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Flat knot 6.1140

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,3,1,2,0,0,1,0,2,1,1,2,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1140']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+46t^5+130t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1140']
2-strand cable arrow polynomial of the knot is: -176*K1**4 + 480*K1**3*K2*K3 - 352*K1**3*K3 + 64*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 - 2560*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 352*K1**2*K2*K4 + 3776*K1**2*K2 - 432*K1**2*K3**2 - 3324*K1**2 + 1024*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4552*K1*K2*K3 + 768*K1*K3*K4 + 64*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1120*K2**4 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1328*K2**2*K3**2 - 296*K2**2*K4**2 + 1512*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 2394*K2**2 + 760*K2*K3*K5 + 192*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1500*K3**2 - 528*K4**2 - 180*K5**2 - 78*K6**2 - 4*K7**2 - 2*K8**2 + 2680
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1140']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4187', 'vk6.4268', 'vk6.5433', 'vk6.5551', 'vk6.7548', 'vk6.7631', 'vk6.9056', 'vk6.9137', 'vk6.18244', 'vk6.18579', 'vk6.24720', 'vk6.25133', 'vk6.36846', 'vk6.37309', 'vk6.44079', 'vk6.44418', 'vk6.48499', 'vk6.48580', 'vk6.49183', 'vk6.49295', 'vk6.50284', 'vk6.50356', 'vk6.51053', 'vk6.51086', 'vk6.56047', 'vk6.56321', 'vk6.60604', 'vk6.60947', 'vk6.65717', 'vk6.66011', 'vk6.68758', 'vk6.68966']
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 O1O2O3O4U1U2U4O5O6U5U3U6
R3 orbit {'O1O2O3O4U1U2U4O5O6U5U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2U6O5O6U1U3U4
Gauss code of K* O1O2O3U4U5O4O6O5U1U2U6U3
Gauss code of -K* O1O2O3U1U3O4O5O6U4U2U5U6
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 1 2 -1 2],[ 3 0 1 3 2 0 1],[ 1 -1 0 2 1 0 1],[-1 -3 -2 0 0 0 2],[-2 -2 -1 0 0 0 0],[ 1 0 0 0 0 0 1],[-2 -1 -1 -2 0 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 0 0 -1 -2],[-2 0 0 -2 -1 -1 -1],[-1 0 2 0 0 -2 -3],[ 1 0 1 0 0 0 0],[ 1 1 1 2 0 0 -1],[ 3 2 1 3 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,0,0,1,2,2,1,1,1,0,2,3,0,0,1]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,3,1,2,0,0,1,0,2,1,1,2,0,0]
Phi of -K [-3,-1,-1,1,2,2,1,2,1,3,4,0,0,2,2,2,3,2,1,-1,0]
Phi of K* [-2,-2,-1,1,1,3,0,-1,2,2,4,1,2,3,3,0,2,1,0,1,2]
Phi of -K* [-3,-1,-1,1,2,2,0,1,3,1,2,0,0,1,0,2,1,1,2,0,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+26t^4+42t^2+1
Outer characteristic polynomial t^7+46t^5+130t^3+5t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -176*K1**4 + 480*K1**3*K2*K3 - 352*K1**3*K3 + 64*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 - 2560*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 352*K1**2*K2*K4 + 3776*K1**2*K2 - 432*K1**2*K3**2 - 3324*K1**2 + 1024*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4552*K1*K2*K3 + 768*K1*K3*K4 + 64*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1120*K2**4 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1328*K2**2*K3**2 - 296*K2**2*K4**2 + 1512*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 2394*K2**2 + 760*K2*K3*K5 + 192*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 1500*K3**2 - 528*K4**2 - 180*K5**2 - 78*K6**2 - 4*K7**2 - 2*K8**2 + 2680
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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