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Flat knot 6.296

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,0,2,3,3,4,1,2,2,2,1,2,3,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.296']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K3 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.166', '6.225', '6.296', '6.498']
Outer characteristic polynomial of the knot is: t^7+110t^5+96t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.296']
2-strand cable arrow polynomial of the knot is: -192*K1**4 + 384*K1**3*K2*K3 - 96*K1**3*K3 + 1088*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 2880*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 3136*K1**2*K2 - 352*K1**2*K3**2 - 2400*K1**2 + 256*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 + 32*K1*K2*K3**3 - 288*K1*K2*K3*K4 + 3192*K1*K2*K3 + 744*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1200*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 448*K2**2*K3**2 - 304*K2**2*K4**2 + 1376*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 1416*K2**2 - 32*K2*K3**2*K4 + 352*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 888*K3**2 - 414*K4**2 - 80*K5**2 - 16*K6**2 + 1788
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.296']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73981', 'vk6.73984', 'vk6.74502', 'vk6.74505', 'vk6.75960', 'vk6.75965', 'vk6.76714', 'vk6.76717', 'vk6.78948', 'vk6.78957', 'vk6.79495', 'vk6.79504', 'vk6.80479', 'vk6.80487', 'vk6.80954', 'vk6.80959', 'vk6.83008', 'vk6.83101', 'vk6.83659', 'vk6.83788', 'vk6.83942', 'vk6.84121', 'vk6.84270', 'vk6.85183', 'vk6.85538', 'vk6.85869', 'vk6.86255', 'vk6.86589', 'vk6.86750', 'vk6.87453', 'vk6.88307', 'vk6.89746']
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 O1O2O3O4O5U2U1O6U3U4U6U5
R3 orbit {'O1O2O3O4O5U2U1O6U3U4U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U6U2U3O6U5U4
Gauss code of K* O1O2O3O4U5U6U1U2U4O6O5U3
Gauss code of -K* O1O2O3O4U2O5O6U1U3U4U6U5
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 -3 -3 -1 1 4 2],[ 3 0 0 2 3 4 2],[ 3 0 0 1 2 3 2],[ 1 -2 -1 0 1 3 2],[-1 -3 -2 -1 0 2 1],[-4 -4 -3 -3 -2 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 4 2 1 -1 -3 -3],[-4 0 0 -2 -3 -3 -4],[-2 0 0 -1 -2 -2 -2],[-1 2 1 0 -1 -2 -3],[ 1 3 2 1 0 -1 -2],[ 3 3 2 2 1 0 0],[ 3 4 2 3 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,1,3,3,0,2,3,3,4,1,2,2,2,1,2,3,1,2,0]
Phi over symmetry [-4,-2,-1,1,3,3,0,2,3,3,4,1,2,2,2,1,2,3,1,2,0]
Phi of -K [-3,-3,-1,1,2,4,0,0,1,3,3,1,2,3,4,1,1,2,0,1,2]
Phi of K* [-4,-2,-1,1,3,3,2,1,2,3,4,0,1,3,3,1,1,2,0,1,0]
Phi of -K* [-3,-3,-1,1,2,4,0,1,2,2,3,2,3,2,4,1,2,3,1,2,0]
Symmetry type of based matrix c
u-polynomial -t^4+2t^3-t^2
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+70t^4+31t^2
Outer characteristic polynomial t^7+110t^5+96t^3+4t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K3 + K2 + 2
2-strand cable arrow polynomial -192*K1**4 + 384*K1**3*K2*K3 - 96*K1**3*K3 + 1088*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 2880*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 3136*K1**2*K2 - 352*K1**2*K3**2 - 2400*K1**2 + 256*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 1248*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 + 32*K1*K2*K3**3 - 288*K1*K2*K3*K4 + 3192*K1*K2*K3 + 744*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1200*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 448*K2**2*K3**2 - 304*K2**2*K4**2 + 1376*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 1416*K2**2 - 32*K2*K3**2*K4 + 352*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 888*K3**2 - 414*K4**2 - 80*K5**2 - 16*K6**2 + 1788
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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