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Flat knot 6.747

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,0,2,2,3,0,2,0,2,1,1,2,2,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.747']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863']
Outer characteristic polynomial of the knot is: t^7+55t^5+70t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.747']
2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 1504*K1**4*K2 - 4608*K1**4 + 448*K1**3*K2*K3 - 832*K1**3*K3 + 832*K1**2*K2**3 - 7888*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 12816*K1**2*K2 - 512*K1**2*K3**2 - 6668*K1**2 + 256*K1*K2**3*K3 - 1312*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7776*K1*K2*K3 + 600*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1152*K2**4 - 32*K2**3*K6 - 272*K2**2*K3**2 - 16*K2**2*K4**2 + 1224*K2**2*K4 - 5110*K2**2 + 160*K2*K3*K5 + 16*K2*K4*K6 - 1848*K3**2 - 280*K4**2 - 20*K5**2 - 2*K6**2 + 5318
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.747']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16567', 'vk6.16660', 'vk6.18124', 'vk6.18460', 'vk6.22970', 'vk6.23091', 'vk6.24583', 'vk6.24996', 'vk6.34967', 'vk6.35088', 'vk6.36722', 'vk6.37141', 'vk6.42540', 'vk6.42651', 'vk6.43994', 'vk6.44306', 'vk6.54814', 'vk6.54895', 'vk6.55942', 'vk6.56238', 'vk6.59246', 'vk6.59320', 'vk6.60480', 'vk6.60842', 'vk6.64796', 'vk6.64861', 'vk6.65603', 'vk6.65910', 'vk6.68098', 'vk6.68163', 'vk6.68678', 'vk6.68889']
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 O1O2O3O4U2O5U3U1U5O6U4U6
R3 orbit {'O1O2O3O4U2O5U3U1U5O6U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1O5U6U4U2O6U3
Gauss code of K* O1O2O3U4O5O4U2U6U1U5O6U3
Gauss code of -K* O1O2O3U1O4U5U3U4U2O6O5U6
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 -2 -1 2 2 1],[ 2 0 -1 1 4 2 1],[ 2 1 0 1 2 1 1],[ 1 -1 -1 0 2 1 1],[-2 -4 -2 -2 0 0 1],[-2 -2 -1 -1 0 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 0 1 -2 -2 -4],[-2 0 0 0 -1 -1 -2],[-1 -1 0 0 -1 -1 -1],[ 1 2 1 1 0 -1 -1],[ 2 2 1 1 1 0 1],[ 2 4 2 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,0,-1,2,2,4,0,1,1,2,1,1,1,1,1,-1]
Phi over symmetry [-2,-2,-1,1,2,2,-1,0,2,2,3,0,2,0,2,1,1,2,2,1,0]
Phi of -K [-2,-2,-1,1,2,2,-1,0,2,2,3,0,2,0,2,1,1,2,2,1,0]
Phi of K* [-2,-2,-1,1,2,2,0,1,2,2,3,2,1,0,2,1,2,2,0,0,-1]
Phi of -K* [-2,-2,-1,1,2,2,-1,1,1,2,4,1,1,1,2,1,1,2,0,-1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+37t^4+20t^2
Outer characteristic polynomial t^7+55t^5+70t^3+7t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -64*K1**6 - 64*K1**4*K2**2 + 1504*K1**4*K2 - 4608*K1**4 + 448*K1**3*K2*K3 - 832*K1**3*K3 + 832*K1**2*K2**3 - 7888*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 12816*K1**2*K2 - 512*K1**2*K3**2 - 6668*K1**2 + 256*K1*K2**3*K3 - 1312*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7776*K1*K2*K3 + 600*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1152*K2**4 - 32*K2**3*K6 - 272*K2**2*K3**2 - 16*K2**2*K4**2 + 1224*K2**2*K4 - 5110*K2**2 + 160*K2*K3*K5 + 16*K2*K4*K6 - 1848*K3**2 - 280*K4**2 - 20*K5**2 - 2*K6**2 + 5318
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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