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Flat knot 6.2065

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,1,1,0,0,0,1,0,1,0,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.2065', '7.44691']
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+10t^5+15t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.2065', '7.44691']
2-strand cable arrow polynomial of the knot is: -1792*K1**6 - 3136*K1**4*K2**2 + 6624*K1**4*K2 - 7632*K1**4 + 2656*K1**3*K2*K3 - 1184*K1**3*K3 - 1664*K1**2*K2**4 + 5664*K1**2*K2**3 + 576*K1**2*K2**2*K4 - 14832*K1**2*K2**2 - 1312*K1**2*K2*K4 + 11728*K1**2*K2 - 848*K1**2*K3**2 - 112*K1**2*K4**2 - 808*K1**2 + 2080*K1*K2**3*K3 - 2688*K1*K2**2*K3 - 448*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 8016*K1*K2*K3 + 648*K1*K3*K4 + 104*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 3104*K2**4 - 64*K2**3*K6 - 656*K2**2*K3**2 - 128*K2**2*K4**2 + 2104*K2**2*K4 - 1564*K2**2 + 304*K2*K3*K5 + 48*K2*K4*K6 - 872*K3**2 - 216*K4**2 - 32*K5**2 - 4*K6**2 + 2798
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.2065']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.58', 'vk6.115', 'vk6.208', 'vk6.257', 'vk6.291', 'vk6.673', 'vk6.1218', 'vk6.1267', 'vk6.1354', 'vk6.1403', 'vk6.1443', 'vk6.1925', 'vk6.2384', 'vk6.2444', 'vk6.2938', 'vk6.2994', 'vk6.5748', 'vk6.5779', 'vk6.7813', 'vk6.7844', 'vk6.13279', 'vk6.13312', 'vk6.14775', 'vk6.14798', 'vk6.15929', 'vk6.15954', 'vk6.18047', 'vk6.24485', 'vk6.33036', 'vk6.33382', 'vk6.43915', 'vk6.50506']
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 O1O2U1U2O3O4U3U5O6O5U4U6
R3 orbit {'O1O2U1U2O3O4U3U5O6O5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2U1U3O4O3U5U6O5O6U2U4
Gauss code of K* O1O2U3U4O3O4U5U1O5O6U2U6
Gauss code of -K* O1O2U1U2O3O4U5U3O5O6U4U6
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 -1 0 1 0],[ 1 0 1 0 0 1 0],[-1 -1 0 0 0 -1 0],[ 1 0 0 0 1 1 1],[ 0 0 0 -1 0 0 0],[-1 -1 1 -1 0 0 0],[ 0 0 0 -1 0 0 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 0 0 -1 -1],[-1 -1 0 0 0 0 -1],[ 0 0 0 0 0 -1 0],[ 0 0 0 0 0 -1 0],[ 1 1 0 1 1 0 0],[ 1 1 1 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,0,0,1,1,0,0,0,1,0,1,0,1,0,0]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,1,1,0,0,0,1,0,1,0,1,0,0]
Phi of -K [-1,-1,0,0,1,1,0,0,0,1,2,1,1,1,1,0,1,1,1,1,-1]
Phi of K* [-1,-1,0,0,1,1,-1,1,1,1,2,1,1,1,1,0,1,0,1,0,0]
Phi of -K* [-1,-1,0,0,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0,0,-1]
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+6t^4+7t^2
Outer characteristic polynomial t^7+10t^5+15t^3+2t
Flat arrow polynomial 8*K1**3 - 12*K1**2 - 8*K1*K2 - 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -1792*K1**6 - 3136*K1**4*K2**2 + 6624*K1**4*K2 - 7632*K1**4 + 2656*K1**3*K2*K3 - 1184*K1**3*K3 - 1664*K1**2*K2**4 + 5664*K1**2*K2**3 + 576*K1**2*K2**2*K4 - 14832*K1**2*K2**2 - 1312*K1**2*K2*K4 + 11728*K1**2*K2 - 848*K1**2*K3**2 - 112*K1**2*K4**2 - 808*K1**2 + 2080*K1*K2**3*K3 - 2688*K1*K2**2*K3 - 448*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 8016*K1*K2*K3 + 648*K1*K3*K4 + 104*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 3104*K2**4 - 64*K2**3*K6 - 656*K2**2*K3**2 - 128*K2**2*K4**2 + 2104*K2**2*K4 - 1564*K2**2 + 304*K2*K3*K5 + 48*K2*K4*K6 - 872*K3**2 - 216*K4**2 - 32*K5**2 - 4*K6**2 + 2798
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {3, 5}, {4}, {1, 2}]]
If K is slice False
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