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Flat knot 6.1732

Min(phi) over symmetries of the knot is: [-2,0,1,1,1,1,1,1,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1732']
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^5+17t^3+24t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1732']
2-strand cable arrow polynomial of the knot is: -576*K1**6 + 1760*K1**4*K2 - 4032*K1**4 + 544*K1**3*K2*K3 - 1248*K1**3*K3 - 2512*K1**2*K2**2 - 960*K1**2*K2*K4 + 7624*K1**2*K2 - 960*K1**2*K3**2 - 448*K1**2*K4**2 - 5012*K1**2 - 384*K1*K2**2*K3 - 384*K1*K2*K3*K4 + 6056*K1*K2*K3 + 2640*K1*K3*K4 + 600*K1*K4*K5 - 56*K2**4 - 144*K2**2*K3**2 - 112*K2**2*K4**2 + 928*K2**2*K4 - 4332*K2**2 - 96*K2*K3**2*K4 + 200*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2536*K3**2 - 1230*K4**2 - 156*K5**2 - 28*K6**2 + 4692
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1732']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4146', 'vk6.4177', 'vk6.5384', 'vk6.5415', 'vk6.7514', 'vk6.7537', 'vk6.9015', 'vk6.9046', 'vk6.12418', 'vk6.12451', 'vk6.13350', 'vk6.13575', 'vk6.13606', 'vk6.14262', 'vk6.14711', 'vk6.14734', 'vk6.15192', 'vk6.15865', 'vk6.15890', 'vk6.30831', 'vk6.30864', 'vk6.32015', 'vk6.32048', 'vk6.33076', 'vk6.33107', 'vk6.33848', 'vk6.34310', 'vk6.48488', 'vk6.50273', 'vk6.53538', 'vk6.53927', 'vk6.54255']
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 O1O2O3U2U4O5O4U6U1O6U5U3
R3 orbit {'O1O2O3U2U4O5O4U6U1O6U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4O5U3U5O6O4U6U2
Gauss code of K* O1O2U1O3O4U2U5U4O5O6U3U6
Gauss code of -K* O1O2U3O4O3U5U2O5O6U1U6U4
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 2 1 0 -1],[ 1 0 0 2 1 -1 1],[ 1 0 0 1 1 0 1],[-2 -2 -1 0 -1 -1 -2],[-1 -1 -1 1 0 0 -2],[ 0 1 0 1 0 0 0],[ 1 -1 -1 2 2 0 0]]
Primitive based matrix [[ 0 2 0 -1 -1],[-2 0 -1 -2 -2],[ 0 1 0 1 0],[ 1 2 -1 0 1],[ 1 2 0 -1 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,0,1,1,1,2,2,-1,0,-1]
Phi over symmetry [-2,0,1,1,1,1,1,1,2,-1]
Phi of -K [-1,-1,0,2,-1,2,1,1,1,1]
Phi of K* [-2,0,1,1,1,1,1,1,2,-1]
Phi of -K* [-1,-1,0,2,-1,0,2,-1,2,1]
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^4+11t^2+9
Outer characteristic polynomial t^5+17t^3+24t
Flat arrow polynomial -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -576*K1**6 + 1760*K1**4*K2 - 4032*K1**4 + 544*K1**3*K2*K3 - 1248*K1**3*K3 - 2512*K1**2*K2**2 - 960*K1**2*K2*K4 + 7624*K1**2*K2 - 960*K1**2*K3**2 - 448*K1**2*K4**2 - 5012*K1**2 - 384*K1*K2**2*K3 - 384*K1*K2*K3*K4 + 6056*K1*K2*K3 + 2640*K1*K3*K4 + 600*K1*K4*K5 - 56*K2**4 - 144*K2**2*K3**2 - 112*K2**2*K4**2 + 928*K2**2*K4 - 4332*K2**2 - 96*K2*K3**2*K4 + 200*K2*K3*K5 + 136*K2*K4*K6 + 24*K3**2*K6 - 2536*K3**2 - 1230*K4**2 - 156*K5**2 - 28*K6**2 + 4692
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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