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Flat knot 6.1639

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,1,0,2,2,0,1,0,0,0,1,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1639', '7.33447']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.206', '6.236', '6.575', '6.580', '6.613', '6.619', '6.810', '6.819', '6.831', '6.838', '6.957', '6.1018', '6.1028', '6.1046', '6.1073', '6.1279', '6.1507', '6.1532', '6.1556', '6.1639', '6.1688', '6.1924', '6.1931']
Outer characteristic polynomial of the knot is: t^7+28t^5+58t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1639']
2-strand cable arrow polynomial of the knot is: -2048*K1**4*K2**2 + 3904*K1**4*K2 - 3776*K1**4 + 1216*K1**3*K2*K3 - 448*K1**3*K3 - 2688*K1**2*K2**4 + 6464*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 12704*K1**2*K2**2 - 640*K1**2*K2*K4 + 6608*K1**2*K2 - 400*K1**2 + 2688*K1*K2**3*K3 - 3008*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5936*K1*K2*K3 + 144*K1*K3*K4 - 704*K2**6 + 448*K2**4*K4 - 3824*K2**4 - 608*K2**2*K3**2 - 48*K2**2*K4**2 + 2448*K2**2*K4 + 448*K2**2 + 96*K2*K3*K5 - 544*K3**2 - 148*K4**2 + 1378
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1639']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.304', 'vk6.341', 'vk6.690', 'vk6.738', 'vk6.1482', 'vk6.1938', 'vk6.1975', 'vk6.2455', 'vk6.2637', 'vk6.3108', 'vk6.18259', 'vk6.18596', 'vk6.24743', 'vk6.25151', 'vk6.36872', 'vk6.37335', 'vk6.44094', 'vk6.56062', 'vk6.60618', 'vk6.65732']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3U1O4U2U3U4O5O6U5U6
R3 orbit {'O1O2O3U1O4U2U3U4O5O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4O5U6U1U2O6U3
Gauss code of K* O1O2O3U4U5O4O5U6U1U2O6U3
Gauss code of -K* Same
Diagrammatic symmetry type -
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 2 -1 1],[ 2 0 1 2 2 0 0],[ 1 -1 0 1 2 0 0],[-1 -2 -1 0 1 0 0],[-2 -2 -2 -1 0 0 0],[ 1 0 0 0 0 0 1],[-1 0 0 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 0 -2 -2],[-1 0 0 0 -1 0 0],[-1 1 0 0 0 -1 -2],[ 1 0 1 0 0 0 0],[ 1 2 0 1 0 0 -1],[ 2 2 0 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,1,0,2,2,0,1,0,0,0,1,2,0,0,1]
Phi over symmetry [-2,-1,-1,1,1,2,0,1,0,2,2,0,1,0,0,0,1,2,0,0,1]
Phi of -K [-2,-1,-1,1,1,2,0,1,1,3,2,0,1,2,1,2,1,3,0,0,1]
Phi of K* [-2,-1,-1,1,1,2,0,1,1,3,2,0,1,2,1,2,1,3,0,0,1]
Phi of -K* [-2,-1,-1,1,1,2,0,1,0,2,2,0,1,0,0,0,1,2,0,0,1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+16t^4+16t^2+1
Outer characteristic polynomial t^7+28t^5+58t^3+5t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 4*K1*K2 - 4*K1 + 2*K2 + 3
2-strand cable arrow polynomial -2048*K1**4*K2**2 + 3904*K1**4*K2 - 3776*K1**4 + 1216*K1**3*K2*K3 - 448*K1**3*K3 - 2688*K1**2*K2**4 + 6464*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 12704*K1**2*K2**2 - 640*K1**2*K2*K4 + 6608*K1**2*K2 - 400*K1**2 + 2688*K1*K2**3*K3 - 3008*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5936*K1*K2*K3 + 144*K1*K3*K4 - 704*K2**6 + 448*K2**4*K4 - 3824*K2**4 - 608*K2**2*K3**2 - 48*K2**2*K4**2 + 2448*K2**2*K4 + 448*K2**2 + 96*K2*K3*K5 - 544*K3**2 - 148*K4**2 + 1378
Genus of based matrix 0
Fillings of based matrix [[{5, 6}, {1, 4}, {2, 3}]]
If K is slice True
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