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Flat knot 6.746

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,-1,1,2,3,-1,2,2,3,1,0,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.746']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863']
Outer characteristic polynomial of the knot is: t^7+57t^5+55t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.746']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 672*K1**4*K2 - 3504*K1**4 + 512*K1**3*K2*K3 + 32*K1**3*K3*K4 - 960*K1**3*K3 - 192*K1**2*K2**4 + 416*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 4640*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 9688*K1**2*K2 - 1008*K1**2*K3**2 - 112*K1**2*K4**2 - 6228*K1**2 + 288*K1*K2**3*K3 - 608*K1*K2**2*K3 - 256*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7384*K1*K2*K3 + 1368*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 752*K2**4 - 32*K2**3*K6 - 192*K2**2*K3**2 - 16*K2**2*K4**2 + 1120*K2**2*K4 - 4878*K2**2 + 192*K2*K3*K5 + 16*K2*K4*K6 - 2332*K3**2 - 588*K4**2 - 40*K5**2 - 2*K6**2 + 5098
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.746']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16932', 'vk6.17174', 'vk6.20545', 'vk6.21946', 'vk6.23331', 'vk6.23625', 'vk6.28003', 'vk6.29470', 'vk6.35374', 'vk6.35795', 'vk6.39411', 'vk6.41604', 'vk6.42851', 'vk6.43130', 'vk6.45991', 'vk6.47667', 'vk6.55095', 'vk6.55350', 'vk6.57425', 'vk6.58596', 'vk6.59496', 'vk6.59791', 'vk6.62096', 'vk6.63074', 'vk6.64946', 'vk6.65153', 'vk6.66965', 'vk6.67826', 'vk6.68238', 'vk6.68380', 'vk6.69580', 'vk6.70277']
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 O1O2O3O4U2O5U3U1U4O6U5U6
R3 orbit {'O1O2O3O4U2O5U3U1U4O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6O5U1U4U2O6U3
Gauss code of K* O1O2O3U4O5O4U2U6U1U3O6U5
Gauss code of -K* O1O2O3U4O5U1U3U5U2O6O4U6
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 -2 -1 2 2 1],[ 2 0 -1 1 3 3 1],[ 2 1 0 1 2 2 0],[ 1 -1 -1 0 1 2 1],[-2 -3 -2 -1 0 1 1],[-2 -3 -2 -2 -1 0 1],[-1 -1 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -2 -2],[-2 0 1 1 -1 -2 -3],[-2 -1 0 1 -2 -2 -3],[-1 -1 -1 0 -1 0 -1],[ 1 1 2 1 0 -1 -1],[ 2 2 2 0 1 0 1],[ 2 3 3 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,2,2,-1,-1,1,2,3,-1,2,2,3,1,0,1,1,1,-1]
Phi over symmetry [-2,-2,-1,1,2,2,-1,-1,1,2,3,-1,2,2,3,1,0,1,1,1,-1]
Phi of -K [-2,-2,-1,1,2,2,-1,0,3,2,2,0,2,1,1,1,1,2,2,2,1]
Phi of K* [-2,-2,-1,1,2,2,-1,2,1,1,2,2,2,1,2,1,2,3,0,0,-1]
Phi of -K* [-2,-2,-1,1,2,2,-1,1,1,3,3,1,0,2,2,1,1,2,-1,-1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+21z+35
Enhanced Jones-Krushkal polynomial 2w^3z^2+21w^2z+35w
Inner characteristic polynomial t^6+39t^4+13t^2+1
Outer characteristic polynomial t^7+57t^5+55t^3+6t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 4*K1*K2 - K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -64*K1**6 + 672*K1**4*K2 - 3504*K1**4 + 512*K1**3*K2*K3 + 32*K1**3*K3*K4 - 960*K1**3*K3 - 192*K1**2*K2**4 + 416*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 4640*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 9688*K1**2*K2 - 1008*K1**2*K3**2 - 112*K1**2*K4**2 - 6228*K1**2 + 288*K1*K2**3*K3 - 608*K1*K2**2*K3 - 256*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 7384*K1*K2*K3 + 1368*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 752*K2**4 - 32*K2**3*K6 - 192*K2**2*K3**2 - 16*K2**2*K4**2 + 1120*K2**2*K4 - 4878*K2**2 + 192*K2*K3*K5 + 16*K2*K4*K6 - 2332*K3**2 - 588*K4**2 - 40*K5**2 - 2*K6**2 + 5098
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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