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Flat knot 6.1370

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,1,2,3,0,1,0,0,1,1,0,-1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1370']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1*K2 - 2*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.233', '6.340', '6.382', '6.550', '6.656', '6.663', '6.683', '6.698', '6.739', '6.745', '6.759', '6.765', '6.1357', '6.1358', '6.1370']
Outer characteristic polynomial of the knot is: t^7+36t^5+75t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1370', '6.1388']
2-strand cable arrow polynomial of the knot is: 768*K1**4*K2 - 1664*K1**4 + 832*K1**3*K2*K3 - 128*K1**2*K2**4 + 384*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3328*K1**2*K2**2 - 128*K1**2*K2*K4 + 3824*K1**2*K2 - 1024*K1**2*K3**2 - 32*K1**2*K4**2 - 1768*K1**2 + 256*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 64*K1*K2**2*K5 + 3264*K1*K2*K3 + 1168*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 624*K2**4 - 384*K2**2*K3**2 - 8*K2**2*K4**2 + 752*K2**2*K4 - 1576*K2**2 + 176*K2*K3*K5 - 936*K3**2 - 356*K4**2 - 16*K5**2 + 1810
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1370']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20149', 'vk6.20157', 'vk6.21439', 'vk6.21443', 'vk6.27269', 'vk6.27285', 'vk6.28929', 'vk6.28945', 'vk6.38692', 'vk6.38706', 'vk6.40884', 'vk6.47271', 'vk6.47279', 'vk6.56982', 'vk6.56993', 'vk6.58132', 'vk6.62687', 'vk6.67465', 'vk6.70042', 'vk6.70049']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3U4O5O4U1U3U2O6U5U6
R3 orbit {'O1O2O3U4O5O4U1U3U2O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U2U1U3O6O5U6
Gauss code of K* Same
Gauss code of -K* O1O2O3U4U5O4U2U1U3O6O5U6
Diagrammatic symmetry type +
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 0 0 1 1 1],[ 3 0 2 1 3 2 1],[ 0 -2 0 0 0 1 1],[ 0 -1 0 0 0 0 1],[-1 -3 0 0 0 1 0],[-1 -2 -1 0 -1 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 1 0 0 0 -3],[-1 -1 0 1 0 -1 -2],[-1 0 -1 0 -1 -1 -1],[ 0 0 0 1 0 0 -1],[ 0 0 1 1 0 0 -2],[ 3 3 2 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,-1,0,0,0,3,-1,0,1,2,1,1,1,0,1,2]
Phi over symmetry [-3,0,0,1,1,1,1,2,1,2,3,0,1,0,0,1,1,0,-1,0,-1]
Phi of -K [-3,0,0,1,1,1,1,2,1,2,3,0,1,0,0,1,1,0,-1,0,-1]
Phi of K* [-1,-1,-1,0,0,3,-1,0,0,0,3,-1,0,1,2,1,1,1,0,1,2]
Phi of -K* [-3,0,0,1,1,1,1,2,1,2,3,0,1,0,0,1,1,0,-1,0,-1]
Symmetry type of based matrix +
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+24t^4+43t^2+4
Outer characteristic polynomial t^7+36t^5+75t^3+8t
Flat arrow polynomial 4*K1**3 - 2*K1*K2 - 2*K1 + 1
2-strand cable arrow polynomial 768*K1**4*K2 - 1664*K1**4 + 832*K1**3*K2*K3 - 128*K1**2*K2**4 + 384*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3328*K1**2*K2**2 - 128*K1**2*K2*K4 + 3824*K1**2*K2 - 1024*K1**2*K3**2 - 32*K1**2*K4**2 - 1768*K1**2 + 256*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 64*K1*K2**2*K5 + 3264*K1*K2*K3 + 1168*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 624*K2**4 - 384*K2**2*K3**2 - 8*K2**2*K4**2 + 752*K2**2*K4 - 1576*K2**2 + 176*K2*K3*K5 - 936*K3**2 - 356*K4**2 - 16*K5**2 + 1810
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice False
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