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Flat knot 6.1704

Min(phi) over symmetries of the knot is: [-2,0,0,2,0,1,3,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1515', '6.1704']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1209', '6.1245', '6.1509', '6.1541', '6.1704', '6.1778', '6.1914']
Outer characteristic polynomial of the knot is: t^5+16t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1704']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 960*K1**4*K2**2 + 2016*K1**4*K2 - 3408*K1**4 + 640*K1**3*K2*K3 - 192*K1**3*K3 + 1504*K1**2*K2**3 - 6944*K1**2*K2**2 - 352*K1**2*K2*K4 + 8072*K1**2*K2 - 848*K1**2*K3**2 - 288*K1**2*K4**2 - 3988*K1**2 + 864*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 160*K1*K2**2*K5 - 672*K1*K2*K3*K4 + 6992*K1*K2*K3 - 96*K1*K2*K4*K5 + 1912*K1*K3*K4 + 512*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1632*K2**4 - 32*K2**3*K6 - 1296*K2**2*K3**2 - 320*K2**2*K4**2 + 2048*K2**2*K4 - 3514*K2**2 - 64*K2*K3**2*K4 + 1048*K2*K3*K5 + 232*K2*K4*K6 - 2120*K3**2 - 1012*K4**2 - 284*K5**2 - 22*K6**2 + 4186
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1704']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4692', 'vk6.4995', 'vk6.6178', 'vk6.6649', 'vk6.8175', 'vk6.8593', 'vk6.9561', 'vk6.9900', 'vk6.17384', 'vk6.20920', 'vk6.20984', 'vk6.22330', 'vk6.22406', 'vk6.23551', 'vk6.23888', 'vk6.28396', 'vk6.36144', 'vk6.40058', 'vk6.40185', 'vk6.42109', 'vk6.43055', 'vk6.43359', 'vk6.46586', 'vk6.46690', 'vk6.48724', 'vk6.49512', 'vk6.49715', 'vk6.51424', 'vk6.55542', 'vk6.58926', 'vk6.65288', 'vk6.69772']
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 O1O2O3U1U3O4O5U6U2O6U4U5
R3 orbit {'O1O2O3U1U3O4O5U6U2O6U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6U2U6O4O5U1U3
Gauss code of K* O1O2U1O3O4U5U2U6O5O6U3U4
Gauss code of -K* O1O2U3O4O3U1U2O5O6U5U4U6
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 0 1 0 2 -1],[ 2 0 2 1 1 1 1],[ 0 -2 0 0 0 1 0],[-1 -1 0 0 0 0 -1],[ 0 -1 0 0 0 1 0],[-2 -1 -1 0 -1 0 -2],[ 1 -1 0 1 0 2 0]]
Primitive based matrix [[ 0 2 0 0 -2],[-2 0 -1 -1 -1],[ 0 1 0 0 -1],[ 0 1 0 0 -2],[ 2 1 1 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,0,0,2,1,1,1,0,1,2]
Phi over symmetry [-2,0,0,2,0,1,3,0,1,1]
Phi of -K [-2,0,0,2,0,1,3,0,1,1]
Phi of K* [-2,0,0,2,1,1,3,0,0,1]
Phi of -K* [-2,0,0,2,1,2,1,0,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^4+8t^2+1
Outer characteristic polynomial t^5+16t^3+5t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
2-strand cable arrow polynomial -256*K1**6 - 960*K1**4*K2**2 + 2016*K1**4*K2 - 3408*K1**4 + 640*K1**3*K2*K3 - 192*K1**3*K3 + 1504*K1**2*K2**3 - 6944*K1**2*K2**2 - 352*K1**2*K2*K4 + 8072*K1**2*K2 - 848*K1**2*K3**2 - 288*K1**2*K4**2 - 3988*K1**2 + 864*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 - 160*K1*K2**2*K5 - 672*K1*K2*K3*K4 + 6992*K1*K2*K3 - 96*K1*K2*K4*K5 + 1912*K1*K3*K4 + 512*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 1632*K2**4 - 32*K2**3*K6 - 1296*K2**2*K3**2 - 320*K2**2*K4**2 + 2048*K2**2*K4 - 3514*K2**2 - 64*K2*K3**2*K4 + 1048*K2*K3*K5 + 232*K2*K4*K6 - 2120*K3**2 - 1012*K4**2 - 284*K5**2 - 22*K6**2 + 4186
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {5}, {4}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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