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Flat knot 6.1675

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,4,0,0,1,1,1,2,2,0,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1675']
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+48t^5+94t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1675']
2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 6656*K1**4*K2 - 9664*K1**4 - 768*K1**3*K2**2*K3 + 3072*K1**3*K2*K3 - 1664*K1**3*K3 - 256*K1**2*K2**4 + 3264*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 512*K1**2*K2**2*K4 - 15008*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 1216*K1**2*K2*K4 + 12192*K1**2*K2 - 2112*K1**2*K3**2 - 64*K1**2*K4**2 - 1168*K1**2 + 1344*K1*K2**3*K3 - 3456*K1*K2**2*K3 - 448*K1*K2**2*K5 + 256*K1*K2*K3**3 - 576*K1*K2*K3*K4 + 10496*K1*K2*K3 + 1632*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 2160*K2**4 - 64*K2**3*K6 - 1440*K2**2*K3**2 - 32*K2**2*K4**2 + 2112*K2**2*K4 - 3132*K2**2 - 64*K2*K3**2*K4 + 720*K2*K3*K5 + 32*K2*K4*K6 - 128*K3**4 + 48*K3**2*K6 - 1672*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 :['6.1675']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3216', 'vk6.3226', 'vk6.3244', 'vk6.3327', 'vk6.3342', 'vk6.3356', 'vk6.3440', 'vk6.3501', 'vk6.15218', 'vk6.15230', 'vk6.15249', 'vk6.33868', 'vk6.33882', 'vk6.33899', 'vk6.34327', 'vk6.34337', 'vk6.48091', 'vk6.48158', 'vk6.48161', 'vk6.54450']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3U2O4U5U4U6O5O6U1U3
R3 orbit {'O1O2O3U2O4U5U4U6O5O6U1U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U3O4O5U4U6U5O6U2
Gauss code of K* O1O2O3U1U3O4O5U4U6U5O6U2
Gauss code of -K* Same
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 -1 2],[ 1 0 0 1 0 1 1],[-2 -2 -1 0 1 -4 -1],[-1 -1 0 -1 0 -1 0],[ 2 1 -1 4 1 0 2],[-1 -2 -1 1 0 -2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 1 -1 -1 -2 -4],[-1 -1 0 0 0 -1 -1],[-1 1 0 0 -1 -2 -2],[ 1 1 0 1 0 0 1],[ 1 2 1 2 0 0 -1],[ 2 4 1 2 -1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,-1,1,1,2,4,0,0,1,1,1,2,2,0,-1,1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,1,2,4,0,0,1,1,1,2,2,0,-1,1]
Phi of -K [-2,-1,-1,1,1,2,0,2,1,2,0,0,0,1,1,1,2,2,0,0,2]
Phi of K* [-2,-1,-1,1,1,2,0,2,1,2,0,0,0,1,1,1,2,2,0,0,2]
Phi of -K* [-2,-1,-1,1,1,2,-1,1,1,2,4,0,0,1,1,1,2,2,0,-1,1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+36t^4+68t^2+4
Outer characteristic polynomial t^7+48t^5+94t^3+8t
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 + 6656*K1**4*K2 - 9664*K1**4 - 768*K1**3*K2**2*K3 + 3072*K1**3*K2*K3 - 1664*K1**3*K3 - 256*K1**2*K2**4 + 3264*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 512*K1**2*K2**2*K4 - 15008*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 1216*K1**2*K2*K4 + 12192*K1**2*K2 - 2112*K1**2*K3**2 - 64*K1**2*K4**2 - 1168*K1**2 + 1344*K1*K2**3*K3 - 3456*K1*K2**2*K3 - 448*K1*K2**2*K5 + 256*K1*K2*K3**3 - 576*K1*K2*K3*K4 + 10496*K1*K2*K3 + 1632*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 2160*K2**4 - 64*K2**3*K6 - 1440*K2**2*K3**2 - 32*K2**2*K4**2 + 2112*K2**2*K4 - 3132*K2**2 - 64*K2*K3**2*K4 + 720*K2*K3*K5 + 32*K2*K4*K6 - 128*K3**4 + 48*K3**2*K6 - 1672*K3**2 - 396*K4**2 - 40*K5**2 - 4*K6**2 + 3578
Genus of based matrix 0
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}]]
If K is slice True
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