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Flat knot 6.412

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,0,1,3,1,1,1,1,1,0,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.412', '7.19582']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.406', '6.410', '6.412', '6.1151', '6.1175', '6.1176']
Outer characteristic polynomial of the knot is: t^7+38t^5+67t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.412', '7.19582']
2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 2464*K1**4*K2 - 4544*K1**4 - 384*K1**3*K2**2*K3 + 1376*K1**3*K2*K3 - 832*K1**3*K3 + 384*K1**2*K2**5 - 2752*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 5792*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 960*K1**2*K2**2*K4 - 13712*K1**2*K2**2 - 1632*K1**2*K2*K4 + 9696*K1**2*K2 - 384*K1**2*K3**2 - 32*K1**2*K3*K5 - 2148*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 3552*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 2144*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 960*K1*K2**2*K5 - 384*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7936*K1*K2*K3 - 32*K1*K2*K4*K5 + 560*K1*K3*K4 + 96*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1664*K2**4*K4 - 4352*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 976*K2**2*K3**2 - 472*K2**2*K4**2 + 2896*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 434*K2**2 + 480*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 - 972*K3**2 - 350*K4**2 - 64*K5**2 - 14*K6**2 + 2644
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.412']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.479', 'vk6.545', 'vk6.583', 'vk6.945', 'vk6.1043', 'vk6.1085', 'vk6.1632', 'vk6.1737', 'vk6.1813', 'vk6.2123', 'vk6.2226', 'vk6.2265', 'vk6.2551', 'vk6.2870', 'vk6.3043', 'vk6.3171', 'vk6.20416', 'vk6.20712', 'vk6.21779', 'vk6.22156', 'vk6.27767', 'vk6.28261', 'vk6.29290', 'vk6.29686', 'vk6.39193', 'vk6.39718', 'vk6.41966', 'vk6.46281', 'vk6.57284', 'vk6.57643', 'vk6.58533', 'vk6.61950']
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 O1O2O3O4O5U4U5O6U3U1U2U6
R3 orbit {'O1O2O3O4O5U4U5O6U3U1U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U4U5U3O6U1U2
Gauss code of K* O1O2O3O4U2U3U1U5U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U6U4U2U3
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 -1 -1 1 3],[ 2 0 1 0 -1 1 3],[ 0 -1 0 0 -1 1 2],[ 1 0 0 0 -1 1 1],[ 1 1 1 1 0 1 0],[-1 -1 -1 -1 -1 0 0],[-3 -3 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 0 -2 0 -1 -3],[-1 0 0 -1 -1 -1 -1],[ 0 2 1 0 -1 0 -1],[ 1 0 1 1 0 1 1],[ 1 1 1 0 -1 0 0],[ 2 3 1 1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,0,2,0,1,3,1,1,1,1,1,0,1,-1,-1,0]
Phi over symmetry [-3,-1,0,1,1,2,0,2,0,1,3,1,1,1,1,1,0,1,-1,-1,0]
Phi of -K [-2,-1,-1,0,1,3,1,2,1,2,2,1,1,1,3,0,1,4,0,1,2]
Phi of K* [-3,-1,0,1,1,2,2,1,3,4,2,0,1,1,2,1,0,1,-1,1,2]
Phi of -K* [-2,-1,-1,0,1,3,-1,0,1,1,3,1,1,1,0,0,1,1,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+22t^4+30t^2+1
Outer characteristic polynomial t^7+38t^5+67t^3+10t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -1152*K1**4*K2**2 + 2464*K1**4*K2 - 4544*K1**4 - 384*K1**3*K2**2*K3 + 1376*K1**3*K2*K3 - 832*K1**3*K3 + 384*K1**2*K2**5 - 2752*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 5792*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 960*K1**2*K2**2*K4 - 13712*K1**2*K2**2 - 1632*K1**2*K2*K4 + 9696*K1**2*K2 - 384*K1**2*K3**2 - 32*K1**2*K3*K5 - 2148*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 3552*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 2144*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 960*K1*K2**2*K5 - 384*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7936*K1*K2*K3 - 32*K1*K2*K4*K5 + 560*K1*K3*K4 + 96*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1664*K2**4*K4 - 4352*K2**4 + 224*K2**3*K3*K5 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 976*K2**2*K3**2 - 472*K2**2*K4**2 + 2896*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 434*K2**2 + 480*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 - 972*K3**2 - 350*K4**2 - 64*K5**2 - 14*K6**2 + 2644
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice False
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