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Flat knot 6.1129

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,0,0,2,2,-1,1,1,3,1,1,1,-1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1129']
Arrow polynomial of the knot is: -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.546', '6.591', '6.598', '6.666', '6.680', '6.742', '6.778', '6.805', '6.822', '6.824', '6.1129', '6.1512', '6.1647', '6.1678', '6.1705', '6.1847', '6.1857']
Outer characteristic polynomial of the knot is: t^7+38t^5+129t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1129']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 512*K1**4*K2**2 + 1024*K1**4*K2 - 3168*K1**4 + 576*K1**3*K2*K3 - 4320*K1**2*K2**2 + 5184*K1**2*K2 - 896*K1**2*K3**2 - 1240*K1**2 + 4512*K1*K2*K3 + 784*K1*K3*K4 + 16*K1*K4*K5 - 1200*K2**4 - 544*K2**2*K3**2 - 16*K2**2*K4**2 + 1200*K2**2*K4 - 2108*K2**2 + 704*K2*K3*K5 + 32*K2*K4*K6 - 32*K3**4 + 80*K3**2*K6 - 1432*K3**2 - 524*K4**2 - 224*K5**2 - 44*K6**2 + 2658
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1129']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11566', 'vk6.11906', 'vk6.12917', 'vk6.13223', 'vk6.20956', 'vk6.22374', 'vk6.28419', 'vk6.31358', 'vk6.31765', 'vk6.32524', 'vk6.32923', 'vk6.40137', 'vk6.42148', 'vk6.46648', 'vk6.52351', 'vk6.52614', 'vk6.53485', 'vk6.58951', 'vk6.64484', 'vk6.69789']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2O3O4U5U2O5U1U4O6U3U6
R3 orbit {'O1O2O3O4U5U2O5U1U4O6U3U6'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2U1O3O4U5O6O5U3U2U6U4
Gauss code of -K* O1O2U1O3O4U5O6O5U3U2U6U4
Diagrammatic symmetry type r
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -1 1 2 -1 1],[ 2 0 1 3 2 1 1],[ 1 -1 0 1 0 1 1],[-1 -3 -1 0 0 -1 1],[-2 -2 0 0 0 -2 0],[ 1 -1 -1 1 2 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 0 0 -2 -2],[-1 0 0 1 -1 -1 -3],[-1 0 -1 0 -1 -1 -1],[ 1 0 1 1 0 1 -1],[ 1 2 1 1 -1 0 -1],[ 2 2 3 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,0,0,2,2,-1,1,1,3,1,1,1,-1,1,1]
Phi over symmetry [-2,-1,-1,1,1,2,0,0,0,2,2,-1,1,1,3,1,1,1,-1,1,1]
Phi of -K [-2,-1,-1,1,1,2,0,0,0,2,2,-1,1,1,3,1,1,1,-1,1,1]
Phi of K* [-2,-1,-1,1,1,2,1,1,1,3,2,-1,1,1,2,1,1,0,-1,0,0]
Phi of -K* [-2,-1,-1,1,1,2,1,1,1,3,2,-1,1,1,2,1,1,0,-1,0,0]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 16z+33
Enhanced Jones-Krushkal polynomial 16w^2z+33w
Inner characteristic polynomial t^6+26t^4+69t^2
Outer characteristic polynomial t^7+38t^5+129t^3
Flat arrow polynomial -12*K1**2 - 4*K1*K2 + 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -512*K1**6 - 512*K1**4*K2**2 + 1024*K1**4*K2 - 3168*K1**4 + 576*K1**3*K2*K3 - 4320*K1**2*K2**2 + 5184*K1**2*K2 - 896*K1**2*K3**2 - 1240*K1**2 + 4512*K1*K2*K3 + 784*K1*K3*K4 + 16*K1*K4*K5 - 1200*K2**4 - 544*K2**2*K3**2 - 16*K2**2*K4**2 + 1200*K2**2*K4 - 2108*K2**2 + 704*K2*K3*K5 + 32*K2*K4*K6 - 32*K3**4 + 80*K3**2*K6 - 1432*K3**2 - 524*K4**2 - 224*K5**2 - 44*K6**2 + 2658
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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