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Flat knot 6.1245

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,1,1,1,3,0,1,1,1,0,0,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.1245']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1209', '6.1245', '6.1509', '6.1541', '6.1704', '6.1778', '6.1914']
Outer characteristic polynomial of the knot is: t^7+31t^5+33t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1245']
2-strand cable arrow polynomial of the knot is: -320*K1**6 - 320*K1**4*K2**2 + 1696*K1**4*K2 - 4576*K1**4 + 416*K1**3*K2*K3 + 32*K1**3*K3*K4 - 416*K1**3*K3 + 512*K1**2*K2**3 - 4944*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 9296*K1**2*K2 - 1248*K1**2*K3**2 - 224*K1**2*K3*K5 - 352*K1**2*K4**2 - 6016*K1**2 + 288*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 928*K1*K2**2*K3 - 288*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 7608*K1*K2*K3 + 3232*K1*K3*K4 + 928*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 864*K2**4 - 32*K2**3*K6 - 576*K2**2*K3**2 - 256*K2**2*K4**2 + 1840*K2**2*K4 - 5346*K2**2 - 128*K2*K3**2*K4 + 904*K2*K3*K5 + 264*K2*K4*K6 + 32*K3**2*K6 - 3168*K3**2 - 1768*K4**2 - 512*K5**2 - 54*K6**2 + 6150
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1245']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16580', 'vk6.16671', 'vk6.18137', 'vk6.18471', 'vk6.22983', 'vk6.23102', 'vk6.24596', 'vk6.25007', 'vk6.34972', 'vk6.35091', 'vk6.36727', 'vk6.37144', 'vk6.42545', 'vk6.42654', 'vk6.43999', 'vk6.44309', 'vk6.54811', 'vk6.54893', 'vk6.55939', 'vk6.56233', 'vk6.59243', 'vk6.59318', 'vk6.60477', 'vk6.60837', 'vk6.64785', 'vk6.64848', 'vk6.65592', 'vk6.65897', 'vk6.68087', 'vk6.68150', 'vk6.68667', 'vk6.68876']
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 O1O2O3O4U3U5U2O6O5U1U4U6
R3 orbit {'O1O2O3O4U3U5U2O6O5U1U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U4O6O5U3U6U2
Gauss code of K* O1O2O3U4U2O5O6O4U5U3U1U6
Gauss code of -K* O1O2O3U4U1O5O4O6U2U6U5U3
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 0 -1 2 0 1],[ 2 0 1 0 3 1 1],[ 0 -1 0 0 1 0 0],[ 1 0 0 0 1 1 1],[-2 -3 -1 -1 0 -2 0],[ 0 -1 0 -1 2 0 1],[-1 -1 0 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 0 -1 -2 -1 -3],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 0 0 -1],[ 0 2 1 0 0 -1 -1],[ 1 1 1 0 1 0 0],[ 2 3 1 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,0,1,2,1,3,0,1,1,1,0,0,1,1,1,0]
Phi over symmetry [-2,-1,0,0,1,2,0,1,1,1,3,0,1,1,1,0,0,1,1,2,0]
Phi of -K [-2,-1,0,0,1,2,1,1,1,2,1,0,1,1,2,0,0,0,1,1,1]
Phi of K* [-2,-1,0,0,1,2,1,0,1,2,1,0,1,1,2,0,0,1,1,1,1]
Phi of -K* [-2,-1,0,0,1,2,0,1,1,1,3,0,1,1,1,0,0,1,1,2,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+21t^4+19t^2+1
Outer characteristic polynomial t^7+31t^5+33t^3+4t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
2-strand cable arrow polynomial -320*K1**6 - 320*K1**4*K2**2 + 1696*K1**4*K2 - 4576*K1**4 + 416*K1**3*K2*K3 + 32*K1**3*K3*K4 - 416*K1**3*K3 + 512*K1**2*K2**3 - 4944*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 9296*K1**2*K2 - 1248*K1**2*K3**2 - 224*K1**2*K3*K5 - 352*K1**2*K4**2 - 6016*K1**2 + 288*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 928*K1*K2**2*K3 - 288*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 7608*K1*K2*K3 + 3232*K1*K3*K4 + 928*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 864*K2**4 - 32*K2**3*K6 - 576*K2**2*K3**2 - 256*K2**2*K4**2 + 1840*K2**2*K4 - 5346*K2**2 - 128*K2*K3**2*K4 + 904*K2*K3*K5 + 264*K2*K4*K6 + 32*K3**2*K6 - 3168*K3**2 - 1768*K4**2 - 512*K5**2 - 54*K6**2 + 6150
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{6}, {4, 5}, {2, 3}, {1}]]
If K is slice False
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