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Flat knot 6.1592

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,1,2,1,0,1,1,-1,2,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1592']
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^7+37t^5+185t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1592']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 512*K1**4*K2 - 2016*K1**4 + 64*K1**3*K2*K3 - 1328*K1**2*K2**2 + 3832*K1**2*K2 - 480*K1**2*K3**2 - 160*K1**2*K4**2 - 2796*K1**2 + 2664*K1*K2*K3 + 872*K1*K3*K4 + 272*K1*K4*K5 - 80*K2**4 - 144*K2**2*K3**2 - 16*K2**2*K4**2 + 568*K2**2*K4 - 2796*K2**2 + 304*K2*K3*K5 + 32*K2*K4*K6 - 1320*K3**2 - 652*K4**2 - 212*K5**2 - 12*K6**2 + 2954
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1592']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16946', 'vk6.17189', 'vk6.20539', 'vk6.21939', 'vk6.23346', 'vk6.23641', 'vk6.27996', 'vk6.29462', 'vk6.35386', 'vk6.35807', 'vk6.39403', 'vk6.41596', 'vk6.42863', 'vk6.43142', 'vk6.45983', 'vk6.47659', 'vk6.55109', 'vk6.55370', 'vk6.57401', 'vk6.58577', 'vk6.59511', 'vk6.59811', 'vk6.62072', 'vk6.63054', 'vk6.64954', 'vk6.65162', 'vk6.66949', 'vk6.67809', 'vk6.68247', 'vk6.68390', 'vk6.69563', 'vk6.70259']
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 O1O2O3U4O5U6U2O6O4U1U3U5
R3 orbit {'O1O2O3U4O5U6U2O6O4U1U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1U3O5O6U2U6O4U5
Gauss code of K* O1O2O3U1U4U2O5U3O6O4U6U5
Gauss code of -K* O1O2O3U4U5O6O5U1O4U2U6U3
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 0 1 0 2 -1],[ 2 0 2 2 1 2 1],[ 0 -2 0 -1 1 0 0],[-1 -2 1 0 -1 0 -1],[ 0 -1 -1 1 0 2 -1],[-2 -2 0 0 -2 0 -2],[ 1 -1 0 1 1 2 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 0 -2 -2 -2],[-1 0 0 1 -1 -1 -2],[ 0 0 -1 0 1 0 -2],[ 0 2 1 -1 0 -1 -1],[ 1 2 1 0 1 0 -1],[ 2 2 2 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,0,2,2,2,-1,1,1,2,-1,0,2,1,1,1]
Phi over symmetry [-2,-1,0,0,1,2,0,0,1,1,2,1,0,1,1,-1,2,2,0,0,1]
Phi of -K [-2,-1,0,0,1,2,0,0,1,1,2,1,0,1,1,-1,2,2,0,0,1]
Phi of K* [-2,-1,0,0,1,2,1,0,2,1,2,0,2,1,1,-1,0,1,1,0,0]
Phi of -K* [-2,-1,0,0,1,2,1,1,2,2,2,1,0,1,2,-1,1,2,-1,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 14z+29
Enhanced Jones-Krushkal polynomial -2w^3z+16w^2z+29w
Inner characteristic polynomial t^6+27t^4+123t^2
Outer characteristic polynomial t^7+37t^5+185t^3
Flat arrow polynomial -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -64*K1**6 + 512*K1**4*K2 - 2016*K1**4 + 64*K1**3*K2*K3 - 1328*K1**2*K2**2 + 3832*K1**2*K2 - 480*K1**2*K3**2 - 160*K1**2*K4**2 - 2796*K1**2 + 2664*K1*K2*K3 + 872*K1*K3*K4 + 272*K1*K4*K5 - 80*K2**4 - 144*K2**2*K3**2 - 16*K2**2*K4**2 + 568*K2**2*K4 - 2796*K2**2 + 304*K2*K3*K5 + 32*K2*K4*K6 - 1320*K3**2 - 652*K4**2 - 212*K5**2 - 12*K6**2 + 2954
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]]
If K is slice False
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