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Flat knot 6.1613

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,2,3,0,1,2,1,0,1,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1613']
Arrow polynomial of the knot is: -10*K1**2 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.947', '6.1027', '6.1399', '6.1430', '6.1433', '6.1442', '6.1465', '6.1469', '6.1476', '6.1505', '6.1529', '6.1606', '6.1612', '6.1613', '6.1616', '6.1649', '6.1694', '6.1736', '6.1768', '6.1771', '6.1774', '6.1884', '6.1886', '6.1887', '6.1889', '6.1960', '6.1962']
Outer characteristic polynomial of the knot is: t^7+32t^5+62t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1613']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 192*K1**4*K2**2 + 1536*K1**4*K2 - 4720*K1**4 + 256*K1**3*K2*K3 - 960*K1**3*K3 - 2752*K1**2*K2**2 - 96*K1**2*K2*K4 + 7600*K1**2*K2 - 176*K1**2*K3**2 - 2432*K1**2 + 2784*K1*K2*K3 + 160*K1*K3*K4 - 104*K2**4 + 72*K2**2*K4 - 2448*K2**2 - 624*K3**2 - 42*K4**2 + 2520
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1613']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4440', 'vk6.4535', 'vk6.5826', 'vk6.5953', 'vk6.7886', 'vk6.7998', 'vk6.9313', 'vk6.9432', 'vk6.13410', 'vk6.13507', 'vk6.13698', 'vk6.14080', 'vk6.15053', 'vk6.15173', 'vk6.17793', 'vk6.17824', 'vk6.18825', 'vk6.19432', 'vk6.19727', 'vk6.24340', 'vk6.25420', 'vk6.25451', 'vk6.26610', 'vk6.33256', 'vk6.33317', 'vk6.37544', 'vk6.44885', 'vk6.48643', 'vk6.50545', 'vk6.53648', 'vk6.55824', 'vk6.65488']
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 O1O2O3U2O4U5U6O5U3O6U1U4
R3 orbit {'O1O2O3U2O4U5U6O5U3O6U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5U1O6U5U6O4U2
Gauss code of K* O1O2U1O3U2O4O5U4U6U3O6U5
Gauss code of -K* O1O2U3O4U5O3O5U1O6U4U6U2
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 2 -1 0],[ 1 0 -1 2 2 0 1],[ 1 1 0 1 1 1 0],[-1 -2 -1 0 0 -1 0],[-2 -2 -1 0 0 -3 -1],[ 1 0 -1 1 3 0 0],[ 0 -1 0 0 1 0 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 -1 -2 -3],[-1 0 0 0 -1 -2 -1],[ 0 1 0 0 0 -1 0],[ 1 1 1 0 0 1 1],[ 1 2 2 1 -1 0 0],[ 1 3 1 0 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,0,1,1,2,3,0,1,2,1,0,1,0,-1,-1,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,2,3,0,1,2,1,0,1,0,-1,-1,0]
Phi of -K [-1,-1,-1,0,1,2,-1,-1,1,1,2,0,0,0,1,1,1,0,1,1,1]
Phi of K* [-2,-1,0,1,1,1,1,1,0,1,2,1,1,0,1,1,0,1,0,-1,-1]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,0,1,3,1,0,1,1,1,2,2,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 17z+35
Enhanced Jones-Krushkal polynomial 17w^2z+35w
Inner characteristic polynomial t^6+24t^4+45t^2+9
Outer characteristic polynomial t^7+32t^5+62t^3+13t
Flat arrow polynomial -10*K1**2 + 5*K2 + 6
2-strand cable arrow polynomial -384*K1**6 - 192*K1**4*K2**2 + 1536*K1**4*K2 - 4720*K1**4 + 256*K1**3*K2*K3 - 960*K1**3*K3 - 2752*K1**2*K2**2 - 96*K1**2*K2*K4 + 7600*K1**2*K2 - 176*K1**2*K3**2 - 2432*K1**2 + 2784*K1*K2*K3 + 160*K1*K3*K4 - 104*K2**4 + 72*K2**2*K4 - 2448*K2**2 - 624*K3**2 - 42*K4**2 + 2520
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {5}, {1, 4}, {2, 3}]]
If K is slice False
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