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Flat knot 6.1779

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,0,1,0,1,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1763', '6.1779', '7.45946', '7.45965']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.927', '6.1364', '6.1367', '6.1540', '6.1675', '6.1779', '6.1811', '6.1876', '6.2075']
Outer characteristic polynomial of the knot is: t^7+16t^5+23t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1763', '6.1779']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 1152*K1**4*K2**2 + 4224*K1**4*K2 - 9120*K1**4 + 1536*K1**3*K2*K3 - 1184*K1**3*K3 - 896*K1**2*K2**4 + 3776*K1**2*K2**3 + 640*K1**2*K2**2*K4 - 13632*K1**2*K2**2 - 1312*K1**2*K2*K4 + 13232*K1**2*K2 - 640*K1**2*K3**2 - 96*K1**2*K4**2 - 1008*K1**2 + 1152*K1*K2**3*K3 - 2112*K1*K2**2*K3 - 416*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 8128*K1*K2*K3 + 816*K1*K3*K4 + 160*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2032*K2**4 - 64*K2**3*K6 - 288*K2**2*K3**2 - 128*K2**2*K4**2 + 1616*K2**2*K4 - 2356*K2**2 + 224*K2*K3*K5 + 48*K2*K4*K6 - 1000*K3**2 - 284*K4**2 - 40*K5**2 - 4*K6**2 + 3074
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1779']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.18', 'vk6.28', 'vk6.35', 'vk6.151', 'vk6.152', 'vk6.166', 'vk6.169', 'vk6.1196', 'vk6.1211', 'vk6.1295', 'vk6.1304', 'vk6.1315', 'vk6.2356', 'vk6.2388', 'vk6.2401', 'vk6.2959', 'vk6.3544', 'vk6.3545', 'vk6.6920', 'vk6.6921', 'vk6.6952', 'vk6.6953', 'vk6.15383', 'vk6.15384', 'vk6.15502', 'vk6.33432', 'vk6.33436', 'vk6.33489', 'vk6.33493', 'vk6.33598', 'vk6.49943', 'vk6.53743']
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 O1O2O3U4U3O5O4U6U5O6U1U2
R3 orbit {'O1O2O3U4U3O5O4U6U5O6U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U3O4U5U4O6O5U1U6
Gauss code of K* O1O2U1O3O4U3U4U5O6O5U2U6
Gauss code of -K* O1O2U3O4O3U5U4O6O5U6U1U2
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 1 1 0 0 -1],[ 1 0 1 1 1 0 0],[-1 -1 0 1 -1 0 -2],[-1 -1 -1 0 -1 -1 -1],[ 0 -1 1 1 0 0 0],[ 0 0 0 1 0 0 0],[ 1 0 2 1 0 0 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 0 -1 -1 -2],[-1 -1 0 -1 -1 -1 -1],[ 0 0 1 0 0 0 0],[ 0 1 1 0 0 -1 0],[ 1 1 1 0 1 0 0],[ 1 2 1 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,0,1,1,2,1,1,1,1,0,0,0,1,0,0]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,0,1,0,1,0,1,1,0]
Phi of -K [-1,-1,0,0,1,1,0,0,1,1,1,1,1,0,1,0,0,0,1,0,-1]
Phi of K* [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,0,1,0,1,0,1,1,0]
Phi of -K* [-1,-1,0,0,1,1,0,0,0,1,2,0,1,1,1,0,1,0,1,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+12t^4+15t^2+4
Outer characteristic polynomial t^7+16t^5+23t^3+8t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -512*K1**6 - 1152*K1**4*K2**2 + 4224*K1**4*K2 - 9120*K1**4 + 1536*K1**3*K2*K3 - 1184*K1**3*K3 - 896*K1**2*K2**4 + 3776*K1**2*K2**3 + 640*K1**2*K2**2*K4 - 13632*K1**2*K2**2 - 1312*K1**2*K2*K4 + 13232*K1**2*K2 - 640*K1**2*K3**2 - 96*K1**2*K4**2 - 1008*K1**2 + 1152*K1*K2**3*K3 - 2112*K1*K2**2*K3 - 416*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 8128*K1*K2*K3 + 816*K1*K3*K4 + 160*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2032*K2**4 - 64*K2**3*K6 - 288*K2**2*K3**2 - 128*K2**2*K4**2 + 1616*K2**2*K4 - 2356*K2**2 + 224*K2*K3*K5 + 48*K2*K4*K6 - 1000*K3**2 - 284*K4**2 - 40*K5**2 - 4*K6**2 + 3074
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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