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Flat knot 6.1488

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,1,2,4,-1,0,2,1,0,1,0,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1488']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.395', '6.430', '6.440', '6.548', '6.551', '6.774', '6.832', '6.887', '6.908', '6.911', '6.1205', '6.1332', '6.1339', '6.1341', '6.1346', '6.1382', '6.1488', '6.1651', '6.1655', '6.1686']
Outer characteristic polynomial of the knot is: t^7+31t^5+68t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1488']
2-strand cable arrow polynomial of the knot is: -64*K1**3*K3 - 192*K1**2*K2**4 + 704*K1**2*K2**3 - 4048*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 4000*K1**2*K2 - 128*K1**2*K3**2 - 3096*K1**2 + 512*K1*K2**3*K3 - 832*K1*K2**2*K3 - 256*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4696*K1*K2*K3 + 416*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1152*K2**4 - 32*K2**3*K6 - 560*K2**2*K3**2 - 16*K2**2*K4**2 + 1184*K2**2*K4 - 2030*K2**2 + 488*K2*K3*K5 + 16*K2*K4*K6 - 1248*K3**2 - 260*K4**2 - 72*K5**2 - 2*K6**2 + 2258
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1488']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4731', 'vk6.5053', 'vk6.6263', 'vk6.6708', 'vk6.8232', 'vk6.8677', 'vk6.9621', 'vk6.9941', 'vk6.20655', 'vk6.22088', 'vk6.28141', 'vk6.29572', 'vk6.39583', 'vk6.41816', 'vk6.46198', 'vk6.47818', 'vk6.48763', 'vk6.48967', 'vk6.49567', 'vk6.49778', 'vk6.50777', 'vk6.50986', 'vk6.51261', 'vk6.51463', 'vk6.57579', 'vk6.58747', 'vk6.62249', 'vk6.63197', 'vk6.67049', 'vk6.67924', 'vk6.69674', 'vk6.70357']
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 O1O2O3U1O4U3U2O5O6U5U4U6
R3 orbit {'O1O2O3U1O4U3U2O5O6U5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5U6O4O6U2U1O5U3
Gauss code of K* O1O2O3U4U5U6O4U2O6O5U1U3
Gauss code of -K* O1O2O3U1U3O4O5U2O6U5U4U6
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 1 -1 2],[ 2 0 2 1 2 0 0],[ 0 -2 0 0 2 0 1],[ 0 -1 0 0 1 0 1],[-1 -2 -2 -1 0 0 2],[ 1 0 0 0 0 0 1],[-2 0 -1 -1 -2 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -2 -1 -1 -1 0],[-1 2 0 -1 -2 0 -2],[ 0 1 1 0 0 0 -1],[ 0 1 2 0 0 0 -2],[ 1 1 0 0 0 0 0],[ 2 0 2 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,2,1,1,1,0,1,2,0,2,0,0,1,0,2,0]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,1,2,4,-1,0,2,1,0,1,0,1,1,1]
Phi of -K [-2,-1,0,0,1,2,1,0,1,1,4,1,1,2,2,0,-1,1,0,1,-1]
Phi of K* [-2,-1,0,0,1,2,-1,1,1,2,4,-1,0,2,1,0,1,0,1,1,1]
Phi of -K* [-2,-1,0,0,1,2,0,1,2,2,0,0,0,0,1,0,1,1,2,1,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+21t^4+20t^2
Outer characteristic polynomial t^7+31t^5+68t^3+4t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
2-strand cable arrow polynomial -64*K1**3*K3 - 192*K1**2*K2**4 + 704*K1**2*K2**3 - 4048*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 4000*K1**2*K2 - 128*K1**2*K3**2 - 3096*K1**2 + 512*K1*K2**3*K3 - 832*K1*K2**2*K3 - 256*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4696*K1*K2*K3 + 416*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1152*K2**4 - 32*K2**3*K6 - 560*K2**2*K3**2 - 16*K2**2*K4**2 + 1184*K2**2*K4 - 2030*K2**2 + 488*K2*K3*K5 + 16*K2*K4*K6 - 1248*K3**2 - 260*K4**2 - 72*K5**2 - 2*K6**2 + 2258
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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