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Flat knot 6.106

Min(phi) over symmetries of the knot is: [-4,-2,1,1,2,2,0,3,4,2,3,2,2,1,2,0,1,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.106']
Arrow polynomial of the knot is: 4*K1**2*K2 - 2*K1*K3 - K2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.64', '6.74', '6.106', '6.178', '6.300', '6.397', '6.479', '6.481', '6.500']
Outer characteristic polynomial of the knot is: t^7+79t^5+34t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.106']
2-strand cable arrow polynomial of the knot is: 1152*K1**4*K2 - 1984*K1**4 - 256*K1**3*K3 + 480*K1**2*K2**3 - 2688*K1**2*K2**2 - 640*K1**2*K2*K4 + 3944*K1**2*K2 - 128*K1**2*K3**2 - 192*K1**2*K3*K5 - 480*K1**2*K4**2 - 32*K1**2*K4*K6 - 64*K1**2*K5**2 - 2840*K1**2 + 96*K1*K2**2*K3*K4 - 288*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 2840*K1*K2*K3 - 288*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 1744*K1*K3*K4 + 1328*K1*K4*K5 + 240*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 544*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 128*K2**2*K3**2 - 336*K2**2*K4**2 + 1144*K2**2*K4 - 128*K2**2*K5**2 - 8*K2**2*K6**2 - 2216*K2**2 + 776*K2*K3*K5 + 384*K2*K4*K6 + 72*K2*K5*K7 - 1240*K3**2 - 1266*K4**2 - 688*K5**2 - 136*K6**2 + 2880
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.106']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17131', 'vk6.17372', 'vk6.20608', 'vk6.22024', 'vk6.23536', 'vk6.23868', 'vk6.28078', 'vk6.29529', 'vk6.35700', 'vk6.36121', 'vk6.39484', 'vk6.41691', 'vk6.43039', 'vk6.43343', 'vk6.46075', 'vk6.47736', 'vk6.55280', 'vk6.55526', 'vk6.57480', 'vk6.58646', 'vk6.59704', 'vk6.60042', 'vk6.62155', 'vk6.63113', 'vk6.65085', 'vk6.65271', 'vk6.67004', 'vk6.67871', 'vk6.68335', 'vk6.68481', 'vk6.69622', 'vk6.70312']
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 O1O2O3O4O5O6U2U6U5U1U4U3
R3 orbit {'O1O2O3O4O5O6U2U6U5U1U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U3U6U2U1U5
Gauss code of K* O1O2O3O4O5O6U4U1U6U5U3U2
Gauss code of -K* O1O2O3O4O5O6U5U4U2U1U6U3
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 -4 2 2 1 1],[ 2 0 -2 3 2 1 1],[ 4 2 0 4 3 2 1],[-2 -3 -4 0 0 0 0],[-2 -2 -3 0 0 0 0],[-1 -1 -2 0 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 2 1 1 -2 -4],[-2 0 0 0 0 -2 -3],[-2 0 0 0 0 -3 -4],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -1 -2],[ 2 2 3 1 1 0 -2],[ 4 3 4 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,2,4,0,0,0,2,3,0,0,3,4,0,1,1,1,2,2]
Phi over symmetry [-4,-2,1,1,2,2,0,3,4,2,3,2,2,1,2,0,1,1,1,1,0]
Phi of -K [-4,-2,1,1,2,2,0,3,4,2,3,2,2,1,2,0,1,1,1,1,0]
Phi of K* [-2,-2,-1,-1,2,4,0,1,1,1,2,1,1,2,3,0,2,3,2,4,0]
Phi of -K* [-4,-2,1,1,2,2,2,1,2,3,4,1,1,2,3,0,0,0,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+49t^4+9t^2
Outer characteristic polynomial t^7+79t^5+34t^3+3t
Flat arrow polynomial 4*K1**2*K2 - 2*K1*K3 - K2
2-strand cable arrow polynomial 1152*K1**4*K2 - 1984*K1**4 - 256*K1**3*K3 + 480*K1**2*K2**3 - 2688*K1**2*K2**2 - 640*K1**2*K2*K4 + 3944*K1**2*K2 - 128*K1**2*K3**2 - 192*K1**2*K3*K5 - 480*K1**2*K4**2 - 32*K1**2*K4*K6 - 64*K1**2*K5**2 - 2840*K1**2 + 96*K1*K2**2*K3*K4 - 288*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 2840*K1*K2*K3 - 288*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 1744*K1*K3*K4 + 1328*K1*K4*K5 + 240*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 544*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 128*K2**2*K3**2 - 336*K2**2*K4**2 + 1144*K2**2*K4 - 128*K2**2*K5**2 - 8*K2**2*K6**2 - 2216*K2**2 + 776*K2*K3*K5 + 384*K2*K4*K6 + 72*K2*K5*K7 - 1240*K3**2 - 1266*K4**2 - 688*K5**2 - 136*K6**2 + 2880
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice False
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