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Flat knot 6.1364

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,2,1,1,3,1,0,1,1,1,0,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1364']
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+36t^5+60t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1364']
2-strand cable arrow polynomial of the knot is: 1600*K1**4*K2 - 3616*K1**4 + 832*K1**3*K2*K3 - 672*K1**3*K3 - 128*K1**2*K2**4 + 800*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6912*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 8664*K1**2*K2 - 1152*K1**2*K3**2 - 32*K1**2*K4**2 - 3832*K1**2 + 480*K1*K2**3*K3 - 2080*K1*K2**2*K3 - 256*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 7440*K1*K2*K3 + 1536*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1264*K2**4 - 64*K2**3*K6 - 704*K2**2*K3**2 - 128*K2**2*K4**2 + 1872*K2**2*K4 - 3748*K2**2 + 632*K2*K3*K5 + 80*K2*K4*K6 - 1848*K3**2 - 604*K4**2 - 112*K5**2 - 12*K6**2 + 3738
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1364']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13897', 'vk6.13994', 'vk6.14186', 'vk6.14427', 'vk6.14968', 'vk6.15091', 'vk6.15654', 'vk6.16110', 'vk6.16710', 'vk6.16737', 'vk6.16832', 'vk6.18812', 'vk6.19281', 'vk6.19575', 'vk6.23148', 'vk6.23215', 'vk6.25410', 'vk6.26472', 'vk6.33716', 'vk6.33793', 'vk6.34272', 'vk6.35136', 'vk6.37539', 'vk6.42721', 'vk6.44698', 'vk6.54133', 'vk6.54915', 'vk6.54940', 'vk6.56398', 'vk6.56618', 'vk6.59342', 'vk6.64608']
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 O1O2O3U2O4O5U4U1U3O6U5U6
R3 orbit {'O1O2O3U2O4O5U4U1U3O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U1U3U6O5O6U2
Gauss code of K* O1O2O3U4O5O4U2U6U3O6U1U5
Gauss code of -K* O1O2O3U4U3O5U1U5U2O6O4U6
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 1 -1 2 1],[ 2 0 0 2 0 3 1],[ 1 0 0 1 0 1 0],[-1 -2 -1 0 0 2 1],[ 1 0 0 0 0 1 1],[-2 -3 -1 -2 -1 0 1],[-1 -1 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 1 -2 -1 -1 -3],[-1 -1 0 -1 0 -1 -1],[-1 2 1 0 -1 0 -2],[ 1 1 0 1 0 0 0],[ 1 1 1 0 0 0 0],[ 2 3 1 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,-1,2,1,1,3,1,0,1,1,1,0,2,0,0,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,2,1,1,3,1,0,1,1,1,0,2,0,0,0]
Phi of -K [-2,-1,-1,1,1,2,1,1,1,2,1,0,1,2,2,2,1,2,-1,-1,2]
Phi of K* [-2,-1,-1,1,1,2,-1,2,2,2,1,1,1,2,1,2,1,2,0,1,1]
Phi of -K* [-2,-1,-1,1,1,2,0,0,1,2,3,0,0,1,1,1,0,1,-1,-1,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+27z+27
Enhanced Jones-Krushkal polynomial 7w^3z^2+27w^2z+27w
Inner characteristic polynomial t^6+24t^4+20t^2
Outer characteristic polynomial t^7+36t^5+60t^3+6t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial 1600*K1**4*K2 - 3616*K1**4 + 832*K1**3*K2*K3 - 672*K1**3*K3 - 128*K1**2*K2**4 + 800*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6912*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 8664*K1**2*K2 - 1152*K1**2*K3**2 - 32*K1**2*K4**2 - 3832*K1**2 + 480*K1*K2**3*K3 - 2080*K1*K2**2*K3 - 256*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 7440*K1*K2*K3 + 1536*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1264*K2**4 - 64*K2**3*K6 - 704*K2**2*K3**2 - 128*K2**2*K4**2 + 1872*K2**2*K4 - 3748*K2**2 + 632*K2*K3*K5 + 80*K2*K4*K6 - 1848*K3**2 - 604*K4**2 - 112*K5**2 - 12*K6**2 + 3738
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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