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Flat knot 6.798

Min(phi) over symmetries of the knot is: [-3,0,1,2,1,1,3,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.798', '7.33847']
Arrow polynomial of the knot is: -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']
Outer characteristic polynomial of the knot is: t^5+27t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.798', '7.33847']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 256*K1**4*K2**2 + 608*K1**4*K2 - 1056*K1**4 + 288*K1**3*K2*K3 - 1200*K1**2*K2**2 + 1744*K1**2*K2 - 192*K1**2*K3**2 - 852*K1**2 + 1536*K1*K2*K3 + 248*K1*K3*K4 + 8*K1*K4*K5 - 152*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 464*K2**2*K4 - 1366*K2**2 + 264*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 724*K3**2 - 298*K4**2 - 120*K5**2 - 18*K6**2 + 1360
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.798']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11039', 'vk6.11117', 'vk6.11559', 'vk6.11898', 'vk6.12205', 'vk6.12312', 'vk6.13212', 'vk6.19233', 'vk6.19339', 'vk6.19528', 'vk6.19632', 'vk6.22391', 'vk6.22709', 'vk6.22810', 'vk6.26041', 'vk6.26103', 'vk6.26525', 'vk6.28435', 'vk6.30608', 'vk6.30703', 'vk6.31340', 'vk6.31346', 'vk6.31751', 'vk6.31916', 'vk6.32514', 'vk6.32913', 'vk6.34763', 'vk6.38108', 'vk6.40150', 'vk6.40161', 'vk6.42378', 'vk6.44628', 'vk6.44757', 'vk6.46667', 'vk6.52338', 'vk6.52606', 'vk6.52800', 'vk6.56645', 'vk6.64720', 'vk6.66286']
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 O1O2O3O4U5O6U3U6U2O5U1U4
R3 orbit {'O1O2O3O4U5O6U3U6U2O5U1U4', 'O1O2O3O4U5U2O6U3U6O5U1U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U1U4O5U3U6U2O6U5
Gauss code of K* O1O2O3U4O5O6U5U3U1U6O4U2
Gauss code of -K* O1O2O3U2O4U5U3U1U6O5O6U4
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 -1 0 -1 3 -2 1],[ 1 0 1 0 3 -1 1],[ 0 -1 0 -1 1 -1 1],[ 1 0 1 0 1 0 1],[-3 -3 -1 -1 0 -3 0],[ 2 1 1 0 3 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 3 0 -1 -2],[-3 0 -1 -1 -3],[ 0 1 0 -1 -1],[ 1 1 1 0 0],[ 2 3 1 0 0]]
If based matrix primitive False
Phi of primitive based matrix [-3,0,1,2,1,1,3,1,1,0]
Phi over symmetry [-3,0,1,2,1,1,3,1,1,0]
Phi of -K [-2,-1,0,3,1,1,2,0,3,2]
Phi of K* [-3,0,1,2,2,3,2,0,1,1]
Phi of -K* [-2,-1,0,3,0,1,3,1,1,1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial -2w^3z+13w^2z+23w
Inner characteristic polynomial t^4+13t^2+4
Outer characteristic polynomial t^5+27t^3+11t
Flat arrow polynomial -6*K1**2 - 2*K1*K2 + K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -256*K1**6 - 256*K1**4*K2**2 + 608*K1**4*K2 - 1056*K1**4 + 288*K1**3*K2*K3 - 1200*K1**2*K2**2 + 1744*K1**2*K2 - 192*K1**2*K3**2 - 852*K1**2 + 1536*K1*K2*K3 + 248*K1*K3*K4 + 8*K1*K4*K5 - 152*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 464*K2**2*K4 - 1366*K2**2 + 264*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 724*K3**2 - 298*K4**2 - 120*K5**2 - 18*K6**2 + 1360
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {2, 4}, {3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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