4U2
Gauss code of K* O1O2O3O4U2U5U1U6U3O6O5U4
Gauss code of -K* O1O2O3O4U1O5O6U2U6U4U5U3
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -2 -1 -1 3 3],[ 2 0 -1 1 0 4 3],[ 2 1 0 1 0 3 2],[ 1 -1 -1 0 0 2 2],[ 1 0 0 0 0 1 1],[-3 -4 -3 -2 -1 0 1],[-3 -3 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 3 -1 -1 -2 -2],[-3 0 1 -1 -2 -3 -4],[-3 -1 0 -1 -2 -2 -3],[ 1 1 1 0 0 0 0],[ 1 2 2 0 0 -1 -1],[ 2 3 2 0 1 0 1],[ 2 4 3 0 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,1,1,2,2,-1,1,2,3,4,1,2,2,3,0,0,0,1,1,-1]
Phi over symmetry [-3,-3,1,1,2,2,-1,1,2,3,4,1,2,2,3,0,0,0,1,1,-1]
Phi of -K [-2,-2,-1,-1,3,3,-1,0,1,2,3,0,1,1,2,0,2,2,3,3,-1]
Phi of K* [-3,-3,1,1,2,2,-1,2,3,2,3,2,3,1,2,0,0,0,1,1,-1]
Phi of -K* [-2,-2,-1,-1,3,3,-1,0,1,3,4,0,1,2,3,0,1,1,2,2,-1]
Symmetry type of based matrix c
u-polynomial -2t^3+2t^2+2t
Normalized Jones-Krushkal polynomial 13z+27
Enhanced Jones-Krushkal polynomial 13w^2z+27w
Inner characteristic polynomial t^6+52t^4+12t^2
Outer characteristic polynomial t^7+80t^5+48t^3
Flat arrow polynomial -4*K1**2 - 4*K1*K2 + 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -64*K1**6 + 128*K1**4*K2 - 1360*K1**4 + 64*K1**3*K2*K3 + 64*K1**3*K3*K4 - 592*K1**2*K2**2 + 1944*K1**2*K2 - 688*K1**2*K3**2 - 352*K1**2*K4**2 - 1420*K1**2 + 1904*K1*K2*K3 + 1152*K1*K3*K4 + 304*K1*K4*K5 + 16*K1*K5*K6 - 48*K2**4 - 48*K2**2*K3**2 - 16*K2**2*K4**2 + 232*K2**2*K4 - 1428*K2**2 + 128*K2*K3*K5 + 24*K2*K4*K6 + 8*K3**2*K6 - 1056*K3**2 - 520*K4**2 - 132*K5**2 - 20*K6**2 + 1774
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice False
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