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Flat knot 6.666

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,3,0,1,1,1,1,0,1,0,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.666']
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+31t^5+54t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.666']
2-strand cable arrow polynomial of the knot is: -128*K1**6 - 64*K1**4*K2**2 + 2240*K1**4*K2 - 7936*K1**4 + 384*K1**3*K2*K3 - 1280*K1**3*K3 + 256*K1**2*K2**3 - 6528*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 14216*K1**2*K2 - 1088*K1**2*K3**2 - 32*K1**2*K3*K5 - 6092*K1**2 - 832*K1*K2**2*K3 - 32*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 8216*K1*K2*K3 + 1448*K1*K3*K4 + 72*K1*K4*K5 - 432*K2**4 - 176*K2**2*K3**2 - 16*K2**2*K4**2 + 880*K2**2*K4 - 5964*K2**2 + 272*K2*K3*K5 + 32*K2*K4*K6 - 2440*K3**2 - 576*K4**2 - 92*K5**2 - 12*K6**2 + 6086
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.666']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3645', 'vk6.3742', 'vk6.3931', 'vk6.4028', 'vk6.4502', 'vk6.4597', 'vk6.5884', 'vk6.6011', 'vk6.7130', 'vk6.7303', 'vk6.7396', 'vk6.7933', 'vk6.8052', 'vk6.9363', 'vk6.17922', 'vk6.18019', 'vk6.18753', 'vk6.24457', 'vk6.24874', 'vk6.25335', 'vk6.37492', 'vk6.43884', 'vk6.44221', 'vk6.44524', 'vk6.48285', 'vk6.48350', 'vk6.50068', 'vk6.50178', 'vk6.50578', 'vk6.50641', 'vk6.55883', 'vk6.60713']
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 O1O2O3O4U2O5U4U5O6U3U1U6
R3 orbit {'O1O2O3O4U2O5U4U5O6U3U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U2O5U6U1O6U3
Gauss code of K* O1O2O3U2U4U1U5O4U6O5O6U3
Gauss code of -K* O1O2O3U1O4O5U4O6U5U3U6U2
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 -2 0 0 1 2],[ 1 0 -2 1 0 1 2],[ 2 2 0 2 1 1 1],[ 0 -1 -2 0 -1 1 1],[ 0 0 -1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -1 -2 -1],[-1 0 0 -1 -1 -1 -1],[ 0 0 1 0 1 0 -1],[ 0 1 1 -1 0 -1 -2],[ 1 2 1 0 1 0 -2],[ 2 1 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,0,1,2,1,1,1,1,1,-1,0,1,1,2,2]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,1,2,3,0,1,1,1,1,0,1,0,2,1]
Phi of -K [-2,-1,0,0,1,2,-1,0,1,2,3,0,1,1,1,1,0,1,0,2,1]
Phi of K* [-2,-1,0,0,1,2,1,1,2,1,3,0,0,1,2,-1,0,0,1,1,-1]
Phi of -K* [-2,-1,0,0,1,2,2,1,2,1,1,0,1,1,2,1,1,0,1,1,0]
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+21t^4+24t^2+1
Outer characteristic polynomial t^7+31t^5+54t^3+5t
Flat arrow polynomial -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -128*K1**6 - 64*K1**4*K2**2 + 2240*K1**4*K2 - 7936*K1**4 + 384*K1**3*K2*K3 - 1280*K1**3*K3 + 256*K1**2*K2**3 - 6528*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 14216*K1**2*K2 - 1088*K1**2*K3**2 - 32*K1**2*K3*K5 - 6092*K1**2 - 832*K1*K2**2*K3 - 32*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 8216*K1*K2*K3 + 1448*K1*K3*K4 + 72*K1*K4*K5 - 432*K2**4 - 176*K2**2*K3**2 - 16*K2**2*K4**2 + 880*K2**2*K4 - 5964*K2**2 + 272*K2*K3*K5 + 32*K2*K4*K6 - 2440*K3**2 - 576*K4**2 - 92*K5**2 - 12*K6**2 + 6086
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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