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Flat knot 6.1021

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,-1,1,2,3,0,0,1,1,0,2,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1021']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 + 2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922']
Outer characteristic polynomial of the knot is: t^7+36t^5+47t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1021']
2-strand cable arrow polynomial of the knot is: 1344*K1**4*K2 - 3856*K1**4 + 384*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1152*K1**3*K3 + 128*K1**2*K2**2*K4 - 3232*K1**2*K2**2 - 800*K1**2*K2*K4 + 8128*K1**2*K2 - 784*K1**2*K3**2 - 240*K1**2*K4**2 - 5168*K1**2 - 672*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6000*K1*K2*K3 + 2168*K1*K3*K4 + 248*K1*K4*K5 - 64*K2**4 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 1240*K2**2*K4 - 4796*K2**2 + 352*K2*K3*K5 + 32*K2*K4*K6 - 2412*K3**2 - 1164*K4**2 - 180*K5**2 - 4*K6**2 + 4802
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1021']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4875', 'vk6.5220', 'vk6.6465', 'vk6.6886', 'vk6.8430', 'vk6.8851', 'vk6.9774', 'vk6.10067', 'vk6.11667', 'vk6.12018', 'vk6.13013', 'vk6.20490', 'vk6.20775', 'vk6.21847', 'vk6.27892', 'vk6.29398', 'vk6.29744', 'vk6.32664', 'vk6.33005', 'vk6.39327', 'vk6.39807', 'vk6.46371', 'vk6.47597', 'vk6.47948', 'vk6.48833', 'vk6.49104', 'vk6.51356', 'vk6.51569', 'vk6.53274', 'vk6.57359', 'vk6.64343', 'vk6.66912']
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 O1O2O3O4U3U2O5U4O6U1U6U5
R3 orbit {'O1O2O3O4U3U2O5U4O6U1U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U4O6U1O5U3U2
Gauss code of K* O1O2O3U1U4U5U6O5O4U3O6U2
Gauss code of -K* O1O2O3U2O4U1O5O6U4U6U5U3
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 -1 -1 1 2 1],[ 2 0 -1 -1 2 3 1],[ 1 1 0 0 2 1 0],[ 1 1 0 0 1 1 0],[-1 -2 -2 -1 0 1 0],[-2 -3 -1 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 -1 -1 -3],[-1 0 0 0 0 0 -1],[-1 1 0 0 -1 -2 -2],[ 1 1 0 1 0 0 1],[ 1 1 0 2 0 0 1],[ 2 3 1 2 -1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,1,1,1,3,0,0,0,1,1,2,2,0,-1,-1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,-1,1,2,3,0,0,1,1,0,2,1,0,0,1]
Phi of -K [-2,-1,-1,1,1,2,2,2,1,2,1,0,0,2,2,1,2,2,0,0,1]
Phi of K* [-2,-1,-1,1,1,2,0,1,2,2,1,0,0,1,1,2,2,2,0,2,2]
Phi of -K* [-2,-1,-1,1,1,2,-1,-1,1,2,3,0,0,1,1,0,2,1,0,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+24t^4+19t^2+1
Outer characteristic polynomial t^7+36t^5+47t^3+4t
Flat arrow polynomial -4*K1*K2 + 2*K1 + 2*K3 + 1
2-strand cable arrow polynomial 1344*K1**4*K2 - 3856*K1**4 + 384*K1**3*K2*K3 + 96*K1**3*K3*K4 - 1152*K1**3*K3 + 128*K1**2*K2**2*K4 - 3232*K1**2*K2**2 - 800*K1**2*K2*K4 + 8128*K1**2*K2 - 784*K1**2*K3**2 - 240*K1**2*K4**2 - 5168*K1**2 - 672*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6000*K1*K2*K3 + 2168*K1*K3*K4 + 248*K1*K4*K5 - 64*K2**4 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 1240*K2**2*K4 - 4796*K2**2 + 352*K2*K3*K5 + 32*K2*K4*K6 - 2412*K3**2 - 1164*K4**2 - 180*K5**2 - 4*K6**2 + 4802
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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