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Flat knot 6.1494

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,2,2,-1,0,0,0,1,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1494']
Arrow polynomial of the knot is: -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.425', '6.655', '6.755', '6.769', '6.792', '6.1240', '6.1494', '6.1522', '6.1534', '6.1587', '6.1707', '6.1746', '6.1747', '6.1786', '6.1814', '6.1828', '6.1835', '6.1854', '6.1870']
Outer characteristic polynomial of the knot is: t^7+19t^5+35t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1494']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 192*K1**4*K2**2 + 2080*K1**4*K2 - 6704*K1**4 + 896*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1984*K1**3*K3 + 128*K1**2*K2**2*K4 - 4608*K1**2*K2**2 - 896*K1**2*K2*K4 + 12384*K1**2*K2 - 1424*K1**2*K3**2 - 256*K1**2*K4**2 - 6172*K1**2 - 448*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 8096*K1*K2*K3 + 2328*K1*K3*K4 + 272*K1*K4*K5 - 104*K2**4 - 80*K2**2*K3**2 - 48*K2**2*K4**2 + 864*K2**2*K4 - 5668*K2**2 + 144*K2*K3*K5 + 32*K2*K4*K6 - 2776*K3**2 - 970*K4**2 - 84*K5**2 - 4*K6**2 + 5896
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1494']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3668', 'vk6.3763', 'vk6.3958', 'vk6.4053', 'vk6.4490', 'vk6.4585', 'vk6.5872', 'vk6.5999', 'vk6.7151', 'vk6.7330', 'vk6.7421', 'vk6.7929', 'vk6.8048', 'vk6.9359', 'vk6.17913', 'vk6.18008', 'vk6.18750', 'vk6.24452', 'vk6.24869', 'vk6.25332', 'vk6.37497', 'vk6.43879', 'vk6.44224', 'vk6.44529', 'vk6.48292', 'vk6.48355', 'vk6.50073', 'vk6.50187', 'vk6.50590', 'vk6.50653', 'vk6.55856', 'vk6.60708']
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 O1O2O3U1O4U3U5O6O5U4U6U2
R3 orbit {'O1O2O3U1O4U3U5O6O5U4U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U5O6O4U6U1O5U3
Gauss code of K* O1O2O3U4U3U5O4U1O5O6U2U6
Gauss code of -K* O1O2O3U4U2O4O5U3O6U5U1U6
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 1 0 0 1 0],[ 2 0 2 1 1 2 0],[-1 -2 0 -1 0 0 0],[ 0 -1 1 0 1 0 1],[ 0 -1 0 -1 0 0 0],[-1 -2 0 0 0 0 0],[ 0 0 0 -1 0 0 0]]
Primitive based matrix [[ 0 1 1 0 0 0 -2],[-1 0 0 0 0 0 -2],[-1 0 0 0 0 -1 -2],[ 0 0 0 0 0 -1 0],[ 0 0 0 0 0 -1 -1],[ 0 0 1 1 1 0 -1],[ 2 2 2 0 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,0,2,0,0,0,0,2,0,0,1,2,0,1,0,1,1,1]
Phi over symmetry [-2,0,0,0,1,1,0,1,1,2,2,-1,0,0,0,1,0,1,0,0,0]
Phi of -K [-2,0,0,0,1,1,1,1,2,1,1,-1,-1,0,1,0,1,1,1,1,0]
Phi of K* [-1,-1,0,0,0,2,0,0,1,1,1,1,1,1,1,1,1,1,0,1,2]
Phi of -K* [-2,0,0,0,1,1,0,1,1,2,2,-1,0,0,0,1,0,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+13t^4+18t^2
Outer characteristic polynomial t^7+19t^5+35t^3+3t
Flat arrow polynomial -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -512*K1**6 - 192*K1**4*K2**2 + 2080*K1**4*K2 - 6704*K1**4 + 896*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1984*K1**3*K3 + 128*K1**2*K2**2*K4 - 4608*K1**2*K2**2 - 896*K1**2*K2*K4 + 12384*K1**2*K2 - 1424*K1**2*K3**2 - 256*K1**2*K4**2 - 6172*K1**2 - 448*K1*K2**2*K3 - 32*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 8096*K1*K2*K3 + 2328*K1*K3*K4 + 272*K1*K4*K5 - 104*K2**4 - 80*K2**2*K3**2 - 48*K2**2*K4**2 + 864*K2**2*K4 - 5668*K2**2 + 144*K2*K3*K5 + 32*K2*K4*K6 - 2776*K3**2 - 970*K4**2 - 84*K5**2 - 4*K6**2 + 5896
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]]
If K is slice False
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