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Flat knot 6.2070

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,1,1,-1,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.2070']
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+12t^5+25t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2070']
2-strand cable arrow polynomial of the knot is: -640*K1**6 - 448*K1**4*K2**2 + 2720*K1**4*K2 - 6112*K1**4 + 1088*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1376*K1**3*K3 - 192*K1**2*K2**4 + 1152*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 8672*K1**2*K2**2 - 1056*K1**2*K2*K4 + 13072*K1**2*K2 - 1056*K1**2*K3**2 - 64*K1**2*K3*K5 - 192*K1**2*K4**2 - 5896*K1**2 + 480*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 256*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 9488*K1*K2*K3 + 1824*K1*K3*K4 + 296*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1024*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 1576*K2**2*K4 - 5410*K2**2 + 352*K2*K3*K5 + 104*K2*K4*K6 - 2580*K3**2 - 800*K4**2 - 140*K5**2 - 22*K6**2 + 5670
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2070']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4461', 'vk6.4558', 'vk6.5843', 'vk6.5972', 'vk6.7895', 'vk6.8013', 'vk6.9322', 'vk6.9443', 'vk6.13408', 'vk6.13505', 'vk6.13692', 'vk6.14065', 'vk6.15036', 'vk6.15158', 'vk6.17791', 'vk6.17822', 'vk6.18837', 'vk6.19421', 'vk6.19714', 'vk6.24334', 'vk6.25432', 'vk6.25463', 'vk6.26597', 'vk6.33258', 'vk6.33319', 'vk6.37564', 'vk6.44882', 'vk6.48666', 'vk6.50554', 'vk6.53642', 'vk6.55822', 'vk6.65494']
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 O1O2U1U3O4O3U2U5O6O5U4U6
R3 orbit {'O1O2U1U3O4O3U2U5O6O5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2U1U3O4O3U5U2O6O5U6U4
Gauss code of K* O1O2U3U4O3O5U1U5O4O6U2U6
Gauss code of -K* O1O2U3U1O3O4U5U2O5O6U4U6
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 1 0],[ 1 0 1 1 1 1 0],[ 0 -1 0 0 0 0 1],[-1 -1 0 0 -1 0 -1],[ 1 -1 0 1 0 1 0],[-1 -1 0 0 -1 0 0],[ 0 0 -1 1 0 0 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 0 0 0 -1 -1],[-1 0 0 0 -1 -1 -1],[ 0 0 0 0 1 0 -1],[ 0 0 1 -1 0 0 0],[ 1 1 1 0 0 0 -1],[ 1 1 1 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,0,0,0,1,1,0,1,1,1,-1,0,1,0,0,1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,1,1,-1,0,1,0,0,0]
Phi of -K [-1,-1,0,0,1,1,-1,0,1,1,1,1,1,1,1,-1,1,1,0,1,0]
Phi of K* [-1,-1,0,0,1,1,0,0,1,1,1,1,1,1,1,-1,1,1,0,1,1]
Phi of -K* [-1,-1,0,0,1,1,-1,0,0,1,1,0,1,1,1,-1,0,1,0,0,0]
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+8t^4+13t^2+1
Outer characteristic polynomial t^7+12t^5+25t^3+8t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 8*K1*K2 + K1 + 6*K2 + 3*K3 + 7
2-strand cable arrow polynomial -640*K1**6 - 448*K1**4*K2**2 + 2720*K1**4*K2 - 6112*K1**4 + 1088*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1376*K1**3*K3 - 192*K1**2*K2**4 + 1152*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 8672*K1**2*K2**2 - 1056*K1**2*K2*K4 + 13072*K1**2*K2 - 1056*K1**2*K3**2 - 64*K1**2*K3*K5 - 192*K1**2*K4**2 - 5896*K1**2 + 480*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 256*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 9488*K1*K2*K3 + 1824*K1*K3*K4 + 296*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1024*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 1576*K2**2*K4 - 5410*K2**2 + 352*K2*K3*K5 + 104*K2*K4*K6 - 2580*K3**2 - 800*K4**2 - 140*K5**2 - 22*K6**2 + 5670
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}]]
If K is slice False
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