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Flat knot 6.1535

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,2,1,3,1,1,0,1,0,0,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1535']
Arrow polynomial of the knot is: -16*K1**2 - 4*K1*K2 + 2*K1 + 8*K2 + 2*K3 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.610', '6.1535']
Outer characteristic polynomial of the knot is: t^7+31t^5+36t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1535', '6.1830']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 2112*K1**4*K2 - 5824*K1**4 + 1024*K1**3*K2*K3 + 64*K1**3*K3*K4 - 864*K1**3*K3 + 128*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6240*K1**2*K2**2 - 704*K1**2*K2*K4 + 11784*K1**2*K2 - 832*K1**2*K3**2 - 128*K1**2*K4**2 - 6372*K1**2 - 1056*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 8416*K1*K2*K3 + 1704*K1*K3*K4 + 168*K1*K4*K5 - 320*K2**4 - 144*K2**2*K3**2 - 48*K2**2*K4**2 + 1456*K2**2*K4 - 6348*K2**2 + 344*K2*K3*K5 + 32*K2*K4*K6 - 2848*K3**2 - 952*K4**2 - 124*K5**2 - 4*K6**2 + 6182
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1535']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3572', 'vk6.3598', 'vk6.3602', 'vk6.3819', 'vk6.3823', 'vk6.3852', 'vk6.3856', 'vk6.6985', 'vk6.6997', 'vk6.7018', 'vk6.7030', 'vk6.7203', 'vk6.7215', 'vk6.7233', 'vk6.15332', 'vk6.15335', 'vk6.15459', 'vk6.15460', 'vk6.33969', 'vk6.34012', 'vk6.34015', 'vk6.34427', 'vk6.48226', 'vk6.48238', 'vk6.48381', 'vk6.49962', 'vk6.49988', 'vk6.49992', 'vk6.53996', 'vk6.53997', 'vk6.54050', 'vk6.54497']
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 O1O2O3U2O4U5U4O6O5U1U6U3
R3 orbit {'O1O2O3U2O4U5U4O6O5U1U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U3O5O4U6U5O6U2
Gauss code of K* O1O2O3U1U4U3O4U5O6O5U2U6
Gauss code of -K* O1O2O3U4U2O5O4U5O6U1U6U3
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 0 0],[ 2 0 0 3 1 1 0],[ 1 0 0 1 0 1 0],[-2 -3 -1 0 1 -2 -1],[-1 -1 0 -1 0 -1 -1],[ 0 -1 -1 2 1 0 0],[ 0 0 0 1 1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 1 -1 -2 -1 -3],[-1 -1 0 -1 -1 0 -1],[ 0 1 1 0 0 0 0],[ 0 2 1 0 0 -1 -1],[ 1 1 0 0 1 0 0],[ 2 3 1 0 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,-1,1,2,1,3,1,1,0,1,0,0,0,1,1,0]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,2,1,3,1,1,0,1,0,0,0,1,1,0]
Phi of -K [-2,-1,0,0,1,2,1,1,2,2,1,0,1,2,2,0,0,0,0,1,2]
Phi of K* [-2,-1,0,0,1,2,2,0,1,2,1,0,0,2,2,0,0,1,1,2,1]
Phi of -K* [-2,-1,0,0,1,2,0,0,1,1,3,0,1,0,1,0,1,1,1,2,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+21t^4+16t^2+1
Outer characteristic polynomial t^7+31t^5+36t^3+4t
Flat arrow polynomial -16*K1**2 - 4*K1*K2 + 2*K1 + 8*K2 + 2*K3 + 9
2-strand cable arrow polynomial -384*K1**4*K2**2 + 2112*K1**4*K2 - 5824*K1**4 + 1024*K1**3*K2*K3 + 64*K1**3*K3*K4 - 864*K1**3*K3 + 128*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6240*K1**2*K2**2 - 704*K1**2*K2*K4 + 11784*K1**2*K2 - 832*K1**2*K3**2 - 128*K1**2*K4**2 - 6372*K1**2 - 1056*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 8416*K1*K2*K3 + 1704*K1*K3*K4 + 168*K1*K4*K5 - 320*K2**4 - 144*K2**2*K3**2 - 48*K2**2*K4**2 + 1456*K2**2*K4 - 6348*K2**2 + 344*K2*K3*K5 + 32*K2*K4*K6 - 2848*K3**2 - 952*K4**2 - 124*K5**2 - 4*K6**2 + 6182
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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