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Flat knot 6.149

Min(phi) over symmetries of the knot is: [-4,-3,1,1,2,3,0,2,3,4,3,1,3,3,2,1,0,1,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.149']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.96', '6.149', '6.269', '6.441', '6.457']
Outer characteristic polynomial of the knot is: t^7+110t^5+156t^3+12t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.149']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 224*K1**4*K2 - 400*K1**4 + 160*K1**3*K2*K3 + 32*K1**3*K3*K4 - 320*K1**3*K3 - 768*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2368*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 6544*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 7008*K1**2*K2 - 272*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 5720*K1**2 + 1600*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7024*K1*K2*K3 - 32*K1*K2*K4*K5 + 1128*K1*K3*K4 + 224*K1*K4*K5 - 32*K2**6 - 32*K2**4*K4**2 + 352*K2**4*K4 - 2768*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 976*K2**2*K3**2 - 544*K2**2*K4**2 + 2552*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3074*K2**2 + 568*K2*K3*K5 + 144*K2*K4*K6 + 8*K2*K5*K7 - 2100*K3**2 - 810*K4**2 - 140*K5**2 - 14*K6**2 + 4232
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.149']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71564', 'vk6.71671', 'vk6.72085', 'vk6.72302', 'vk6.74038', 'vk6.74601', 'vk6.76086', 'vk6.76798', 'vk6.77184', 'vk6.77283', 'vk6.77479', 'vk6.77645', 'vk6.79026', 'vk6.79604', 'vk6.80561', 'vk6.81013', 'vk6.81099', 'vk6.81145', 'vk6.81165', 'vk6.81207', 'vk6.81316', 'vk6.81461', 'vk6.82265', 'vk6.83501', 'vk6.83828', 'vk6.83978', 'vk6.85389', 'vk6.86318', 'vk6.87097', 'vk6.88030', 'vk6.88334', 'vk6.88957']
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 O1O2O3O4O5U1O6U2U5U6U3U4
R3 orbit {'O1O2O3O4O5U1O6U2U5U6U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U3U6U1U4O6U5
Gauss code of K* O1O2O3O4O5U6U1U4U5U2O6U3
Gauss code of -K* O1O2O3O4O5U3O6U4U1U2U5U6
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 -4 -3 1 3 1 2],[ 4 0 1 3 4 2 2],[ 3 -1 0 3 4 1 2],[-1 -3 -3 0 1 -1 1],[-3 -4 -4 -1 0 -1 1],[-1 -2 -1 1 1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 2 1 1 -3 -4],[-3 0 1 -1 -1 -4 -4],[-2 -1 0 -1 -1 -2 -2],[-1 1 1 0 1 -1 -2],[-1 1 1 -1 0 -3 -3],[ 3 4 2 1 3 0 -1],[ 4 4 2 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,-1,3,4,-1,1,1,4,4,1,1,2,2,-1,1,2,3,3,1]
Phi over symmetry [-4,-3,1,1,2,3,0,2,3,4,3,1,3,3,2,1,0,1,0,1,2]
Phi of -K [-4,-3,1,1,2,3,0,2,3,4,3,1,3,3,2,1,0,1,0,1,2]
Phi of K* [-3,-2,-1,-1,3,4,2,1,1,2,3,0,0,3,4,-1,1,2,3,3,0]
Phi of -K* [-4,-3,1,1,2,3,1,2,3,2,4,1,3,2,4,1,1,1,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2-4w^3z+27w^2z+23w
Inner characteristic polynomial t^6+70t^4+35t^2
Outer characteristic polynomial t^7+110t^5+156t^3+12t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -128*K1**4*K2**2 + 224*K1**4*K2 - 400*K1**4 + 160*K1**3*K2*K3 + 32*K1**3*K3*K4 - 320*K1**3*K3 - 768*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2368*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 6544*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 7008*K1**2*K2 - 272*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 5720*K1**2 + 1600*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7024*K1*K2*K3 - 32*K1*K2*K4*K5 + 1128*K1*K3*K4 + 224*K1*K4*K5 - 32*K2**6 - 32*K2**4*K4**2 + 352*K2**4*K4 - 2768*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 976*K2**2*K3**2 - 544*K2**2*K4**2 + 2552*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3074*K2**2 + 568*K2*K3*K5 + 144*K2*K4*K6 + 8*K2*K5*K7 - 2100*K3**2 - 810*K4**2 - 140*K5**2 - 14*K6**2 + 4232
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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