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Flat knot 6.1748

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,2,2,2,0,1,1,2,0,1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1748']
Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866']
Outer characteristic polynomial of the knot is: t^7+28t^5+40t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1748']
2-strand cable arrow polynomial of the knot is: -512*K1**6 + 1056*K1**4*K2 - 4160*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 992*K1**3*K3 - 1600*K1**2*K2**2 - 544*K1**2*K2*K4 + 6720*K1**2*K2 - 1504*K1**2*K3**2 - 912*K1**2*K4**2 - 4368*K1**2 - 256*K1*K2**2*K3 - 224*K1*K2*K3*K4 + 5408*K1*K2*K3 + 3480*K1*K3*K4 + 976*K1*K4*K5 + 24*K1*K5*K6 - 24*K2**4 - 96*K2**2*K3**2 - 112*K2**2*K4**2 + 704*K2**2*K4 - 3780*K2**2 + 216*K2*K3*K5 + 136*K2*K4*K6 - 2692*K3**2 - 1550*K4**2 - 308*K5**2 - 52*K6**2 + 4676
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1748']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4457', 'vk6.4554', 'vk6.5839', 'vk6.5968', 'vk6.7899', 'vk6.8017', 'vk6.9326', 'vk6.9447', 'vk6.13419', 'vk6.13514', 'vk6.13707', 'vk6.14057', 'vk6.15028', 'vk6.15150', 'vk6.17784', 'vk6.17817', 'vk6.18838', 'vk6.19420', 'vk6.19715', 'vk6.24331', 'vk6.25431', 'vk6.25464', 'vk6.26598', 'vk6.33265', 'vk6.33324', 'vk6.37565', 'vk6.44881', 'vk6.48662', 'vk6.50558', 'vk6.53665', 'vk6.55806', 'vk6.65472']
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 O1O2O3U4U2O4O5U6U3O6U1U5
R3 orbit {'O1O2O3U4U2O4O5U6U3O6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5U1U5O4O6U2U6
Gauss code of K* O1O2U1O3O4U3U5U2O6O5U6U4
Gauss code of -K* O1O2U3O4O3U1U5O6O5U4U6U2
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 0 1 -1 2 -1],[ 1 0 0 2 0 2 0],[ 0 0 0 0 0 1 -1],[-1 -2 0 0 -1 0 -1],[ 1 0 0 1 0 2 0],[-2 -2 -1 0 -2 0 -2],[ 1 0 1 1 0 2 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 -2 -2 -2],[-1 0 0 0 -1 -1 -2],[ 0 1 0 0 0 -1 0],[ 1 2 1 0 0 0 0],[ 1 2 1 1 0 0 0],[ 1 2 2 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,0,1,2,2,2,0,1,1,2,0,1,0,0,0,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,2,2,2,0,1,1,2,0,1,0,0,0,0]
Phi of -K [-1,-1,-1,0,1,2,0,0,0,1,1,0,1,0,1,1,1,1,1,1,1]
Phi of K* [-2,-1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,0,1,0,0,0]
Phi of -K* [-1,-1,-1,0,1,2,0,0,0,1,2,0,0,2,2,1,1,2,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+20t^4+27t^2+4
Outer characteristic polynomial t^7+28t^5+40t^3+7t
Flat arrow polynomial -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -512*K1**6 + 1056*K1**4*K2 - 4160*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 992*K1**3*K3 - 1600*K1**2*K2**2 - 544*K1**2*K2*K4 + 6720*K1**2*K2 - 1504*K1**2*K3**2 - 912*K1**2*K4**2 - 4368*K1**2 - 256*K1*K2**2*K3 - 224*K1*K2*K3*K4 + 5408*K1*K2*K3 + 3480*K1*K3*K4 + 976*K1*K4*K5 + 24*K1*K5*K6 - 24*K2**4 - 96*K2**2*K3**2 - 112*K2**2*K4**2 + 704*K2**2*K4 - 3780*K2**2 + 216*K2*K3*K5 + 136*K2*K4*K6 - 2692*K3**2 - 1550*K4**2 - 308*K5**2 - 52*K6**2 + 4676
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {4}, {1, 3}, {2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice False
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