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Flat knot 6.1234

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,1,1,2,1,0,1,0,0,1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1234', '7.41328']
Arrow polynomial of the knot is: 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.140', '6.569', '6.943', '6.970', '6.1234', '6.1298', '6.1311', '6.1326', '6.1500', '6.1506', '6.1708', '6.1712', '6.1720', '6.1859']
Outer characteristic polynomial of the knot is: t^7+18t^5+36t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1234', '6.1645', '7.41328']
2-strand cable arrow polynomial of the knot is: 1536*K1**4*K2 - 2528*K1**4 + 512*K1**3*K2*K3 - 704*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3120*K1**2*K2**2 - 512*K1**2*K2*K4 + 2808*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K4**2 - 88*K1**2 + 224*K1*K2**3*K3 - 352*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2208*K1*K2*K3 + 352*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 488*K2**4 - 112*K2**2*K3**2 - 48*K2**2*K4**2 + 448*K2**2*K4 - 542*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 340*K3**2 - 118*K4**2 - 20*K5**2 - 2*K6**2 + 700
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1234']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.516', 'vk6.610', 'vk6.638', 'vk6.1020', 'vk6.1113', 'vk6.1153', 'vk6.1873', 'vk6.2293', 'vk6.2517', 'vk6.2565', 'vk6.2593', 'vk6.2803', 'vk6.2898', 'vk6.2918', 'vk6.3081', 'vk6.3202', 'vk6.4624', 'vk6.5913', 'vk6.6034', 'vk6.6553', 'vk6.8079', 'vk6.9392', 'vk6.17848', 'vk6.17865', 'vk6.19063', 'vk6.19865', 'vk6.22552', 'vk6.24369', 'vk6.25675', 'vk6.26308', 'vk6.26753', 'vk6.28575', 'vk6.29808', 'vk6.39903', 'vk6.43790', 'vk6.45053', 'vk6.46848', 'vk6.48010', 'vk6.48085', 'vk6.50668']
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 O1O2O3O4U3U4U2O5O6U1U6U5
R3 orbit {'O1O2O3O4U3U4U2O5O6U1U6U5', 'O1O2O3U2O4U3U4O5O6U1U6U5'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U5U6U4O6O5U3U1U2
Gauss code of K* O1O2O3U4U5O6O5O4U6U3U1U2
Gauss code of -K* O1O2O3U2U1O4O5O6U5U6U4U3
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 1],[ 2 0 0 -1 1 2 1],[ 0 0 0 -1 1 0 0],[ 1 1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[-1 0 0 0 -1 -1 -1],[ 0 0 0 1 0 -1 0],[ 1 0 0 1 1 0 1],[ 2 1 2 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,0,0,0,0,1,0,0,0,2,1,1,1,1,0,-1]
Phi over symmetry [-2,-1,0,1,1,1,-1,0,1,1,2,1,0,1,0,0,1,0,0,0,0]
Phi of -K [-2,-1,0,1,1,1,2,2,1,2,2,0,2,1,2,1,0,1,0,0,0]
Phi of K* [-1,-1,-1,0,1,2,0,0,0,1,2,0,1,2,1,1,2,2,0,2,2]
Phi of -K* [-2,-1,0,1,1,1,-1,0,1,1,2,1,0,1,0,0,1,0,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+19z+15
Enhanced Jones-Krushkal polynomial 6w^3z^2+19w^2z+15w
Inner characteristic polynomial t^6+10t^4+15t^2
Outer characteristic polynomial t^7+18t^5+36t^3+3t
Flat arrow polynomial 4*K1**3 + 2*K1**2 - 4*K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial 1536*K1**4*K2 - 2528*K1**4 + 512*K1**3*K2*K3 - 704*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3120*K1**2*K2**2 - 512*K1**2*K2*K4 + 2808*K1**2*K2 - 352*K1**2*K3**2 - 32*K1**2*K4**2 - 88*K1**2 + 224*K1*K2**3*K3 - 352*K1*K2**2*K3 - 64*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 2208*K1*K2*K3 + 352*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 488*K2**4 - 112*K2**2*K3**2 - 48*K2**2*K4**2 + 448*K2**2*K4 - 542*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 340*K3**2 - 118*K4**2 - 20*K5**2 - 2*K6**2 + 700
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {5}, {3, 4}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{6}, {1, 5}, {3, 4}, {2}]]
If K is slice False
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