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Flat knot 6.1222

Min(phi) over symmetries of the knot is: [-2,-1,1,2,0,0,2,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1222', '7.24324', '7.24432']
Arrow polynomial of the knot is: -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.235', '6.379', '6.411', '6.547', '6.811', '6.818', '6.823', '6.897', '6.898', '6.1008', '6.1053', '6.1109', '6.1110', '6.1130', '6.1222', '6.1239', '6.1303', '6.1307', '6.1342', '6.1491', '6.1495', '6.1496', '6.1519', '6.1592', '6.1593', '6.1642', '6.1652', '6.1653', '6.1671', '6.1673', '6.1717']
Outer characteristic polynomial of the knot is: t^5+15t^3+20t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1222', '7.24324']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 192*K1**4*K2**2 + 384*K1**4*K2 - 1328*K1**4 + 128*K1**3*K2*K3 - 1760*K1**2*K2**2 + 2000*K1**2*K2 - 656*K1**2*K3**2 - 96*K1**2*K4**2 - 112*K1**2 + 1824*K1*K2*K3 + 512*K1*K3*K4 + 64*K1*K4*K5 - 560*K2**4 - 384*K2**2*K3**2 - 48*K2**2*K4**2 + 400*K2**2*K4 - 348*K2**2 + 240*K2*K3*K5 + 32*K2*K4*K6 - 336*K3**2 - 132*K4**2 - 32*K5**2 - 4*K6**2 + 658
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1222']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.64', 'vk6.121', 'vk6.218', 'vk6.267', 'vk6.288', 'vk6.674', 'vk6.703', 'vk6.748', 'vk6.1224', 'vk6.1273', 'vk6.1364', 'vk6.1413', 'vk6.1497', 'vk6.1566', 'vk6.1926', 'vk6.2043', 'vk6.2438', 'vk6.2485', 'vk6.2651', 'vk6.2988', 'vk6.5738', 'vk6.5769', 'vk6.7807', 'vk6.7838', 'vk6.10266', 'vk6.10409', 'vk6.13300', 'vk6.13331', 'vk6.14781', 'vk6.14817', 'vk6.15939', 'vk6.15975', 'vk6.18044', 'vk6.24488', 'vk6.25848', 'vk6.33049', 'vk6.37392', 'vk6.37955', 'vk6.38018', 'vk6.44853']
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 O1O2O3O4U3U1U4O5O6U5U6U2
R3 orbit {'O1O2O3O4U3U1U4O5O6U5U6U2', 'O1O2O3U4U1U3O5O6U5U6O4U2'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U3U5U6O5O6U1U4U2
Gauss code of K* O1O2O3U4U5O4O5O6U2U6U1U3
Gauss code of -K* O1O2O3U2U3O4O5O6U4U6U1U5
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 1 -1 2 -1 1],[ 2 0 2 0 2 0 0],[-1 -2 0 -1 1 -1 1],[ 1 0 1 0 1 0 0],[-2 -2 -1 -1 0 0 0],[ 1 0 1 0 0 0 1],[-1 0 -1 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 -1 -2],[-2 0 0 0 -2],[-1 0 0 -1 0],[ 1 0 1 0 0],[ 2 2 0 0 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,1,2,0,0,2,1,0,0]
Phi over symmetry [-2,-1,1,2,0,0,2,1,0,0]
Phi of -K [-2,-1,1,2,1,3,2,1,3,1]
Phi of K* [-2,-1,1,2,1,3,2,1,3,1]
Phi of -K* [-2,-1,1,2,0,0,2,1,0,0]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 10z+21
Enhanced Jones-Krushkal polynomial 10w^2z+21w
Inner characteristic polynomial t^4+5t^2+4
Outer characteristic polynomial t^5+15t^3+20t
Flat arrow polynomial -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -256*K1**6 - 192*K1**4*K2**2 + 384*K1**4*K2 - 1328*K1**4 + 128*K1**3*K2*K3 - 1760*K1**2*K2**2 + 2000*K1**2*K2 - 656*K1**2*K3**2 - 96*K1**2*K4**2 - 112*K1**2 + 1824*K1*K2*K3 + 512*K1*K3*K4 + 64*K1*K4*K5 - 560*K2**4 - 384*K2**2*K3**2 - 48*K2**2*K4**2 + 400*K2**2*K4 - 348*K2**2 + 240*K2*K3*K5 + 32*K2*K4*K6 - 336*K3**2 - 132*K4**2 - 32*K5**2 - 4*K6**2 + 658
Genus of based matrix 0
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}]]
If K is slice True
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