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Flat knot 6.1667

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,1,1,1,0,1,-1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1667']
Arrow polynomial of the knot is: -12*K1**2 + 6*K2 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.612', '6.1246', '6.1402', '6.1410', '6.1411', '6.1468', '6.1617', '6.1666', '6.1667', '6.1815', '6.1904', '6.1994', '6.1995', '6.1996', '6.2001', '6.2014', '6.2015', '6.2016', '6.2017', '6.2020', '6.2022']
Outer characteristic polynomial of the knot is: t^7+16t^5+17t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1667']
2-strand cable arrow polynomial of the knot is: -1152*K1**6 - 576*K1**4*K2**2 + 2656*K1**4*K2 - 8288*K1**4 + 736*K1**3*K2*K3 - 512*K1**3*K3 - 5952*K1**2*K2**2 - 64*K1**2*K2*K4 + 11560*K1**2*K2 - 288*K1**2*K3**2 - 1888*K1**2 - 128*K1*K2**2*K3 + 4792*K1*K2*K3 + 136*K1*K3*K4 - 304*K2**4 + 352*K2**2*K4 - 3600*K2**2 - 1032*K3**2 - 104*K4**2 + 3654
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1667']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4445', 'vk6.4542', 'vk6.5827', 'vk6.5956', 'vk6.7883', 'vk6.7997', 'vk6.9306', 'vk6.9427', 'vk6.13409', 'vk6.13504', 'vk6.13693', 'vk6.14063', 'vk6.15038', 'vk6.15160', 'vk6.17799', 'vk6.17830', 'vk6.18835', 'vk6.19437', 'vk6.19730', 'vk6.24342', 'vk6.25434', 'vk6.25465', 'vk6.26613', 'vk6.33259', 'vk6.33318', 'vk6.37562', 'vk6.44898', 'vk6.48650', 'vk6.50542', 'vk6.53643', 'vk6.55814', 'vk6.65486']
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 O1O2O3U2O4U3U5U1O5O6U4U6
R3 orbit {'O1O2O3U2O4U3U5U1O5O6U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4O6U3U6U1O5U2
Gauss code of K* O1O2O3U2U4O5O4U3U6U1O6U5
Gauss code of -K* O1O2O3U4O5U3U5U1O6O4U6U2
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 0 -1 0 1 -1 1],[ 0 0 -1 1 1 0 1],[ 1 1 0 1 1 1 0],[ 0 -1 -1 0 1 0 1],[-1 -1 -1 -1 0 -1 1],[ 1 0 -1 0 1 0 1],[-1 -1 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 -1 -1 -1 -1],[-1 -1 0 -1 -1 0 -1],[ 0 1 1 0 1 -1 0],[ 0 1 1 -1 0 -1 0],[ 1 1 0 1 1 0 1],[ 1 1 1 0 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,1,1,1,1,1,1,0,1,-1,1,0,1,0,-1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,1,1,1,0,1,-1,1,1,1,1,-1]
Phi of -K [-1,-1,0,0,1,1,-1,0,0,1,2,1,1,1,1,-1,0,0,0,0,-1]
Phi of K* [-1,-1,0,0,1,1,-1,0,0,1,2,0,0,1,1,-1,1,0,1,0,-1]
Phi of -K* [-1,-1,0,0,1,1,-1,0,0,1,1,1,1,0,1,-1,1,1,1,1,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+12t^4+5t^2
Outer characteristic polynomial t^7+16t^5+17t^3+2t
Flat arrow polynomial -12*K1**2 + 6*K2 + 7
2-strand cable arrow polynomial -1152*K1**6 - 576*K1**4*K2**2 + 2656*K1**4*K2 - 8288*K1**4 + 736*K1**3*K2*K3 - 512*K1**3*K3 - 5952*K1**2*K2**2 - 64*K1**2*K2*K4 + 11560*K1**2*K2 - 288*K1**2*K3**2 - 1888*K1**2 - 128*K1*K2**2*K3 + 4792*K1*K2*K3 + 136*K1*K3*K4 - 304*K2**4 + 352*K2**2*K4 - 3600*K2**2 - 1032*K3**2 - 104*K4**2 + 3654
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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