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Flat knot 6.513

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,0,1,3,2,5,0,0,0,1,1,1,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.513']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 14*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 6*K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.513']
Outer characteristic polynomial of the knot is: t^7+87t^5+95t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.513']
2-strand cable arrow polynomial of the knot is: -448*K1**4*K2**2 + 1312*K1**4*K2 - 3088*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 928*K1**3*K2*K3 - 544*K1**3*K3 - 768*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3584*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 12416*K1**2*K2**2 - 1472*K1**2*K2*K4 + 11800*K1**2*K2 - 272*K1**2*K3**2 - 64*K1**2*K4**2 - 6596*K1**2 + 2688*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 2368*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 704*K1*K2**2*K5 + 32*K1*K2*K3**3 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10512*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1328*K1*K3*K4 + 352*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 640*K2**4*K4 - 4216*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1536*K2**2*K3**2 - 736*K2**2*K4**2 + 3672*K2**2*K4 - 208*K2**2*K5**2 - 8*K2**2*K6**2 - 4072*K2**2 + 1056*K2*K3*K5 + 216*K2*K4*K6 + 48*K2*K5*K7 - 2620*K3**2 - 1104*K4**2 - 288*K5**2 - 16*K6**2 + 5750
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.513']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4683', 'vk6.4984', 'vk6.6161', 'vk6.6636', 'vk6.8160', 'vk6.8578', 'vk6.9552', 'vk6.9893', 'vk6.20691', 'vk6.22129', 'vk6.28212', 'vk6.29635', 'vk6.39672', 'vk6.41911', 'vk6.46256', 'vk6.47861', 'vk6.48715', 'vk6.48928', 'vk6.49495', 'vk6.49706', 'vk6.50741', 'vk6.50948', 'vk6.51222', 'vk6.51417', 'vk6.57626', 'vk6.58782', 'vk6.62306', 'vk6.63237', 'vk6.67100', 'vk6.67962', 'vk6.69700', 'vk6.70381']
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 O1O2O3O4O5U1U6U5O6U4U2U3
R3 orbit {'O1O2O3O4O5U1U6U5O6U4U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U4U2O6U1U6U5
Gauss code of K* O1O2O3U2O4O5O6U1U5U6U4U3
Gauss code of -K* O1O2O3U4O5O4O6U5U3U1U2U6
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 0 2 1 2 -1],[ 4 0 3 4 2 1 3],[ 0 -3 0 1 0 1 -1],[-2 -4 -1 0 0 1 -3],[-1 -2 0 0 0 1 -2],[-2 -1 -1 -1 -1 0 -2],[ 1 -3 1 3 2 2 0]]
Primitive based matrix [[ 0 2 2 1 0 -1 -4],[-2 0 1 0 -1 -3 -4],[-2 -1 0 -1 -1 -2 -1],[-1 0 1 0 0 -2 -2],[ 0 1 1 0 0 -1 -3],[ 1 3 2 2 1 0 -3],[ 4 4 1 2 3 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,1,4,-1,0,1,3,4,1,1,2,1,0,2,2,1,3,3]
Phi over symmetry [-4,-1,0,1,2,2,0,1,3,2,5,0,0,0,1,1,1,1,1,0,-1]
Phi of -K [-4,-1,0,1,2,2,0,1,3,2,5,0,0,0,1,1,1,1,1,0,-1]
Phi of K* [-2,-2,-1,0,1,4,-1,0,1,1,5,1,1,0,2,1,0,3,0,1,0]
Phi of -K* [-4,-1,0,1,2,2,3,3,2,1,4,1,2,2,3,0,1,1,1,0,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+61t^4+49t^2+4
Outer characteristic polynomial t^7+87t^5+95t^3+14t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 14*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 6*K2 + 2*K3 + 7
2-strand cable arrow polynomial -448*K1**4*K2**2 + 1312*K1**4*K2 - 3088*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 928*K1**3*K2*K3 - 544*K1**3*K3 - 768*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3584*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 12416*K1**2*K2**2 - 1472*K1**2*K2*K4 + 11800*K1**2*K2 - 272*K1**2*K3**2 - 64*K1**2*K4**2 - 6596*K1**2 + 2688*K1*K2**3*K3 + 544*K1*K2**2*K3*K4 - 2368*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 704*K1*K2**2*K5 + 32*K1*K2*K3**3 - 576*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 10512*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1328*K1*K3*K4 + 352*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 640*K2**4*K4 - 4216*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1536*K2**2*K3**2 - 736*K2**2*K4**2 + 3672*K2**2*K4 - 208*K2**2*K5**2 - 8*K2**2*K6**2 - 4072*K2**2 + 1056*K2*K3*K5 + 216*K2*K4*K6 + 48*K2*K5*K7 - 2620*K3**2 - 1104*K4**2 - 288*K5**2 - 16*K6**2 + 5750
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}]]
If K is slice False
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