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Flat knot 6.1345

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,1,1,1,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1345']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 4*K1*K2 - K1 + 6*K2 + K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1345', '6.1510', '6.1565', '6.1691', '6.1812']
Outer characteristic polynomial of the knot is: t^7+36t^5+33t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1345']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 320*K1**4*K2**2 + 3328*K1**4*K2 - 8112*K1**4 + 1088*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1824*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 8800*K1**2*K2**2 - 928*K1**2*K2*K4 + 14056*K1**2*K2 - 912*K1**2*K3**2 - 32*K1**2*K3*K5 - 112*K1**2*K4**2 - 4776*K1**2 + 448*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 224*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7920*K1*K2*K3 + 1048*K1*K3*K4 + 88*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 928*K2**4 - 32*K2**3*K6 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 920*K2**2*K4 - 4558*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 1688*K3**2 - 308*K4**2 - 16*K5**2 - 2*K6**2 + 4874
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1345']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4469', 'vk6.4566', 'vk6.5851', 'vk6.5980', 'vk6.7901', 'vk6.8021', 'vk6.9330', 'vk6.9451', 'vk6.13412', 'vk6.13509', 'vk6.13696', 'vk6.14081', 'vk6.15052', 'vk6.15174', 'vk6.17795', 'vk6.17826', 'vk6.18821', 'vk6.19423', 'vk6.19716', 'vk6.24338', 'vk6.25416', 'vk6.25447', 'vk6.26595', 'vk6.33254', 'vk6.33315', 'vk6.37548', 'vk6.44880', 'vk6.48658', 'vk6.50548', 'vk6.53646', 'vk6.55826', 'vk6.65490']
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 O1O2O3U1O4O5U4U6U3O6U5U2
R3 orbit {'O1O2O3U1O4O5U4U6U3O6U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4O5U1U5U6O4O6U3
Gauss code of K* O1O2O3U2O4O5U6U5U3O6U1U4
Gauss code of -K* O1O2O3U4U3O5U1U6U5O6O4U2
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 0 1 1],[-1 -2 0 0 -1 2 -2],[-1 -1 0 0 0 1 -1],[ 1 0 1 0 0 1 1],[-2 -1 -2 -1 -1 0 -2],[ 1 -1 2 1 -1 2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -2 -1 -2 -1],[-1 1 0 0 0 -1 -1],[-1 2 0 0 -1 -2 -2],[ 1 1 0 1 0 1 0],[ 1 2 1 2 -1 0 -1],[ 2 1 1 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,1,2,1,2,1,0,0,1,1,1,2,2,-1,0,1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,1,1,1,2,2,-1,0,1]
Phi of -K [-2,-1,-1,1,1,2,0,1,1,2,3,1,0,1,1,1,2,2,0,-1,0]
Phi of K* [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,1,1,1,2,2,-1,0,1]
Phi of -K* [-2,-1,-1,1,1,2,0,1,1,2,1,1,0,1,1,1,2,2,0,1,2]
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+24t^4+15t^2
Outer characteristic polynomial t^7+36t^5+33t^3+4t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 4*K1*K2 - K1 + 6*K2 + K3 + 7
2-strand cable arrow polynomial -384*K1**6 - 320*K1**4*K2**2 + 3328*K1**4*K2 - 8112*K1**4 + 1088*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1824*K1**3*K3 - 192*K1**2*K2**4 + 1248*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 8800*K1**2*K2**2 - 928*K1**2*K2*K4 + 14056*K1**2*K2 - 912*K1**2*K3**2 - 32*K1**2*K3*K5 - 112*K1**2*K4**2 - 4776*K1**2 + 448*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 224*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7920*K1*K2*K3 + 1048*K1*K3*K4 + 88*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 928*K2**4 - 32*K2**3*K6 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 920*K2**2*K4 - 4558*K2**2 + 88*K2*K3*K5 + 16*K2*K4*K6 - 1688*K3**2 - 308*K4**2 - 16*K5**2 - 2*K6**2 + 4874
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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