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Flat knot 6.1378

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,1,1,1,2,1,1,1,2,0,0,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1378']
Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.313', '6.623', '6.1031', '6.1201', '6.1327', '6.1378', '6.1640', '6.1697', '6.1797', '6.1833']
Outer characteristic polynomial of the knot is: t^7+45t^5+27t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1378', '7.21951']
2-strand cable arrow polynomial of the knot is: -1536*K1**4*K2**2 + 3008*K1**4*K2 - 3648*K1**4 + 640*K1**3*K2*K3 - 224*K1**3*K3 - 448*K1**2*K2**4 + 2528*K1**2*K2**3 - 9440*K1**2*K2**2 - 448*K1**2*K2*K4 + 7128*K1**2*K2 - 1796*K1**2 + 768*K1*K2**3*K3 - 640*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4920*K1*K2*K3 + 8*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 1496*K2**4 - 272*K2**2*K3**2 - 48*K2**2*K4**2 + 712*K2**2*K4 - 1248*K2**2 + 16*K2*K3*K5 - 532*K3**2 - 62*K4**2 + 2076
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1378']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16909', 'vk6.17151', 'vk6.20519', 'vk6.21905', 'vk6.23301', 'vk6.23600', 'vk6.27962', 'vk6.29437', 'vk6.35315', 'vk6.35751', 'vk6.39368', 'vk6.41548', 'vk6.42820', 'vk6.43102', 'vk6.45935', 'vk6.47620', 'vk6.55056', 'vk6.55301', 'vk6.57388', 'vk6.58556', 'vk6.59452', 'vk6.59741', 'vk6.62045', 'vk6.63039', 'vk6.64897', 'vk6.65110', 'vk6.66933', 'vk6.67786', 'vk6.68206', 'vk6.68350', 'vk6.69538', 'vk6.70241']
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 O1O2O3U4O5O6U2U3U6O4U1U5
R3 orbit {'O1O2O3U4O5O6U2U3U6O4U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5U6U1U2O6O4U5
Gauss code of K* O1O2O3U4O5O6U5U1U2O4U6U3
Gauss code of -K* O1O2O3U1U4O5U2U3U6O4O6U5
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 -1 -2 0 -1 2 2],[ 1 0 -1 1 -1 2 2],[ 2 1 0 1 0 2 2],[ 0 -1 -1 0 -1 1 1],[ 1 1 0 1 0 2 2],[-2 -2 -2 -1 -2 0 0],[-2 -2 -2 -1 -2 0 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 0 -1 -2 -2 -2],[-2 0 0 -1 -2 -2 -2],[ 0 1 1 0 -1 -1 -1],[ 1 2 2 1 0 1 0],[ 1 2 2 1 -1 0 -1],[ 2 2 2 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,0,1,2,2,2,1,2,2,2,1,1,1,-1,0,1]
Phi over symmetry [-2,-2,0,1,1,2,0,1,1,1,2,1,1,1,2,0,0,1,-1,0,1]
Phi of -K [-2,-1,-1,0,2,2,0,1,1,2,2,1,0,1,1,0,1,1,1,1,0]
Phi of K* [-2,-2,0,1,1,2,0,1,1,1,2,1,1,1,2,0,0,1,-1,0,1]
Phi of -K* [-2,-1,-1,0,2,2,0,1,1,2,2,1,1,2,2,1,2,2,1,1,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+31t^4+4t^2
Outer characteristic polynomial t^7+45t^5+27t^3+2t
Flat arrow polynomial 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2
2-strand cable arrow polynomial -1536*K1**4*K2**2 + 3008*K1**4*K2 - 3648*K1**4 + 640*K1**3*K2*K3 - 224*K1**3*K3 - 448*K1**2*K2**4 + 2528*K1**2*K2**3 - 9440*K1**2*K2**2 - 448*K1**2*K2*K4 + 7128*K1**2*K2 - 1796*K1**2 + 768*K1*K2**3*K3 - 640*K1*K2**2*K3 - 32*K1*K2**2*K5 + 4920*K1*K2*K3 + 8*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 1496*K2**4 - 272*K2**2*K3**2 - 48*K2**2*K4**2 + 712*K2**2*K4 - 1248*K2**2 + 16*K2*K3*K5 - 532*K3**2 - 62*K4**2 + 2076
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}]]
If K is slice False
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