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Flat knot 6.1293

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,1,3,0,0,1,1,0,0,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1293', '7.37159', '7.37256']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923']
Outer characteristic polynomial of the knot is: t^7+28t^5+36t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1293', '7.37256']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 640*K1**4*K2**2 + 3200*K1**4*K2 - 7456*K1**4 + 1792*K1**3*K2*K3 + 256*K1**3*K3*K4 - 2240*K1**3*K3 - 384*K1**2*K2**4 + 2432*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 9888*K1**2*K2**2 - 1728*K1**2*K2*K4 + 11408*K1**2*K2 - 1824*K1**2*K3**2 - 64*K1**2*K3*K5 - 288*K1**2*K4**2 - 1992*K1**2 + 1280*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 - 256*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8064*K1*K2*K3 - 64*K1*K2*K4*K5 + 1344*K1*K3*K4 + 160*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 1952*K2**4 - 832*K2**2*K3**2 - 224*K2**2*K4**2 + 1520*K2**2*K4 - 2292*K2**2 + 448*K2*K3*K5 + 96*K2*K4*K6 - 1240*K3**2 - 312*K4**2 - 16*K5**2 - 4*K6**2 + 3102
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1293']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19', 'vk6.31', 'vk6.148', 'vk6.165', 'vk6.1199', 'vk6.1292', 'vk6.1303', 'vk6.2362', 'vk6.2400', 'vk6.2962', 'vk6.3548', 'vk6.6932', 'vk6.6965', 'vk6.15380', 'vk6.15501', 'vk6.33431', 'vk6.33486', 'vk6.33601', 'vk6.49940', 'vk6.53749']
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 O1O2O3U2O4O5U4U5O6U1U6U3
R3 orbit {'O1O2O3U2O4O5U4U5O6U1U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U3O4U5U6O5O6U2
Gauss code of K* O1O2O3U1U4U3O4U5U6O5O6U2
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 2 -1 1 1],[ 2 0 0 3 -1 1 1],[ 1 0 0 1 0 0 0],[-2 -3 -1 0 -1 1 0],[ 1 1 0 1 0 1 0],[-1 -1 0 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 1 0 -1 -1 -3],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 0 0 -1],[ 1 1 0 0 0 0 0],[ 1 1 1 0 0 0 1],[ 2 3 1 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,-1,0,1,1,3,0,0,1,1,0,0,1,0,0,-1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,1,3,0,0,1,1,0,0,1,0,0,-1]
Phi of -K [-2,-1,-1,1,1,2,1,2,2,2,1,0,2,2,2,1,2,2,0,2,1]
Phi of K* [-2,-1,-1,1,1,2,1,2,2,2,1,0,2,2,2,1,2,2,0,2,1]
Phi of -K* [-2,-1,-1,1,1,2,-1,0,1,1,3,0,0,1,1,0,0,1,0,0,-1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+24z+33
Enhanced Jones-Krushkal polynomial 4w^3z^2+24w^2z+33w
Inner characteristic polynomial t^6+16t^4+16t^2+1
Outer characteristic polynomial t^7+28t^5+36t^3+7t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -512*K1**6 - 640*K1**4*K2**2 + 3200*K1**4*K2 - 7456*K1**4 + 1792*K1**3*K2*K3 + 256*K1**3*K3*K4 - 2240*K1**3*K3 - 384*K1**2*K2**4 + 2432*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 9888*K1**2*K2**2 - 1728*K1**2*K2*K4 + 11408*K1**2*K2 - 1824*K1**2*K3**2 - 64*K1**2*K3*K5 - 288*K1**2*K4**2 - 1992*K1**2 + 1280*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 960*K1*K2**2*K3 - 256*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8064*K1*K2*K3 - 64*K1*K2*K4*K5 + 1344*K1*K3*K4 + 160*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 1952*K2**4 - 832*K2**2*K3**2 - 224*K2**2*K4**2 + 1520*K2**2*K4 - 2292*K2**2 + 448*K2*K3*K5 + 96*K2*K4*K6 - 1240*K3**2 - 312*K4**2 - 16*K5**2 - 4*K6**2 + 3102
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice True
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