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Flat knot 6.1676

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,3,0,1,0,1,1,-1,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1676']
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+31t^5+95t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1676']
2-strand cable arrow polynomial of the knot is: -704*K1**4*K2**2 + 1760*K1**4*K2 - 3456*K1**4 + 800*K1**3*K2*K3 - 512*K1**3*K3 - 896*K1**2*K2**4 + 4288*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 12144*K1**2*K2**2 - 800*K1**2*K2*K4 + 11576*K1**2*K2 - 352*K1**2*K3**2 - 5396*K1**2 + 1632*K1*K2**3*K3 - 3072*K1*K2**2*K3 - 384*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 9168*K1*K2*K3 + 848*K1*K3*K4 + 128*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 3488*K2**4 - 64*K2**3*K6 - 800*K2**2*K3**2 - 128*K2**2*K4**2 + 3008*K2**2*K4 - 3548*K2**2 + 568*K2*K3*K5 + 48*K2*K4*K6 - 1908*K3**2 - 596*K4**2 - 120*K5**2 - 4*K6**2 + 4642
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1676']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4684', 'vk6.4985', 'vk6.6160', 'vk6.6635', 'vk6.8161', 'vk6.8579', 'vk6.9551', 'vk6.9892', 'vk6.20693', 'vk6.22131', 'vk6.28214', 'vk6.29637', 'vk6.39670', 'vk6.41909', 'vk6.46254', 'vk6.47859', 'vk6.48716', 'vk6.48929', 'vk6.49494', 'vk6.49705', 'vk6.50742', 'vk6.50949', 'vk6.51221', 'vk6.51416', 'vk6.57628', 'vk6.58784', 'vk6.62308', 'vk6.63239', 'vk6.67098', 'vk6.67960', 'vk6.69698', 'vk6.70379']
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 O1O2O3U2O4U5U6U1O5O6U4U3
R3 orbit {'O1O2O3U2O4U5U6U1O5O6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4O5O6U3U5U6O4U2
Gauss code of K* O1O2O3U1U2O4O5U3U6U5O6U4
Gauss code of -K* O1O2O3U4O5U6U5U1O6O4U2U3
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 0 -1 2 1 -2 0],[ 0 0 0 1 -1 0 1],[ 1 0 0 1 0 1 1],[-2 -1 -1 0 0 -3 -1],[-1 1 0 0 0 -2 0],[ 2 0 -1 3 2 0 1],[ 0 -1 -1 1 0 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -1 -1 -3],[-1 0 0 1 0 0 -2],[ 0 1 -1 0 1 0 0],[ 0 1 0 -1 0 -1 -1],[ 1 1 0 0 1 0 1],[ 2 3 2 0 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,1,1,1,3,-1,0,0,2,-1,0,0,1,1,-1]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,1,2,3,0,1,0,1,1,-1,1,0,1,0]
Phi of -K [-2,-1,0,0,1,2,2,1,2,1,1,0,1,2,2,1,1,1,2,1,1]
Phi of K* [-2,-1,0,0,1,2,1,1,1,2,1,1,2,2,1,-1,0,1,1,2,2]
Phi of -K* [-2,-1,0,0,1,2,-1,0,1,2,3,0,1,0,1,1,-1,1,0,1,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+21t^4+57t^2+1
Outer characteristic polynomial t^7+31t^5+95t^3+7t
Flat arrow polynomial 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -704*K1**4*K2**2 + 1760*K1**4*K2 - 3456*K1**4 + 800*K1**3*K2*K3 - 512*K1**3*K3 - 896*K1**2*K2**4 + 4288*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 12144*K1**2*K2**2 - 800*K1**2*K2*K4 + 11576*K1**2*K2 - 352*K1**2*K3**2 - 5396*K1**2 + 1632*K1*K2**3*K3 - 3072*K1*K2**2*K3 - 384*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 9168*K1*K2*K3 + 848*K1*K3*K4 + 128*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 3488*K2**4 - 64*K2**3*K6 - 800*K2**2*K3**2 - 128*K2**2*K4**2 + 3008*K2**2*K4 - 3548*K2**2 + 568*K2*K3*K5 + 48*K2*K4*K6 - 1908*K3**2 - 596*K4**2 - 120*K5**2 - 4*K6**2 + 4642
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}]]
If K is slice False
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