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Flat knot 6.1908

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,1,2,1,1,2,2,1,1,0,1,0,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1908']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 5*K2 + K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1908']
Outer characteristic polynomial of the knot is: t^7+37t^5+74t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1908']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 288*K1**4*K2 - 832*K1**4 + 320*K1**3*K2*K3 - 64*K1**3*K3 + 768*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 288*K1**2*K2**2*K4 - 4832*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 704*K1**2*K2*K4 + 6152*K1**2*K2 - 256*K1**2*K3**2 - 176*K1**2*K4**2 - 4972*K1**2 + 896*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 + 224*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 + 64*K1*K2*K3**3 + 160*K1*K2*K3*K4**2 - 672*K1*K2*K3*K4 + 5712*K1*K2*K3 - 96*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 1376*K1*K3*K4 + 432*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1328*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 944*K2**2*K3**2 + 32*K2**2*K4**3 - 592*K2**2*K4**2 - 32*K2**2*K4*K8 + 1896*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 3326*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 568*K2*K3*K5 + 264*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 40*K3**2*K6 - 1900*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 962*K4**2 - 216*K5**2 - 42*K6**2 - 2*K8**2 + 3962
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1908']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3620', 'vk6.3697', 'vk6.3888', 'vk6.4003', 'vk6.7046', 'vk6.7093', 'vk6.7268', 'vk6.7377', 'vk6.17697', 'vk6.17744', 'vk6.24248', 'vk6.24307', 'vk6.36539', 'vk6.36614', 'vk6.43649', 'vk6.43754', 'vk6.48248', 'vk6.48321', 'vk6.48404', 'vk6.48427', 'vk6.50008', 'vk6.50047', 'vk6.50130', 'vk6.50151', 'vk6.55721', 'vk6.55776', 'vk6.60297', 'vk6.60378', 'vk6.65429', 'vk6.65456', 'vk6.68561', 'vk6.68588']
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 O1O2O3U1U2U4O5O4O6U5U3U6
R3 orbit {'O1O2O3U1U2U4O5O4O6U5U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U5O6O4O5U1U6U3
Gauss code of K* O1O2O3U4U3U5O4O6O5U1U2U6
Gauss code of -K* O1O2O3U1U4U3O4O5O6U2U5U6
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 1 -2 2],[ 2 0 1 2 2 0 1],[ 0 -1 0 1 0 0 1],[-1 -2 -1 0 0 -1 2],[-1 -2 0 0 0 -2 1],[ 2 0 0 1 2 0 1],[-2 -1 -1 -2 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -2 -2],[-2 0 -1 -2 -1 -1 -1],[-1 1 0 0 0 -2 -2],[-1 2 0 0 -1 -1 -2],[ 0 1 0 1 0 0 -1],[ 2 1 2 1 0 0 0],[ 2 1 2 2 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,2,2,1,2,1,1,1,0,0,2,2,1,1,2,0,1,0]
Phi over symmetry [-2,-2,0,1,1,2,0,0,1,2,1,1,2,2,1,1,0,1,0,2,1]
Phi of -K [-2,-2,0,1,1,2,0,1,1,1,3,2,1,2,3,1,0,1,0,0,-1]
Phi of K* [-2,-1,-1,0,2,2,-1,0,1,3,3,0,0,1,2,1,1,1,1,2,0]
Phi of -K* [-2,-2,0,1,1,2,0,0,1,2,1,1,2,2,1,1,0,1,0,2,1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 3z^2+20z+29
Enhanced Jones-Krushkal polynomial 3w^3z^2+20w^2z+29w
Inner characteristic polynomial t^6+23t^4+39t^2+1
Outer characteristic polynomial t^7+37t^5+74t^3+5t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 - 2*K2**2 + 5*K2 + K3 + K4 + 7
2-strand cable arrow polynomial -192*K1**4*K2**2 + 288*K1**4*K2 - 832*K1**4 + 320*K1**3*K2*K3 - 64*K1**3*K3 + 768*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 288*K1**2*K2**2*K4 - 4832*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 704*K1**2*K2*K4 + 6152*K1**2*K2 - 256*K1**2*K3**2 - 176*K1**2*K4**2 - 4972*K1**2 + 896*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 + 224*K1*K2**2*K4*K5 - 192*K1*K2**2*K5 + 64*K1*K2*K3**3 + 160*K1*K2*K3*K4**2 - 672*K1*K2*K3*K4 + 5712*K1*K2*K3 - 96*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 1376*K1*K3*K4 + 432*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1328*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 944*K2**2*K3**2 + 32*K2**2*K4**3 - 592*K2**2*K4**2 - 32*K2**2*K4*K8 + 1896*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 3326*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 568*K2*K3*K5 + 264*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 40*K3**2*K6 - 1900*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 962*K4**2 - 216*K5**2 - 42*K6**2 - 2*K8**2 + 3962
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}]]
If K is slice False
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