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Flat knot 6.103

Min(phi) over symmetries of the knot is: [-4,-1,0,0,1,4,0,1,2,4,4,0,0,1,1,0,1,2,1,3,3]
Flat knots (up to 7 crossings) with same phi are :['6.103']
Arrow polynomial of the knot is: -4*K1*K2 - 4*K1*K3 + 2*K1 + 2*K2 + 2*K3 + 2*K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.103']
Outer characteristic polynomial of the knot is: t^7+97t^5+93t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.103']
2-strand cable arrow polynomial of the knot is: -1312*K1**4 + 704*K1**3*K2*K3 + 64*K1**3*K3*K4 - 448*K1**3*K3 + 192*K1**2*K2**2*K4 - 1728*K1**2*K2**2 - 704*K1**2*K2*K4 + 3952*K1**2*K2 - 608*K1**2*K3**2 - 256*K1**2*K4**2 - 3224*K1**2 - 768*K1*K2**2*K3 - 384*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4080*K1*K2*K3 + 1872*K1*K3*K4 + 608*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**4 - 128*K2**3*K6 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 1232*K2**2*K4 - 96*K2**2*K5**2 - 48*K2**2*K6**2 - 3412*K2**2 - 128*K2*K3**2*K4 + 896*K2*K3*K5 + 496*K2*K4*K6 + 96*K2*K5*K7 + 48*K2*K6*K8 + 96*K3**2*K6 - 2032*K3**2 - 1292*K4**2 - 560*K5**2 - 236*K6**2 - 24*K7**2 - 12*K8**2 + 3462
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.103']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16975', 'vk6.17216', 'vk6.20887', 'vk6.22294', 'vk6.23378', 'vk6.23678', 'vk6.28354', 'vk6.35432', 'vk6.35864', 'vk6.39998', 'vk6.42061', 'vk6.43171', 'vk6.46534', 'vk6.55128', 'vk6.55385', 'vk6.57694', 'vk6.58885', 'vk6.59840', 'vk6.68399', 'vk6.69744']
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 O1O2O3O4O5O6U2U6U4U3U1U5
R3 orbit {'O1O2O3O4O5O6U2U6U4U3U1U5'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3O4O5O6U5U1U4U3U6U2
Gauss code of -K* O1O2O3O4O5O6U5U1U4U3U6U2
Diagrammatic symmetry type r
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 -4 0 0 4 1],[ 1 0 -3 1 1 4 1],[ 4 3 0 3 2 4 1],[ 0 -1 -3 0 0 2 0],[ 0 -1 -2 0 0 1 0],[-4 -4 -4 -2 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 4 1 0 0 -1 -4],[-4 0 0 -1 -2 -4 -4],[-1 0 0 0 0 -1 -1],[ 0 1 0 0 0 -1 -2],[ 0 2 0 0 0 -1 -3],[ 1 4 1 1 1 0 -3],[ 4 4 1 2 3 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,0,1,4,0,1,2,4,4,0,0,1,1,0,1,2,1,3,3]
Phi over symmetry [-4,-1,0,0,1,4,0,1,2,4,4,0,0,1,1,0,1,2,1,3,3]
Phi of -K [-4,-1,0,0,1,4,0,1,2,4,4,0,0,1,1,0,1,2,1,3,3]
Phi of K* [-4,-1,0,0,1,4,3,2,3,1,4,1,1,1,4,0,0,1,0,2,0]
Phi of -K* [-4,-1,0,0,1,4,3,2,3,1,4,1,1,1,4,0,0,1,0,2,0]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+63t^4+23t^2
Outer characteristic polynomial t^7+97t^5+93t^3+4t
Flat arrow polynomial -4*K1*K2 - 4*K1*K3 + 2*K1 + 2*K2 + 2*K3 + 2*K4 + 1
2-strand cable arrow polynomial -1312*K1**4 + 704*K1**3*K2*K3 + 64*K1**3*K3*K4 - 448*K1**3*K3 + 192*K1**2*K2**2*K4 - 1728*K1**2*K2**2 - 704*K1**2*K2*K4 + 3952*K1**2*K2 - 608*K1**2*K3**2 - 256*K1**2*K4**2 - 3224*K1**2 - 768*K1*K2**2*K3 - 384*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4080*K1*K2*K3 + 1872*K1*K3*K4 + 608*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**4 - 128*K2**3*K6 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 1232*K2**2*K4 - 96*K2**2*K5**2 - 48*K2**2*K6**2 - 3412*K2**2 - 128*K2*K3**2*K4 + 896*K2*K3*K5 + 496*K2*K4*K6 + 96*K2*K5*K7 + 48*K2*K6*K8 + 96*K3**2*K6 - 2032*K3**2 - 1292*K4**2 - 560*K5**2 - 236*K6**2 - 24*K7**2 - 12*K8**2 + 3462
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}]]
If K is slice False
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