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Flat knot 6.335

Min(phi) over symmetries of the knot is: [-3,0,1,2,1,3,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.335']
Arrow polynomial of the knot is: -10*K1**2 - 6*K1*K2 + 3*K1 + 5*K2 + 3*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.335', '6.1283', '6.1316']
Outer characteristic polynomial of the knot is: t^5+31t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['4.2', '6.335', '7.33612']
2-strand cable arrow polynomial of the knot is: -832*K1**6 - 576*K1**4*K2**2 + 1952*K1**4*K2 - 4816*K1**4 + 800*K1**3*K2*K3 - 544*K1**3*K3 - 4912*K1**2*K2**2 - 288*K1**2*K2*K4 + 8752*K1**2*K2 - 1872*K1**2*K3**2 - 64*K1**2*K3*K5 - 464*K1**2*K4**2 - 4300*K1**2 - 416*K1*K2**2*K3 - 96*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 7408*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 2784*K1*K3*K4 + 832*K1*K4*K5 + 128*K1*K5*K6 - 552*K2**4 - 560*K2**2*K3**2 - 216*K2**2*K4**2 + 1384*K2**2*K4 - 4778*K2**2 + 1168*K2*K3*K5 + 328*K2*K4*K6 + 64*K3**2*K6 - 2924*K3**2 - 1406*K4**2 - 616*K5**2 - 158*K6**2 + 5396
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.335']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4134', 'vk6.4165', 'vk6.5372', 'vk6.5403', 'vk6.7494', 'vk6.7523', 'vk6.8995', 'vk6.9026', 'vk6.12430', 'vk6.12463', 'vk6.13336', 'vk6.13557', 'vk6.13588', 'vk6.14267', 'vk6.14714', 'vk6.14754', 'vk6.15212', 'vk6.15870', 'vk6.15910', 'vk6.30835', 'vk6.30868', 'vk6.32019', 'vk6.32052', 'vk6.33062', 'vk6.33093', 'vk6.33859', 'vk6.34322', 'vk6.48480', 'vk6.50259', 'vk6.53524', 'vk6.53940', 'vk6.54271']
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 O1O2O3O4O5U2U5O6U4U6U1U3
R3 orbit {'O1O2O3O4O5U2U5O6U4U6U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U5U6U2O6U1U4
Gauss code of K* O1O2O3O4U3U5U4U1U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U4U1U6U2
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 -3 2 0 1 1],[ 1 0 -2 2 0 1 1],[ 3 2 0 3 2 1 1],[-2 -2 -3 0 -1 0 1],[ 0 0 -2 1 0 0 1],[-1 -1 -1 0 0 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 2 1 0 -3],[-2 0 1 -1 -3],[-1 -1 0 -1 -1],[ 0 1 1 0 -2],[ 3 3 1 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,0,3,-1,1,3,1,1,2]
Phi over symmetry [-3,0,1,2,1,3,2,0,1,2]
Phi of -K [-3,0,1,2,1,3,2,0,1,2]
Phi of K* [-2,-1,0,3,2,1,2,0,3,1]
Phi of -K* [-3,0,1,2,2,1,3,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^4+17t^2
Outer characteristic polynomial t^5+31t^3+7t
Flat arrow polynomial -10*K1**2 - 6*K1*K2 + 3*K1 + 5*K2 + 3*K3 + 6
2-strand cable arrow polynomial -832*K1**6 - 576*K1**4*K2**2 + 1952*K1**4*K2 - 4816*K1**4 + 800*K1**3*K2*K3 - 544*K1**3*K3 - 4912*K1**2*K2**2 - 288*K1**2*K2*K4 + 8752*K1**2*K2 - 1872*K1**2*K3**2 - 64*K1**2*K3*K5 - 464*K1**2*K4**2 - 4300*K1**2 - 416*K1*K2**2*K3 - 96*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 7408*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K3**2*K5 + 2784*K1*K3*K4 + 832*K1*K4*K5 + 128*K1*K5*K6 - 552*K2**4 - 560*K2**2*K3**2 - 216*K2**2*K4**2 + 1384*K2**2*K4 - 4778*K2**2 + 1168*K2*K3*K5 + 328*K2*K4*K6 + 64*K3**2*K6 - 2924*K3**2 - 1406*K4**2 - 616*K5**2 - 158*K6**2 + 5396
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {1, 5}, {4}, {2, 3}]]
If K is slice False
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