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Flat knot 6.1367

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-2,1,1,2,3,0,1,2,2,0,0,-1,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1367']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.927', '6.1364', '6.1367', '6.1540', '6.1675', '6.1779', '6.1811', '6.1876', '6.2075']
Outer characteristic polynomial of the knot is: t^7+42t^5+65t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1367']
2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 7680*K1**4*K2 - 9696*K1**4 - 768*K1**3*K2**2*K3 + 3072*K1**3*K2*K3 - 2688*K1**3*K3 - 256*K1**2*K2**4 + 4800*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 15392*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 1856*K1**2*K2*K4 + 12176*K1**2*K2 - 2432*K1**2*K3**2 - 128*K1**2*K3*K5 - 64*K1**2*K4**2 - 1792*K1**2 + 1856*K1*K2**3*K3 - 3840*K1*K2**2*K3 - 384*K1*K2**2*K5 + 256*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 10912*K1*K2*K3 + 1984*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 2768*K2**4 - 1696*K2**2*K3**2 - 96*K2**2*K4**2 + 2384*K2**2*K4 - 2796*K2**2 + 816*K2*K3*K5 + 32*K2*K4*K6 - 160*K3**4 + 64*K3**2*K6 - 1656*K3**2 - 396*K4**2 - 40*K5**2 - 4*K6**2 + 3578
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1367']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3218', 'vk6.3228', 'vk6.3242', 'vk6.3325', 'vk6.3340', 'vk6.3358', 'vk6.3442', 'vk6.3499', 'vk6.15214', 'vk6.15234', 'vk6.15245', 'vk6.15260', 'vk6.33872', 'vk6.33886', 'vk6.33897', 'vk6.34330', 'vk6.48092', 'vk6.48096', 'vk6.48156', 'vk6.54447']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3U2O4O5U4U6U5O6U1U3
R3 orbit {'O1O2O3U2O4O5U4U6U5O6U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U3O4U5U4U6O5O6U2
Gauss code of K* Same
Gauss code of -K* O1O2O3U1U3O4U5U4U6O5O6U2
Diagrammatic symmetry type +
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -1 2 -1 2 -1],[ 1 0 0 2 -1 2 0],[ 1 0 0 1 0 0 1],[-2 -2 -1 0 -1 2 -3],[ 1 1 0 1 0 1 0],[-2 -2 0 -2 -1 0 -2],[ 1 0 -1 3 0 2 0]]
Primitive based matrix [[ 0 2 2 -1 -1 -1 -1],[-2 0 2 -1 -1 -2 -3],[-2 -2 0 0 -1 -2 -2],[ 1 1 0 0 0 0 1],[ 1 1 1 0 0 1 0],[ 1 2 2 0 -1 0 0],[ 1 3 2 -1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,1,1,1,1,-2,1,1,2,3,0,1,2,2,0,0,-1,-1,0,0]
Phi over symmetry [-2,-2,1,1,1,1,-2,1,1,2,3,0,1,2,2,0,0,-1,-1,0,0]
Phi of -K [-1,-1,-1,-1,2,2,-1,0,0,2,2,0,0,1,1,-1,2,3,0,1,-2]
Phi of K* [-2,-2,1,1,1,1,-2,1,1,2,3,0,1,2,2,0,0,-1,-1,0,0]
Phi of -K* [-1,-1,-1,-1,2,2,-1,0,0,2,2,0,0,1,1,-1,2,3,0,1,-2]
Symmetry type of based matrix +
u-polynomial -2t^2+4t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+30t^4+39t^2
Outer characteristic polynomial t^7+42t^5+65t^3+4t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -1152*K1**4*K2**2 + 7680*K1**4*K2 - 9696*K1**4 - 768*K1**3*K2**2*K3 + 3072*K1**3*K2*K3 - 2688*K1**3*K3 - 256*K1**2*K2**4 + 4800*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 15392*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 1856*K1**2*K2*K4 + 12176*K1**2*K2 - 2432*K1**2*K3**2 - 128*K1**2*K3*K5 - 64*K1**2*K4**2 - 1792*K1**2 + 1856*K1*K2**3*K3 - 3840*K1*K2**2*K3 - 384*K1*K2**2*K5 + 256*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 10912*K1*K2*K3 + 1984*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 2768*K2**4 - 1696*K2**2*K3**2 - 96*K2**2*K4**2 + 2384*K2**2*K4 - 2796*K2**2 + 816*K2*K3*K5 + 32*K2*K4*K6 - 160*K3**4 + 64*K3**2*K6 - 1656*K3**2 - 396*K4**2 - 40*K5**2 - 4*K6**2 + 3578
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice False
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