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Flat knot 5.2

Min(phi) over symmetries of the knot is: [-4,-2,1,2,3,1,2,4,3,1,3,2,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['5.2']
Arrow polynomial of the knot is: -4*K1**3 - 4*K1**2*K2 + 6*K1**2 + 2*K1*K2 + 2*K1*K3 + 2*K1 - 2*K2 - 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.2', '7.1367', '7.1512', '7.2277', '7.2569', '7.2602', '7.3258', '7.3350', '7.3439', '7.4403', '7.4449', '7.4600', '7.4766', '7.4973', '7.5328', '7.6219', '7.6253', '7.6267', '7.6337', '7.6351', '7.6628', '7.6634', '7.6762', '7.6805', '7.7035', '7.7134', '7.7172', '7.7188', '7.7191', '7.7623', '7.7641', '7.8197', '7.8236', '7.8241', '7.8273', '7.8315', '7.8542', '7.8548', '7.8551', '7.8715', '7.8725', '7.8755', '7.9023', '7.9056', '7.9101', '7.9506', '7.9825', '7.10521', '7.10619', '7.11233', '7.11416', '7.11690', '7.12290', '7.12632', '7.12686', '7.13317', '7.14423', '7.15137', '7.15709', '7.15859', '7.15932', '7.16618']
Outer characteristic polynomial of the knot is: t^6+81t^4+56t^2+1
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.2']
2-strand cable arrow polynomial of the knot is: -128*K1**2*K2**4 + 416*K1**2*K2**3 - 1536*K1**2*K2**2 - 64*K1**2*K2*K4 + 1376*K1**2*K2 - 1032*K1**2 + 736*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 480*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 1344*K1*K2*K3 + 112*K1*K3*K4 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 904*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 416*K2**2*K3**2 - 136*K2**2*K4**2 + 648*K2**2*K4 - 8*K2**2*K6**2 - 324*K2**2 + 96*K2*K3*K5 + 40*K2*K4*K6 - 328*K3**2 - 108*K4**2 - 4*K6**2 + 730
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.2']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.2028', 'vk5.2030', 'vk5.2062', 'vk5.2066', 'vk5.2141', 'vk5.2144', 'vk5.2179', 'vk5.2181', 'vk5.2219', 'vk5.2227', 'vk5.2270', 'vk5.2272', 'vk5.2301', 'vk5.2330', 'vk5.2369', 'vk5.2373']
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 O1O2O3O4O5U1U2U4U5U3
R3 orbit {'O1O2O3O4O5U1U2U4U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U1U2U4U5
Gauss code of K* O1O2O3O4O5U1U2U5U3U4
Gauss code of -K* O1O2O3O4O5U2U3U1U4U5
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 -4 -2 2 1 3],[ 4 0 1 4 2 3],[ 2 -1 0 3 1 2],[-2 -4 -3 0 -1 1],[-1 -2 -1 1 0 1],[-3 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 2 1 -2 -4],[-3 0 -1 -1 -2 -3],[-2 1 0 -1 -3 -4],[-1 1 1 0 -1 -2],[ 2 2 3 1 0 -1],[ 4 3 4 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,2,4,1,1,2,3,1,3,4,1,2,1]
Phi over symmetry [-4,-2,1,2,3,1,2,4,3,1,3,2,1,1,1]
Phi of -K [-4,-2,1,2,3,1,3,2,4,2,1,3,0,1,0]
Phi of K* [-3,-2,-1,2,4,0,1,3,4,0,1,2,2,3,1]
Phi of -K* [-4,-2,1,2,3,1,2,4,3,1,3,2,1,1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial -4z^2-15z-13
Enhanced Jones-Krushkal polynomial -4w^3z^2-15w^2z-13w
Inner characteristic polynomial t^5+47t^3+6t
Outer characteristic polynomial t^6+81t^4+56t^2+1
Flat arrow polynomial -4*K1**3 - 4*K1**2*K2 + 6*K1**2 + 2*K1*K2 + 2*K1*K3 + 2*K1 - 2*K2 - 1
2-strand cable arrow polynomial -128*K1**2*K2**4 + 416*K1**2*K2**3 - 1536*K1**2*K2**2 - 64*K1**2*K2*K4 + 1376*K1**2*K2 - 1032*K1**2 + 736*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 480*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 1344*K1*K2*K3 + 112*K1*K3*K4 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 904*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 416*K2**2*K3**2 - 136*K2**2*K4**2 + 648*K2**2*K4 - 8*K2**2*K6**2 - 324*K2**2 + 96*K2*K3*K5 + 40*K2*K4*K6 - 328*K3**2 - 108*K4**2 - 4*K6**2 + 730
Genus of based matrix 2
Fillings of based matrix [[{1, 5}, {2, 4}, {3}], [{1, 5}, {3, 4}, {2}], [{1, 5}, {4}, {2, 3}], [{1, 5}, {4}, {3}, {2}], [{2, 5}, {1, 4}, {3}], [{2, 5}, {3, 4}, {1}], [{2, 5}, {4}, {1, 3}], [{2, 5}, {4}, {3}, {1}], [{3, 5}, {1, 4}, {2}], [{3, 5}, {2, 4}, {1}], [{3, 5}, {4}, {1, 2}], [{3, 5}, {4}, {2}, {1}], [{4, 5}, {1, 3}, {2}], [{4, 5}, {2, 3}, {1}], [{4, 5}, {3}, {1, 2}], [{4, 5}, {3}, {2}, {1}], [{5}, {1, 4}, {2, 3}], [{5}, {1, 4}, {3}, {2}], [{5}, {2, 4}, {1, 3}], [{5}, {2, 4}, {3}, {1}], [{5}, {3, 4}, {1, 2}], [{5}, {3, 4}, {2}, {1}], [{5}, {4}, {1, 3}, {2}], [{5}, {4}, {2, 3}, {1}], [{5}, {4}, {3}, {1, 2}]]
If K is slice False
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