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Flat knot 6.822

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,0,2,1,4,3,1,0,2,1,1,3,3,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.822']
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+84t^5+80t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.822']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 768*K1**4*K2 - 2048*K1**4 + 192*K1**3*K2*K3 + 64*K1**3*K3*K4 + 128*K1**2*K2**3 - 2592*K1**2*K2**2 + 3360*K1**2*K2 - 608*K1**2*K3**2 - 64*K1**2*K4**2 - 1344*K1**2 + 2688*K1*K2*K3 + 624*K1*K3*K4 + 64*K1*K4*K5 - 656*K2**4 - 416*K2**2*K3**2 - 16*K2**2*K4**2 + 400*K2**2*K4 - 1164*K2**2 + 272*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 + 32*K3**2*K6 - 776*K3**2 - 244*K4**2 - 56*K5**2 - 12*K6**2 + 1674
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.822']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13957', 'vk6.14053', 'vk6.15024', 'vk6.15146', 'vk6.17449', 'vk6.17468', 'vk6.23957', 'vk6.23988', 'vk6.33761', 'vk6.33837', 'vk6.34301', 'vk6.36252', 'vk6.43410', 'vk6.53889', 'vk6.53923', 'vk6.54434', 'vk6.55599', 'vk6.60086', 'vk6.60102', 'vk6.65311']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2O3O4U2O5U1U5U6U4O6U3
R3 orbit {'O1O2O3O4U2O5U1U5U6U4O6U3'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3O4U3O5U1U6U5U4O6U2
Gauss code of -K* O1O2O3O4U3O5U1U6U5U4O6U2
Diagrammatic symmetry type r
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -2 2 3 1 -1],[ 3 0 0 4 3 1 2],[ 2 0 0 2 1 0 1],[-2 -4 -2 0 1 0 -3],[-3 -3 -1 -1 0 0 -3],[-1 -1 0 0 0 0 -1],[ 1 -2 -1 3 3 1 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 -1 0 -3 -1 -3],[-2 1 0 0 -3 -2 -4],[-1 0 0 0 -1 0 -1],[ 1 3 3 1 0 -1 -2],[ 2 1 2 0 1 0 0],[ 3 3 4 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,1,0,3,1,3,0,3,2,4,1,0,1,1,2,0]
Phi over symmetry [-3,-2,-1,1,2,3,0,2,1,4,3,1,0,2,1,1,3,3,0,0,1]
Phi of -K [-3,-2,-1,1,2,3,1,0,3,1,3,0,3,2,4,1,0,1,1,2,0]
Phi of K* [-3,-2,-1,1,2,3,0,2,1,4,3,1,0,2,1,1,3,3,0,0,1]
Phi of -K* [-3,-2,-1,1,2,3,0,2,1,4,3,1,0,2,1,1,3,3,0,0,1]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 12z+25
Enhanced Jones-Krushkal polynomial 12w^2z+25w
Inner characteristic polynomial t^6+56t^4+24t^2
Outer characteristic polynomial t^7+84t^5+80t^3
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 + 768*K1**4*K2 - 2048*K1**4 + 192*K1**3*K2*K3 + 64*K1**3*K3*K4 + 128*K1**2*K2**3 - 2592*K1**2*K2**2 + 3360*K1**2*K2 - 608*K1**2*K3**2 - 64*K1**2*K4**2 - 1344*K1**2 + 2688*K1*K2*K3 + 624*K1*K3*K4 + 64*K1*K4*K5 - 656*K2**4 - 416*K2**2*K3**2 - 16*K2**2*K4**2 + 400*K2**2*K4 - 1164*K2**2 + 272*K2*K3*K5 + 16*K2*K4*K6 - 32*K3**4 + 32*K3**2*K6 - 776*K3**2 - 244*K4**2 - 56*K5**2 - 12*K6**2 + 1674
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
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