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Flat knot 6.985

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,1,3,3,3,1,1,2,0,1,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.985']
Arrow polynomial of the knot is: -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.574', '6.593', '6.604', '6.649', '6.650', '6.673', '6.690', '6.783', '6.973', '6.985', '6.1033', '6.1035']
Outer characteristic polynomial of the knot is: t^7+46t^5+72t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.985']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 384*K1**4*K2**2 + 1536*K1**4*K2 - 4064*K1**4 + 480*K1**3*K2*K3 - 512*K1**3*K3 + 640*K1**2*K2**3 - 7024*K1**2*K2**2 - 288*K1**2*K2*K4 + 10128*K1**2*K2 - 608*K1**2*K3**2 - 32*K1**2*K4**2 - 5260*K1**2 - 800*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 7808*K1*K2*K3 + 1104*K1*K3*K4 + 120*K1*K4*K5 - 1320*K2**4 - 192*K2**2*K3**2 - 8*K2**2*K4**2 + 1688*K2**2*K4 - 4798*K2**2 + 408*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 2444*K3**2 - 790*K4**2 - 192*K5**2 - 18*K6**2 + 5228
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.985']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11564', 'vk6.11572', 'vk6.11904', 'vk6.11913', 'vk6.12915', 'vk6.13221', 'vk6.13231', 'vk6.20958', 'vk6.20971', 'vk6.22376', 'vk6.22387', 'vk6.28425', 'vk6.31345', 'vk6.31365', 'vk6.31757', 'vk6.32513', 'vk6.32531', 'vk6.32914', 'vk6.32932', 'vk6.40131', 'vk6.40148', 'vk6.42144', 'vk6.46646', 'vk6.46659', 'vk6.52337', 'vk6.52600', 'vk6.52609', 'vk6.53474', 'vk6.53481', 'vk6.58953', 'vk6.64473', 'vk6.69787']
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 O1O2O3O4U1U4O5U3O6U2U6U5
R3 orbit {'O1O2O3O4U1U4O5U3O6U2U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6U3O6U2O5U1U4
Gauss code of K* O1O2O3U4U1U5U6O4O6U3O5U2
Gauss code of -K* O1O2O3U2O4U1O5O6U5U4U3U6
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 -3 -1 0 1 2 1],[ 3 0 3 2 1 2 1],[ 1 -3 0 0 0 3 1],[ 0 -2 0 0 0 1 0],[-1 -1 0 0 0 0 0],[-2 -2 -3 -1 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -1 -3 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[ 0 1 0 0 0 0 -2],[ 1 3 0 1 0 0 -3],[ 3 2 1 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,0,1,3,2,0,0,0,1,0,1,1,0,2,3]
Phi over symmetry [-3,-1,0,1,1,2,-1,1,3,3,3,1,1,2,0,1,1,1,0,1,1]
Phi of -K [-3,-1,0,1,1,2,-1,1,3,3,3,1,1,2,0,1,1,1,0,1,1]
Phi of K* [-2,-1,-1,0,1,3,1,1,1,0,3,0,1,1,3,1,2,3,1,1,-1]
Phi of -K* [-3,-1,0,1,1,2,3,2,1,1,2,0,0,1,3,0,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+30t^4+27t^2+1
Outer characteristic polynomial t^7+46t^5+72t^3+11t
Flat arrow polynomial -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -384*K1**6 - 384*K1**4*K2**2 + 1536*K1**4*K2 - 4064*K1**4 + 480*K1**3*K2*K3 - 512*K1**3*K3 + 640*K1**2*K2**3 - 7024*K1**2*K2**2 - 288*K1**2*K2*K4 + 10128*K1**2*K2 - 608*K1**2*K3**2 - 32*K1**2*K4**2 - 5260*K1**2 - 800*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 7808*K1*K2*K3 + 1104*K1*K3*K4 + 120*K1*K4*K5 - 1320*K2**4 - 192*K2**2*K3**2 - 8*K2**2*K4**2 + 1688*K2**2*K4 - 4798*K2**2 + 408*K2*K3*K5 + 16*K2*K4*K6 + 24*K3**2*K6 - 2444*K3**2 - 790*K4**2 - 192*K5**2 - 18*K6**2 + 5228
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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