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Flat knot 6.981

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,1,0,2,3,-1,1,-1,0,0,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.981']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']
Outer characteristic polynomial of the knot is: t^7+34t^5+77t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.981']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 992*K1**4*K2 - 3488*K1**4 + 352*K1**3*K2*K3 - 896*K1**3*K3 - 192*K1**2*K2**4 + 864*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5088*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 10184*K1**2*K2 - 832*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 6308*K1**2 + 256*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7480*K1*K2*K3 - 32*K1*K2*K4*K5 + 1352*K1*K3*K4 + 40*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1216*K2**4 - 32*K2**3*K6 - 512*K2**2*K3**2 - 40*K2**2*K4**2 + 1552*K2**2*K4 - 4818*K2**2 + 328*K2*K3*K5 + 32*K2*K4*K6 - 2244*K3**2 - 548*K4**2 - 40*K5**2 - 6*K6**2 + 5042
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.981']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4056', 'vk6.4087', 'vk6.5298', 'vk6.5329', 'vk6.7430', 'vk6.7455', 'vk6.8929', 'vk6.8960', 'vk6.10126', 'vk6.10293', 'vk6.10316', 'vk6.14557', 'vk6.15277', 'vk6.15404', 'vk6.15779', 'vk6.16196', 'vk6.29868', 'vk6.29899', 'vk6.33915', 'vk6.33998', 'vk6.34232', 'vk6.34379', 'vk6.48457', 'vk6.49160', 'vk6.50210', 'vk6.50237', 'vk6.51602', 'vk6.53958', 'vk6.54021', 'vk6.54179', 'vk6.54459', 'vk6.63313']
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 O1O2O3O4U1U3O5U4O6U5U6U2
R3 orbit {'O1O2O3O4U1U3O5U4O6U5U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U6O5U1O6U2U4
Gauss code of K* O1O2O3U4U3U5U6O4O5U1O6U2
Gauss code of -K* O1O2O3U2O4U3O5O6U4U5U1U6
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 -3 1 0 1 0 1],[ 3 0 3 1 2 1 0],[-1 -3 0 -1 0 0 1],[ 0 -1 1 0 1 1 0],[-1 -2 0 -1 0 1 1],[ 0 -1 0 -1 -1 0 1],[-1 0 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 1 0 1 -1 -2],[-1 -1 0 -1 -1 0 0],[-1 0 1 0 0 -1 -3],[ 0 -1 1 0 0 -1 -1],[ 0 1 0 1 1 0 -1],[ 3 2 0 3 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,-1,0,-1,1,2,1,1,0,0,0,1,3,1,1,1]
Phi over symmetry [-3,0,0,1,1,1,1,1,0,2,3,-1,1,-1,0,0,1,1,-1,-1,0]
Phi of -K [-3,0,0,1,1,1,2,2,1,2,4,-1,0,0,1,1,2,0,0,-1,-1]
Phi of K* [-1,-1,-1,0,0,3,-1,-1,0,1,4,0,1,0,1,2,0,2,-1,2,2]
Phi of -K* [-3,0,0,1,1,1,1,1,0,2,3,-1,1,-1,0,0,1,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 3z^2+23z+35
Enhanced Jones-Krushkal polynomial 3w^3z^2+23w^2z+35w
Inner characteristic polynomial t^6+22t^4+25t^2+1
Outer characteristic polynomial t^7+34t^5+77t^3+7t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -64*K1**6 + 992*K1**4*K2 - 3488*K1**4 + 352*K1**3*K2*K3 - 896*K1**3*K3 - 192*K1**2*K2**4 + 864*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5088*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 10184*K1**2*K2 - 832*K1**2*K3**2 - 32*K1**2*K3*K5 - 48*K1**2*K4**2 - 6308*K1**2 + 256*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7480*K1*K2*K3 - 32*K1*K2*K4*K5 + 1352*K1*K3*K4 + 40*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 + 96*K2**4*K4 - 1216*K2**4 - 32*K2**3*K6 - 512*K2**2*K3**2 - 40*K2**2*K4**2 + 1552*K2**2*K4 - 4818*K2**2 + 328*K2*K3*K5 + 32*K2*K4*K6 - 2244*K3**2 - 548*K4**2 - 40*K5**2 - 6*K6**2 + 5042
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {3, 5}, {1, 4}, {2}]]
If K is slice False
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