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Flat knot 6.2078

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,1,1,1,0,0,1,1,-1,1,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.2078']
Arrow polynomial of the knot is: 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.218', '6.554', '6.929', '6.932', '6.1014', '6.1024', '6.1068', '6.1526', '6.1664', '6.1676', '6.1755', '6.1763', '6.2065', '6.2078']
Outer characteristic polynomial of the knot is: t^7+14t^5+35t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2078']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1344*K1**4*K2**2 + 2592*K1**4*K2 - 3664*K1**4 - 256*K1**3*K2**2*K3 + 1056*K1**3*K2*K3 - 608*K1**3*K3 - 512*K1**2*K2**4 + 3840*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11264*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 10848*K1**2*K2 - 368*K1**2*K3**2 - 5288*K1**2 + 1088*K1*K2**3*K3 - 2720*K1*K2**2*K3 - 352*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 8752*K1*K2*K3 + 928*K1*K3*K4 + 144*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 3104*K2**4 - 64*K2**3*K6 - 768*K2**2*K3**2 - 128*K2**2*K4**2 + 2872*K2**2*K4 - 3796*K2**2 + 656*K2*K3*K5 + 80*K2*K4*K6 - 1952*K3**2 - 696*K4**2 - 168*K5**2 - 12*K6**2 + 4718
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2078']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19942', 'vk6.20055', 'vk6.21187', 'vk6.21338', 'vk6.26907', 'vk6.27116', 'vk6.28661', 'vk6.28807', 'vk6.38331', 'vk6.38513', 'vk6.40471', 'vk6.40713', 'vk6.45204', 'vk6.45409', 'vk6.47027', 'vk6.47155', 'vk6.56731', 'vk6.56869', 'vk6.57831', 'vk6.58007', 'vk6.61156', 'vk6.61394', 'vk6.62397', 'vk6.62554', 'vk6.66430', 'vk6.66576', 'vk6.67201', 'vk6.67366', 'vk6.69083', 'vk6.69224', 'vk6.69864', 'vk6.69965']
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 O1O2U1U3O4O5U6U2O3O6U4U5
R3 orbit {'O1O2U1U3O4O5U6U2O3O6U4U5'}
R3 orbit length 1
Gauss code of -K O1O2U3U1O4O5U2U6O3O6U4U5
Gauss code of K* O1O2U3U4O3O5U1U2O6O4U5U6
Gauss code of -K* O1O2U3U4O5O3U1U2O4O6U5U6
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 0 -1 1 0],[ 1 0 1 0 1 1 0],[-1 -1 0 -1 -1 0 0],[ 0 0 1 0 0 1 -1],[ 1 -1 1 0 0 1 1],[-1 -1 0 -1 -1 0 -1],[ 0 0 0 1 -1 1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 0 0 -1 -1 -1],[-1 0 0 -1 -1 -1 -1],[ 0 0 1 0 1 0 -1],[ 0 1 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,-1,0,0,1,1,0,0,1,1,1,1,1,1,1,-1,0,1,0,0,-1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,1,1,1,0,0,1,1,-1,1,1,0,1,0]
Phi of -K [-1,-1,0,0,1,1,-1,1,1,1,1,0,1,1,1,-1,0,1,0,0,0]
Phi of K* [-1,-1,0,0,1,1,0,0,0,1,1,0,1,1,1,-1,1,1,0,1,-1]
Phi of -K* [-1,-1,0,0,1,1,-1,0,1,1,1,0,0,1,1,-1,1,1,0,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+10t^4+21t^2+1
Outer characteristic polynomial t^7+14t^5+35t^3+4t
Flat arrow polynomial 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1344*K1**4*K2**2 + 2592*K1**4*K2 - 3664*K1**4 - 256*K1**3*K2**2*K3 + 1056*K1**3*K2*K3 - 608*K1**3*K3 - 512*K1**2*K2**4 + 3840*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11264*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 800*K1**2*K2*K4 + 10848*K1**2*K2 - 368*K1**2*K3**2 - 5288*K1**2 + 1088*K1*K2**3*K3 - 2720*K1*K2**2*K3 - 352*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 8752*K1*K2*K3 + 928*K1*K3*K4 + 144*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 3104*K2**4 - 64*K2**3*K6 - 768*K2**2*K3**2 - 128*K2**2*K4**2 + 2872*K2**2*K4 - 3796*K2**2 + 656*K2*K3*K5 + 80*K2*K4*K6 - 1952*K3**2 - 696*K4**2 - 168*K5**2 - 12*K6**2 + 4718
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]]
If K is slice False
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