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Flat knot 6.1338

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,0,1,0,0,2,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1338']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.906', '6.1223', '6.1338', '6.1351', '6.1571', '6.1670', '6.1718', '6.1743', '6.1765', '6.1793', '6.1852', '6.2070']
Outer characteristic polynomial of the knot is: t^7+50t^5+57t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1338']
2-strand cable arrow polynomial of the knot is: -64*K1**6 - 384*K1**4*K2**2 + 2688*K1**4*K2 - 7008*K1**4 + 1024*K1**3*K2*K3 - 1760*K1**3*K3 - 256*K1**2*K2**4 + 1664*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9600*K1**2*K2**2 - 1312*K1**2*K2*K4 + 14192*K1**2*K2 - 1024*K1**2*K3**2 - 96*K1**2*K3*K5 - 224*K1**2*K4**2 - 6236*K1**2 + 640*K1*K2**3*K3 - 1632*K1*K2**2*K3 - 320*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 10056*K1*K2*K3 + 2096*K1*K3*K4 + 504*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1344*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 1968*K2**2*K4 - 5562*K2**2 + 408*K2*K3*K5 + 104*K2*K4*K6 - 2644*K3**2 - 1028*K4**2 - 200*K5**2 - 22*K6**2 + 5978
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1338']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4258', 'vk6.4262', 'vk6.4339', 'vk6.4342', 'vk6.5535', 'vk6.5539', 'vk6.5656', 'vk6.5660', 'vk6.7726', 'vk6.7729', 'vk6.9124', 'vk6.9127', 'vk6.9204', 'vk6.9208', 'vk6.19822', 'vk6.19830', 'vk6.26259', 'vk6.26265', 'vk6.26702', 'vk6.26708', 'vk6.38215', 'vk6.38223', 'vk6.44988', 'vk6.44996', 'vk6.48568', 'vk6.48572', 'vk6.49277', 'vk6.49281', 'vk6.50415', 'vk6.50418', 'vk6.66361', 'vk6.66367']
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 O1O2O3U1O4O5U3U6U5O6U4U2
R3 orbit {'O1O2O3U1O4O5U3U6U5O6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4O5U6U5U1O6O4U3
Gauss code of K* O1O2O3U2O4O5U6U5U1O6U4U3
Gauss code of -K* O1O2O3U1U4O5U3U6U5O6O4U2
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 -2 1 -1 1 2 -1],[ 2 0 2 1 1 1 2],[-1 -2 0 -2 1 2 -2],[ 1 -1 2 0 2 1 0],[-1 -1 -1 -2 0 1 -2],[-2 -1 -2 -1 -1 0 -2],[ 1 -2 2 0 2 2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -2 -1 -2 -1],[-1 1 0 -1 -2 -2 -1],[-1 2 1 0 -2 -2 -2],[ 1 1 2 2 0 0 -1],[ 1 2 2 2 0 0 -2],[ 2 1 1 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,1,2,1,2,1,1,2,2,1,2,2,2,0,1,2]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,0,1,0,0,2,-1,-1,0]
Phi of -K [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,0,1,0,0,2,-1,-1,0]
Phi of K* [-2,-1,-1,1,1,2,-1,0,1,2,3,1,0,0,1,0,0,2,0,-1,0]
Phi of -K* [-2,-1,-1,1,1,2,1,2,1,2,1,0,2,2,1,2,2,2,-1,1,2]
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+38t^4+35t^2
Outer characteristic polynomial t^7+50t^5+57t^3+4t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -64*K1**6 - 384*K1**4*K2**2 + 2688*K1**4*K2 - 7008*K1**4 + 1024*K1**3*K2*K3 - 1760*K1**3*K3 - 256*K1**2*K2**4 + 1664*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9600*K1**2*K2**2 - 1312*K1**2*K2*K4 + 14192*K1**2*K2 - 1024*K1**2*K3**2 - 96*K1**2*K3*K5 - 224*K1**2*K4**2 - 6236*K1**2 + 640*K1*K2**3*K3 - 1632*K1*K2**2*K3 - 320*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 10056*K1*K2*K3 + 2096*K1*K3*K4 + 504*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1344*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 1968*K2**2*K4 - 5562*K2**2 + 408*K2*K3*K5 + 104*K2*K4*K6 - 2644*K3**2 - 1028*K4**2 - 200*K5**2 - 22*K6**2 + 5978
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
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