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Flat knot 6.192

Min(phi) over symmetries of the knot is: [-4,-3,1,2,2,2,0,2,2,4,5,1,1,2,3,1,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.192']
Arrow polynomial of the knot is: 4*K1**3 - 6*K1*K2 - 2*K1*K3 + K2 + 2*K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.192', '6.251']
Outer characteristic polynomial of the knot is: t^7+107t^5+135t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.192']
2-strand cable arrow polynomial of the knot is: -64*K1**4 + 32*K1**3*K2*K3 - 128*K1**3*K3 + 96*K1**2*K2**3 - 1664*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 32*K1**2*K2*K4 + 3672*K1**2*K2 - 224*K1**2*K3**2 - 3988*K1**2 + 512*K1*K2**3*K3 - 1472*K1*K2**2*K3 - 192*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 4632*K1*K2*K3 - 32*K1*K2*K4*K5 + 920*K1*K3*K4 + 224*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 688*K2**4 + 32*K2**3*K3*K5 - 128*K2**3*K6 - 592*K2**2*K3**2 - 32*K2**2*K3*K7 - 24*K2**2*K4**2 + 1624*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3544*K2**2 + 720*K2*K3*K5 + 160*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 + 16*K3**2*K6 - 1984*K3**2 - 758*K4**2 - 300*K5**2 - 72*K6**2 - 8*K7**2 - 2*K8**2 + 3366
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.192']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71628', 'vk6.71795', 'vk6.72215', 'vk6.72355', 'vk6.73408', 'vk6.73602', 'vk6.73886', 'vk6.74274', 'vk6.74898', 'vk6.75379', 'vk6.75688', 'vk6.75887', 'vk6.76449', 'vk6.77248', 'vk6.77334', 'vk6.77582', 'vk6.77686', 'vk6.78341', 'vk6.78878', 'vk6.79320', 'vk6.80121', 'vk6.80301', 'vk6.80428', 'vk6.80781', 'vk6.82017', 'vk6.82755', 'vk6.85357', 'vk6.86692', 'vk6.86935', 'vk6.87034', 'vk6.87598', 'vk6.89476']
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 O1O2O3O4O5U2O6U1U5U6U4U3
R3 orbit {'O1O2O3O4O5U2O6U1U5U6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U2U6U1U5O6U4
Gauss code of K* O1O2O3O4O5U1U6U5U4U2O6U3
Gauss code of -K* O1O2O3O4O5U3O6U4U2U1U6U5
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 -4 -3 2 2 1 2],[ 4 0 0 5 4 2 2],[ 3 0 0 3 2 1 1],[-2 -5 -3 0 0 -1 1],[-2 -4 -2 0 0 -1 1],[-1 -2 -1 1 1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 2 2 1 -3 -4],[-2 0 1 0 -1 -2 -4],[-2 -1 0 -1 -1 -1 -2],[-2 0 1 0 -1 -3 -5],[-1 1 1 1 0 -1 -2],[ 3 2 1 3 1 0 0],[ 4 4 2 5 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-2,-1,3,4,-1,0,1,2,4,1,1,1,2,1,3,5,1,2,0]
Phi over symmetry [-4,-3,1,2,2,2,0,2,2,4,5,1,1,2,3,1,1,1,-1,-1,0]
Phi of -K [-4,-3,1,2,2,2,1,3,1,2,4,3,2,3,4,0,0,0,0,-1,-1]
Phi of K* [-2,-2,-2,-1,3,4,-1,-1,0,4,4,0,0,2,1,0,3,2,3,3,1]
Phi of -K* [-4,-3,1,2,2,2,0,2,2,4,5,1,1,2,3,1,1,1,-1,-1,0]
Symmetry type of based matrix c
u-polynomial t^4+t^3-3t^2-t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+69t^4+12t^2
Outer characteristic polynomial t^7+107t^5+135t^3+4t
Flat arrow polynomial 4*K1**3 - 6*K1*K2 - 2*K1*K3 + K2 + 2*K3 + K4 + 1
2-strand cable arrow polynomial -64*K1**4 + 32*K1**3*K2*K3 - 128*K1**3*K3 + 96*K1**2*K2**3 - 1664*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 32*K1**2*K2*K4 + 3672*K1**2*K2 - 224*K1**2*K3**2 - 3988*K1**2 + 512*K1*K2**3*K3 - 1472*K1*K2**2*K3 - 192*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 4632*K1*K2*K3 - 32*K1*K2*K4*K5 + 920*K1*K3*K4 + 224*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 688*K2**4 + 32*K2**3*K3*K5 - 128*K2**3*K6 - 592*K2**2*K3**2 - 32*K2**2*K3*K7 - 24*K2**2*K4**2 + 1624*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 3544*K2**2 + 720*K2*K3*K5 + 160*K2*K4*K6 + 40*K2*K5*K7 + 8*K2*K6*K8 + 16*K3**2*K6 - 1984*K3**2 - 758*K4**2 - 300*K5**2 - 72*K6**2 - 8*K7**2 - 2*K8**2 + 3366
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
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