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Flat knot 6.331

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,1,3,-1,0,1,3,4,0,0,1,1,1,2,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.331']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.77', '6.242', '6.331', '6.862']
Outer characteristic polynomial of the knot is: t^7+69t^5+86t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.331']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1152*K1**4*K2**2 + 1600*K1**4*K2 - 2784*K1**4 + 864*K1**3*K2*K3 + 32*K1**3*K3*K4 - 352*K1**3*K3 - 960*K1**2*K2**4 + 3296*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 128*K1**2*K2**2*K4**2 + 320*K1**2*K2**2*K4 - 11232*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 992*K1**2*K2*K4 + 10832*K1**2*K2 - 352*K1**2*K3**2 - 176*K1**2*K4**2 - 6120*K1**2 + 2336*K1*K2**3*K3 + 672*K1*K2**2*K3*K4 - 2432*K1*K2**2*K3 + 224*K1*K2**2*K4*K5 - 704*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9512*K1*K2*K3 + 1064*K1*K3*K4 + 192*K1*K4*K5 - 64*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 768*K2**4*K4 - 3792*K2**4 + 160*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 1392*K2**2*K3**2 - 752*K2**2*K4**2 + 3024*K2**2*K4 - 112*K2**2*K5**2 - 16*K2**2*K6**2 - 3496*K2**2 + 704*K2*K3*K5 + 160*K2*K4*K6 + 8*K3**2*K6 - 2252*K3**2 - 704*K4**2 - 100*K5**2 - 24*K6**2 + 5086
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.331']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11600', 'vk6.11609', 'vk6.11953', 'vk6.11960', 'vk6.12942', 'vk6.12951', 'vk6.13255', 'vk6.20427', 'vk6.20431', 'vk6.21792', 'vk6.27787', 'vk6.27795', 'vk6.29307', 'vk6.31395', 'vk6.31412', 'vk6.32569', 'vk6.32586', 'vk6.32955', 'vk6.39211', 'vk6.39219', 'vk6.41433', 'vk6.47556', 'vk6.53187', 'vk6.53200', 'vk6.53502', 'vk6.57288', 'vk6.57300', 'vk6.61958', 'vk6.61982', 'vk6.64280', 'vk6.64289', 'vk6.64492']
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 O1O2O3O4O5U2U5O6U3U4U1U6
R3 orbit {'O1O2O3O4O5U2U5O6U3U4U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U5U2U3O6U1U4
Gauss code of K* O1O2O3O4U3U5U1U2U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U3U4U6U2
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 -1 1 1 3],[ 1 0 -3 0 2 1 3],[ 3 3 0 2 3 1 2],[ 1 0 -2 0 1 0 2],[-1 -2 -3 -1 0 0 1],[-1 -1 -1 0 0 0 0],[-3 -3 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 3 1 1 -1 -1 -3],[-3 0 0 -1 -2 -3 -2],[-1 0 0 0 0 -1 -1],[-1 1 0 0 -1 -2 -3],[ 1 2 0 1 0 0 -2],[ 1 3 1 2 0 0 -3],[ 3 2 1 3 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,1,3,0,1,2,3,2,0,0,1,1,1,2,3,0,2,3]
Phi over symmetry [-3,-1,-1,1,1,3,-1,0,1,3,4,0,0,1,1,1,2,2,0,1,2]
Phi of -K [-3,-1,-1,1,1,3,-1,0,1,3,4,0,0,1,1,1,2,2,0,1,2]
Phi of K* [-3,-1,-1,1,1,3,1,2,1,2,4,0,0,1,1,1,2,3,0,-1,0]
Phi of -K* [-3,-1,-1,1,1,3,2,3,1,3,2,0,0,1,2,1,2,3,0,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+47t^4+28t^2+1
Outer characteristic polynomial t^7+69t^5+86t^3+7t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + 5
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1152*K1**4*K2**2 + 1600*K1**4*K2 - 2784*K1**4 + 864*K1**3*K2*K3 + 32*K1**3*K3*K4 - 352*K1**3*K3 - 960*K1**2*K2**4 + 3296*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 128*K1**2*K2**2*K4**2 + 320*K1**2*K2**2*K4 - 11232*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 992*K1**2*K2*K4 + 10832*K1**2*K2 - 352*K1**2*K3**2 - 176*K1**2*K4**2 - 6120*K1**2 + 2336*K1*K2**3*K3 + 672*K1*K2**2*K3*K4 - 2432*K1*K2**2*K3 + 224*K1*K2**2*K4*K5 - 704*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9512*K1*K2*K3 + 1064*K1*K3*K4 + 192*K1*K4*K5 - 64*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 768*K2**4*K4 - 3792*K2**4 + 160*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 1392*K2**2*K3**2 - 752*K2**2*K4**2 + 3024*K2**2*K4 - 112*K2**2*K5**2 - 16*K2**2*K6**2 - 3496*K2**2 + 704*K2*K3*K5 + 160*K2*K4*K6 + 8*K3**2*K6 - 2252*K3**2 - 704*K4**2 - 100*K5**2 - 24*K6**2 + 5086
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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