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Flat knot 6.1499

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,0,0,0,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1499', '7.32492']
Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874']
Outer characteristic polynomial of the knot is: t^7+36t^5+52t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1499', '7.32492']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 576*K1**4*K2**2 + 2912*K1**4*K2 - 6256*K1**4 + 928*K1**3*K2*K3 - 1248*K1**3*K3 - 384*K1**2*K2**4 + 2720*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 9408*K1**2*K2**2 - 832*K1**2*K2*K4 + 9608*K1**2*K2 - 592*K1**2*K3**2 - 1396*K1**2 + 928*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5952*K1*K2*K3 + 408*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1976*K2**4 - 432*K2**2*K3**2 - 48*K2**2*K4**2 + 1328*K2**2*K4 - 1630*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 - 720*K3**2 - 130*K4**2 - 12*K5**2 - 2*K6**2 + 2408
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1499']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.500', 'vk6.593', 'vk6.618', 'vk6.1000', 'vk6.1099', 'vk6.1124', 'vk6.1668', 'vk6.1841', 'vk6.2166', 'vk6.2185', 'vk6.2275', 'vk6.2313', 'vk6.2795', 'vk6.2894', 'vk6.3068', 'vk6.3198', 'vk6.5268', 'vk6.6525', 'vk6.8897', 'vk6.9814', 'vk6.20811', 'vk6.21045', 'vk6.22208', 'vk6.22468', 'vk6.28494', 'vk6.29774', 'vk6.39863', 'vk6.40278', 'vk6.46421', 'vk6.46921', 'vk6.49140', 'vk6.58835']
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 O1O2O3U1O4U5U6O5O6U3U4U2
R3 orbit {'O1O2O3U1O4U5U6O5O6U3U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U1O5O6U5U6O4U3
Gauss code of K* O1O2O3U4U3U1O4U2O5O6U5U6
Gauss code of -K* O1O2O3U4U5O4O5U2O6U3U1U6
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 -2 1 0 1 -1 1],[ 2 0 2 1 1 2 2],[-1 -2 0 -1 1 -2 0],[ 0 -1 1 0 1 -1 1],[-1 -1 -1 -1 0 -2 0],[ 1 -2 2 1 2 0 1],[-1 -2 0 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 -1 -2 -2],[-1 -1 0 0 -1 -2 -1],[-1 0 0 0 -1 -1 -2],[ 0 1 1 1 0 -1 -1],[ 1 2 2 1 1 0 -2],[ 2 2 1 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,1,2,2,0,1,2,1,1,1,2,1,1,2]
Phi over symmetry [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,0,0,0,0,0,-1,0]
Phi of -K [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,0,0,0,0,0,-1,0]
Phi of K* [-1,-1,-1,0,1,2,-1,0,0,0,2,0,0,0,1,0,1,1,0,1,-1]
Phi of -K* [-2,-1,0,1,1,1,2,1,1,2,2,1,2,1,2,1,1,1,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+22z+33
Enhanced Jones-Krushkal polynomial 3w^3z^2+22w^2z+33w
Inner characteristic polynomial t^6+28t^4+37t^2+4
Outer characteristic polynomial t^7+36t^5+52t^3+9t
Flat arrow polynomial 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -256*K1**6 - 576*K1**4*K2**2 + 2912*K1**4*K2 - 6256*K1**4 + 928*K1**3*K2*K3 - 1248*K1**3*K3 - 384*K1**2*K2**4 + 2720*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 9408*K1**2*K2**2 - 832*K1**2*K2*K4 + 9608*K1**2*K2 - 592*K1**2*K3**2 - 1396*K1**2 + 928*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5952*K1*K2*K3 + 408*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1976*K2**4 - 432*K2**2*K3**2 - 48*K2**2*K4**2 + 1328*K2**2*K4 - 1630*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 - 720*K3**2 - 130*K4**2 - 12*K5**2 - 2*K6**2 + 2408
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]]
If K is slice False
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