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Flat knot 6.342

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,2,3,3,1,1,2,2,-1,-1,-1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.342']
Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.136', '6.207', '6.342', '6.370', '6.376', '6.442', '6.456', '6.539', '6.631', '6.636', '6.674', '6.679', '6.705', '6.740', '6.760', '6.794', '6.795', '6.1369']
Outer characteristic polynomial of the knot is: t^7+58t^5+41t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.342']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 160*K1**4*K2 - 480*K1**4 + 32*K1**3*K2*K3 - 272*K1**2*K2**2 + 600*K1**2*K2 - 128*K1**2*K3**2 - 48*K1**2*K4**2 - 220*K1**2 + 384*K1*K2*K3 + 168*K1*K3*K4 + 40*K1*K4*K5 - 16*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 32*K2**2*K4 - 294*K2**2 + 24*K2*K3*K5 + 8*K2*K4*K6 - 164*K3**2 - 64*K4**2 - 16*K5**2 - 2*K6**2 + 342
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.342']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11246', 'vk6.11324', 'vk6.12507', 'vk6.12618', 'vk6.13878', 'vk6.13973', 'vk6.14140', 'vk6.14365', 'vk6.14949', 'vk6.15070', 'vk6.15596', 'vk6.16068', 'vk6.16359', 'vk6.16400', 'vk6.17440', 'vk6.22591', 'vk6.22622', 'vk6.22763', 'vk6.23948', 'vk6.24089', 'vk6.24181', 'vk6.25976', 'vk6.26144', 'vk6.26366', 'vk6.28318', 'vk6.30928', 'vk6.31051', 'vk6.31224', 'vk6.31573', 'vk6.33689', 'vk6.34644', 'vk6.34712', 'vk6.34732', 'vk6.35562', 'vk6.36011', 'vk6.36244', 'vk6.37649', 'vk6.38080', 'vk6.39942', 'vk6.40129', 'vk6.42275', 'vk6.44549', 'vk6.44566', 'vk6.44797', 'vk6.52012', 'vk6.54393', 'vk6.56535', 'vk6.59049', 'vk6.59147', 'vk6.64561']
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 O1O2O3O4O5U3U1O6U5U6U4U2
R3 orbit {'O1O2O3O4O5U3U1U4O6U5U6U2', 'O1O2O3O4U5U1O6U4U6U3O5U2', 'O1O2O3O4O5U3U1O6U5U6U4U2', 'O1O2O3O4U5U1U3O6U4U6O5U2'}
R3 orbit length 4
Gauss code of -K O1O2O3O4O5U4U2U6U1O6U5U3
Gauss code of K* O1O2O3O4U5U4U6U3U1O6O5U2
Gauss code of -K* O1O2O3O4U3O5O6U4U2U6U1U5
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 1 -2 2 1 1],[ 3 0 3 0 3 2 1],[-1 -3 0 -2 1 0 1],[ 2 0 2 0 2 1 1],[-2 -3 -1 -2 0 -1 1],[-1 -2 0 -1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 1 -2 -3],[-2 0 1 -1 -1 -2 -3],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 0 -1 -2],[-1 1 1 0 0 -2 -3],[ 2 2 1 1 2 0 0],[ 3 3 1 2 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,2,3,-1,1,1,2,3,1,1,1,1,0,1,2,2,3,0]
Phi over symmetry [-3,-2,1,1,1,2,0,1,2,3,3,1,1,2,2,-1,-1,-1,0,1,1]
Phi of -K [-3,-2,1,1,1,2,1,1,2,3,2,1,2,2,2,0,-1,0,-1,0,2]
Phi of K* [-2,-1,-1,-1,2,3,0,0,2,2,2,0,1,1,1,1,2,2,2,3,1]
Phi of -K* [-3,-2,1,1,1,2,0,1,2,3,3,1,1,2,2,-1,-1,-1,0,1,1]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 6z+13
Enhanced Jones-Krushkal polynomial 6w^2z+13w
Inner characteristic polynomial t^6+38t^4+11t^2
Outer characteristic polynomial t^7+58t^5+41t^3
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -64*K1**6 + 160*K1**4*K2 - 480*K1**4 + 32*K1**3*K2*K3 - 272*K1**2*K2**2 + 600*K1**2*K2 - 128*K1**2*K3**2 - 48*K1**2*K4**2 - 220*K1**2 + 384*K1*K2*K3 + 168*K1*K3*K4 + 40*K1*K4*K5 - 16*K2**4 - 16*K2**2*K3**2 - 8*K2**2*K4**2 + 32*K2**2*K4 - 294*K2**2 + 24*K2*K3*K5 + 8*K2*K4*K6 - 164*K3**2 - 64*K4**2 - 16*K5**2 - 2*K6**2 + 342
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{5, 6}, {4}, {3}, {2}, {1}]]
If K is slice False
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