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Flat knot 6.1565

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,2,1,1,0,2,0,2,-1,1,0,1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1565']
Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 4*K1*K2 - K1 + 6*K2 + K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1345', '6.1510', '6.1565', '6.1691', '6.1812']
Outer characteristic polynomial of the knot is: t^7+41t^5+245t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1565']
2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 384*K1**4*K2 - 1440*K1**4 + 128*K1**3*K2*K3 - 288*K1**3*K3 + 352*K1**2*K2**3 - 3312*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 6736*K1**2*K2 - 160*K1**2*K3**2 - 16*K1**2*K4**2 - 5236*K1**2 + 128*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 160*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5280*K1*K2*K3 + 864*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 608*K2**4 - 32*K2**3*K6 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 1320*K2**2*K4 - 4174*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 1880*K3**2 - 608*K4**2 - 84*K5**2 - 2*K6**2 + 4070
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1565']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71369', 'vk6.71430', 'vk6.71895', 'vk6.71956', 'vk6.72441', 'vk6.72589', 'vk6.72708', 'vk6.72802', 'vk6.72867', 'vk6.73021', 'vk6.74242', 'vk6.74367', 'vk6.74442', 'vk6.74871', 'vk6.75055', 'vk6.76631', 'vk6.76920', 'vk6.77034', 'vk6.77408', 'vk6.77741', 'vk6.77794', 'vk6.79292', 'vk6.79413', 'vk6.79766', 'vk6.79831', 'vk6.79898', 'vk6.80863', 'vk6.80924', 'vk6.81381', 'vk6.85522', 'vk6.87210', 'vk6.89268']
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 O1O2O3U4O5U6U2O4O6U1U5U3
R3 orbit {'O1O2O3U4O5U6U2O4O6U1U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U3O5O6U2U5O4U6
Gauss code of K* O1O2O3U1U4U3O5U2O6O4U5U6
Gauss code of -K* O1O2O3U4U5O6O4U2O5U1U6U3
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 2 -1 1 0],[ 2 0 2 3 0 1 2],[ 0 -2 0 0 0 -1 1],[-2 -3 0 0 -2 -1 -1],[ 1 0 0 2 0 2 0],[-1 -1 1 1 -2 0 -1],[ 0 -2 -1 1 0 1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 0 -1 -2 -3],[-1 1 0 1 -1 -2 -1],[ 0 0 -1 0 1 0 -2],[ 0 1 1 -1 0 0 -2],[ 1 2 2 0 0 0 0],[ 2 3 1 2 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,0,1,2,3,-1,1,2,1,-1,0,2,0,2,0]
Phi over symmetry [-2,-1,0,0,1,2,0,1,2,1,1,0,2,0,2,-1,1,0,1,0,1]
Phi of -K [-2,-1,0,0,1,2,1,0,0,2,1,1,1,0,1,-1,2,2,0,1,0]
Phi of K* [-2,-1,0,0,1,2,0,1,2,1,1,0,2,0,2,-1,1,0,1,0,1]
Phi of -K* [-2,-1,0,0,1,2,0,2,2,1,3,0,0,2,2,-1,1,1,-1,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+31t^4+163t^2
Outer characteristic polynomial t^7+41t^5+245t^3+4t
Flat arrow polynomial 4*K1**3 - 12*K1**2 - 4*K1*K2 - K1 + 6*K2 + K3 + 7
2-strand cable arrow polynomial -64*K1**4*K2**2 + 384*K1**4*K2 - 1440*K1**4 + 128*K1**3*K2*K3 - 288*K1**3*K3 + 352*K1**2*K2**3 - 3312*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 6736*K1**2*K2 - 160*K1**2*K3**2 - 16*K1**2*K4**2 - 5236*K1**2 + 128*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 160*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5280*K1*K2*K3 + 864*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 608*K2**4 - 32*K2**3*K6 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 1320*K2**2*K4 - 4174*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 1880*K3**2 - 608*K4**2 - 84*K5**2 - 2*K6**2 + 4070
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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