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Flat knot 6.1647

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,3,0,0,1,1,1,0,0,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1647']
Arrow polynomial of the knot is: -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.546', '6.591', '6.598', '6.666', '6.680', '6.742', '6.778', '6.805', '6.822', '6.824', '6.1129', '6.1512', '6.1647', '6.1678', '6.1705', '6.1847', '6.1857']
Outer characteristic polynomial of the knot is: t^7+35t^5+53t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1647']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 1600*K1**4*K2 - 5152*K1**4 + 480*K1**3*K2*K3 + 64*K1**3*K3*K4 - 576*K1**3*K3 - 5632*K1**2*K2**2 - 576*K1**2*K2*K4 + 11024*K1**2*K2 - 576*K1**2*K3**2 - 64*K1**2*K3*K5 - 128*K1**2*K4**2 - 6076*K1**2 - 416*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 6888*K1*K2*K3 + 1656*K1*K3*K4 + 216*K1*K4*K5 - 432*K2**4 - 112*K2**2*K3**2 - 16*K2**2*K4**2 + 984*K2**2*K4 - 5236*K2**2 + 184*K2*K3*K5 + 32*K2*K4*K6 - 2344*K3**2 - 884*K4**2 - 108*K5**2 - 12*K6**2 + 5562
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1647']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3569', 'vk6.3591', 'vk6.3603', 'vk6.3818', 'vk6.3830', 'vk6.3849', 'vk6.3861', 'vk6.6988', 'vk6.6992', 'vk6.7019', 'vk6.7023', 'vk6.7210', 'vk6.7214', 'vk6.7236', 'vk6.15337', 'vk6.15338', 'vk6.15462', 'vk6.15465', 'vk6.33979', 'vk6.34025', 'vk6.34026', 'vk6.34437', 'vk6.48227', 'vk6.48231', 'vk6.48384', 'vk6.49959', 'vk6.49981', 'vk6.49993', 'vk6.53983', 'vk6.53986', 'vk6.54040', 'vk6.54487']
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 O1O2O3U1O4U3U5U4O5O6U2U6
R3 orbit {'O1O2O3U1O4U3U5U4O5O6U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O4O5U6U5U1O6U3
Gauss code of K* O1O2O3U2U4O5O4U6U5U1O6U3
Gauss code of -K* O1O2O3U1O4U3U5U4O6O5U6U2
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 0 0 2 -1 1],[ 2 0 2 1 1 2 1],[ 0 -2 0 -1 2 -1 1],[ 0 -1 1 0 1 -1 0],[-2 -1 -2 -1 0 -2 0],[ 1 -2 1 1 2 0 1],[-1 -1 -1 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -2 -2 -1],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 1 -1 -1],[ 0 2 1 -1 0 -1 -2],[ 1 2 1 1 1 0 -2],[ 2 1 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,1,2,2,1,0,1,1,1,-1,1,1,1,2,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,1,2,3,0,0,1,1,1,0,0,1,1,1]
Phi of -K [-2,-1,0,0,1,2,-1,0,1,2,3,0,0,1,1,1,0,0,1,1,1]
Phi of K* [-2,-1,0,0,1,2,1,0,1,1,3,0,1,1,2,-1,0,0,0,1,-1]
Phi of -K* [-2,-1,0,0,1,2,2,1,2,1,1,1,1,1,2,1,0,1,1,2,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+25t^4+35t^2+1
Outer characteristic polynomial t^7+35t^5+53t^3+4t
Flat arrow polynomial -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -256*K1**4*K2**2 + 1600*K1**4*K2 - 5152*K1**4 + 480*K1**3*K2*K3 + 64*K1**3*K3*K4 - 576*K1**3*K3 - 5632*K1**2*K2**2 - 576*K1**2*K2*K4 + 11024*K1**2*K2 - 576*K1**2*K3**2 - 64*K1**2*K3*K5 - 128*K1**2*K4**2 - 6076*K1**2 - 416*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 6888*K1*K2*K3 + 1656*K1*K3*K4 + 216*K1*K4*K5 - 432*K2**4 - 112*K2**2*K3**2 - 16*K2**2*K4**2 + 984*K2**2*K4 - 5236*K2**2 + 184*K2*K3*K5 + 32*K2*K4*K6 - 2344*K3**2 - 884*K4**2 - 108*K5**2 - 12*K6**2 + 5562
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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