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Flat knot 6.379

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,0,1,1,3,3,0,0,1,1,1,2,3,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.379']
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+63t^5+37t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.379']
2-strand cable arrow polynomial of the knot is: -256*K1**6 + 1152*K1**4*K2 - 4256*K1**4 + 320*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1568*K1**3*K3 - 1984*K1**2*K2**2 - 640*K1**2*K2*K4 + 7024*K1**2*K2 - 1184*K1**2*K3**2 - 416*K1**2*K4**2 - 3464*K1**2 - 32*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 4944*K1*K2*K3 + 1824*K1*K3*K4 + 336*K1*K4*K5 - 64*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 448*K2**2*K4 - 3132*K2**2 + 128*K2*K3*K5 + 48*K2*K4*K6 - 1736*K3**2 - 664*K4**2 - 96*K5**2 - 12*K6**2 + 3422
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.379']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.18878', 'vk6.18886', 'vk6.18890', 'vk6.18903', 'vk6.18905', 'vk6.18906', 'vk6.18954', 'vk6.18962', 'vk6.18966', 'vk6.18981', 'vk6.18982', 'vk6.18983', 'vk6.25577', 'vk6.25585', 'vk6.25589', 'vk6.25605', 'vk6.25606', 'vk6.25608', 'vk6.37613', 'vk6.37617', 'vk6.37621', 'vk6.37630', 'vk6.37632', 'vk6.37633', 'vk6.56416', 'vk6.56419', 'vk6.56451', 'vk6.56455', 'vk6.56459', 'vk6.56470', 'vk6.56471', 'vk6.56472']
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 O1O2O3O4O5U4U1O6U3U6U5U2
R3 orbit {'O1O2O3O4O5U4U1O6U3U6U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U1U6U3O6U5U2
Gauss code of K* O1O2O3O4U5U4U1U6U3O6O5U2
Gauss code of -K* O1O2O3O4U3O5O6U2U6U4U1U5
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 -3 1 -1 -1 3 1],[ 3 0 3 1 0 3 1],[-1 -3 0 -2 -1 2 1],[ 1 -1 2 0 0 3 1],[ 1 0 1 0 0 1 0],[-3 -3 -2 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 0 -2 -1 -3 -3],[-1 0 0 -1 0 -1 -1],[-1 2 1 0 -1 -2 -3],[ 1 1 0 1 0 0 0],[ 1 3 1 2 0 0 -1],[ 3 3 1 3 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,0,2,1,3,3,1,0,1,1,1,2,3,0,0,1]
Phi over symmetry [-3,-1,-1,1,1,3,0,1,1,3,3,0,0,1,1,1,2,3,-1,0,2]
Phi of -K [-3,-1,-1,1,1,3,1,2,1,3,3,0,0,1,1,1,2,3,-1,0,2]
Phi of K* [-3,-1,-1,1,1,3,0,2,1,3,3,1,0,1,1,1,2,3,0,1,2]
Phi of -K* [-3,-1,-1,1,1,3,0,1,1,3,3,0,0,1,1,1,2,3,-1,0,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 17z+35
Enhanced Jones-Krushkal polynomial 17w^2z+35w
Inner characteristic polynomial t^6+41t^4+11t^2
Outer characteristic polynomial t^7+63t^5+37t^3+4t
Flat arrow polynomial -8*K1**2 - 4*K1*K2 + 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -256*K1**6 + 1152*K1**4*K2 - 4256*K1**4 + 320*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1568*K1**3*K3 - 1984*K1**2*K2**2 - 640*K1**2*K2*K4 + 7024*K1**2*K2 - 1184*K1**2*K3**2 - 416*K1**2*K4**2 - 3464*K1**2 - 32*K1*K2**2*K3 - 32*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 4944*K1*K2*K3 + 1824*K1*K3*K4 + 336*K1*K4*K5 - 64*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 448*K2**2*K4 - 3132*K2**2 + 128*K2*K3*K5 + 48*K2*K4*K6 - 1736*K3**2 - 664*K4**2 - 96*K5**2 - 12*K6**2 + 3422
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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