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Flat knot 6.1922

Min(phi) over symmetries of the knot is: [-2,-1,1,2,-1,2,3,0,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1922']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 + 2*K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922']
Outer characteristic polynomial of the knot is: t^5+21t^3+21t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1922']
2-strand cable arrow polynomial of the knot is: 1280*K1**4*K2 - 2480*K1**4 - 480*K1**3*K3 + 32*K1**3*K4*K5 - 1760*K1**2*K2**2 - 352*K1**2*K2*K4 + 4640*K1**2*K2 - 16*K1**2*K3**2 - 192*K1**2*K3*K5 - 208*K1**2*K4**2 - 32*K1**2*K4*K6 - 64*K1**2*K5**2 - 3760*K1**2 - 288*K1*K2**2*K3 - 608*K1*K2*K3*K4 + 3160*K1*K2*K3 - 160*K1*K2*K4*K5 + 1992*K1*K3*K4 + 1120*K1*K4*K5 + 128*K1*K5*K6 - 32*K2**4 - 96*K2**2*K3**2 - 112*K2**2*K4**2 + 1256*K2**2*K4 - 3508*K2**2 + 760*K2*K3*K5 + 152*K2*K4*K6 - 1964*K3**2 - 1628*K4**2 - 652*K5**2 - 52*K6**2 + 3922
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1922']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3636', 'vk6.3729', 'vk6.3920', 'vk6.4019', 'vk6.7062', 'vk6.7125', 'vk6.7300', 'vk6.7393', 'vk6.11402', 'vk6.12589', 'vk6.12700', 'vk6.19105', 'vk6.19150', 'vk6.19803', 'vk6.25718', 'vk6.25777', 'vk6.26240', 'vk6.26685', 'vk6.31010', 'vk6.31137', 'vk6.32194', 'vk6.37825', 'vk6.37880', 'vk6.44961', 'vk6.48264', 'vk6.48443', 'vk6.50022', 'vk6.50165', 'vk6.52151', 'vk6.63733', 'vk6.66198', 'vk6.66225']
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 O1O2O3U1U4U3O4O5O6U2U6U5
R3 orbit {'O1O2O3U1U4U3O4O5O6U2U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3U5O4O6O5U2U1U6
Gauss code of K* O1O2O3U2U4U5O6O5O4U1U6U3
Gauss code of -K* O1O2O3U1U4U3O5O6O4U6U5U2
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 -1 2 -1 1 1],[ 2 0 2 1 1 1 1],[ 1 -2 0 1 0 2 1],[-2 -1 -1 0 -2 0 0],[ 1 -1 0 2 0 1 1],[-1 -1 -2 0 -1 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 2 1 -1 -2],[-2 0 0 -1 -1],[-1 0 0 -2 -1],[ 1 1 2 0 -2],[ 2 1 1 2 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,1,2,0,1,1,2,1,2]
Phi over symmetry [-2,-1,1,2,-1,2,3,0,2,1]
Phi of -K [-2,-1,1,2,-1,2,3,0,2,1]
Phi of K* [-2,-1,1,2,1,2,3,0,2,-1]
Phi of -K* [-2,-1,1,2,2,1,1,2,1,0]
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^4+11t^2+1
Outer characteristic polynomial t^5+21t^3+21t
Flat arrow polynomial -4*K1*K2 + 2*K1 + 2*K3 + 1
2-strand cable arrow polynomial 1280*K1**4*K2 - 2480*K1**4 - 480*K1**3*K3 + 32*K1**3*K4*K5 - 1760*K1**2*K2**2 - 352*K1**2*K2*K4 + 4640*K1**2*K2 - 16*K1**2*K3**2 - 192*K1**2*K3*K5 - 208*K1**2*K4**2 - 32*K1**2*K4*K6 - 64*K1**2*K5**2 - 3760*K1**2 - 288*K1*K2**2*K3 - 608*K1*K2*K3*K4 + 3160*K1*K2*K3 - 160*K1*K2*K4*K5 + 1992*K1*K3*K4 + 1120*K1*K4*K5 + 128*K1*K5*K6 - 32*K2**4 - 96*K2**2*K3**2 - 112*K2**2*K4**2 + 1256*K2**2*K4 - 3508*K2**2 + 760*K2*K3*K5 + 152*K2*K4*K6 - 1964*K3**2 - 1628*K4**2 - 652*K5**2 - 52*K6**2 + 3922
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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