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Flat knot 6.52

Min(phi) over symmetries of the knot is: [-5,-1,1,1,1,3,2,1,3,5,4,0,1,2,2,0,0,0,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.52']
Arrow polynomial of the knot is: -8*K1**3*K2 + 12*K1**3 + 4*K1**2*K3 - 4*K1*K2 - 4*K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.52']
Outer characteristic polynomial of the knot is: t^7+107t^5+84t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.52']
2-strand cable arrow polynomial of the knot is: -1920*K1**2*K2**4 + 960*K1**2*K2**3 - 2208*K1**2*K2**2 + 1056*K1**2*K2 - 896*K1**2 + 768*K1*K2**5*K3 + 1728*K1*K2**3*K3 + 1920*K1*K2*K3 + 176*K1*K3*K4 + 80*K1*K4*K5 - 128*K2**6*K4**2 + 384*K2**6*K4 - 1184*K2**6 + 128*K2**5*K4*K6 - 768*K2**4*K3**2 - 320*K2**4*K4**2 + 512*K2**4*K4 - 32*K2**4*K6**2 - 352*K2**4 + 192*K2**3*K3*K5 + 64*K2**3*K4*K6 - 480*K2**2*K3**2 - 176*K2**2*K4**2 + 560*K2**2*K4 - 168*K2**2 + 80*K2*K3*K5 + 64*K2*K4*K6 + 16*K3**2*K6 - 648*K3**2 - 296*K4**2 - 56*K5**2 - 24*K6**2 + 966
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.52']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20135', 'vk6.20139', 'vk6.21427', 'vk6.21430', 'vk6.27239', 'vk6.27247', 'vk6.28903', 'vk6.28909', 'vk6.38655', 'vk6.38669', 'vk6.40868', 'vk6.40877', 'vk6.47254', 'vk6.47260', 'vk6.56956', 'vk6.56964', 'vk6.58112', 'vk6.58119', 'vk6.62650', 'vk6.62662', 'vk6.67451', 'vk6.67458', 'vk6.70025', 'vk6.70029']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4O5O6U1U6U3U4U5U2
R3 orbit {'O1O2O3O4O5O6U1U6U3U4U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U2U3U4U1U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U5U2U3U4U1U6
Diagrammatic symmetry type +
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 1 -1 1 3 1],[ 5 0 5 2 3 4 1],[-1 -5 0 -2 0 2 0],[ 1 -2 2 0 1 2 0],[-1 -3 0 -1 0 1 0],[-3 -4 -2 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 1 1 1 -1 -5],[-3 0 0 -1 -2 -2 -4],[-1 0 0 0 0 0 -1],[-1 1 0 0 0 -1 -3],[-1 2 0 0 0 -2 -5],[ 1 2 0 1 2 0 -2],[ 5 4 1 3 5 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,-1,1,5,0,1,2,2,4,0,0,0,1,0,1,3,2,5,2]
Phi over symmetry [-5,-1,1,1,1,3,2,1,3,5,4,0,1,2,2,0,0,0,0,1,2]
Phi of -K [-5,-1,1,1,1,3,2,1,3,5,4,0,1,2,2,0,0,0,0,1,2]
Phi of K* [-3,-1,-1,-1,1,5,0,1,2,2,4,0,0,0,1,0,1,3,2,5,2]
Phi of -K* [-5,-1,1,1,1,3,2,1,3,5,4,0,1,2,2,0,0,0,0,1,2]
Symmetry type of based matrix +
u-polynomial t^5-t^3-2t
Normalized Jones-Krushkal polynomial 1
Enhanced Jones-Krushkal polynomial 4w^4z-16w^3z+12w^2z+w
Inner characteristic polynomial t^6+69t^4+20t^2
Outer characteristic polynomial t^7+107t^5+84t^3
Flat arrow polynomial -8*K1**3*K2 + 12*K1**3 + 4*K1**2*K3 - 4*K1*K2 - 4*K1 + 1
2-strand cable arrow polynomial -1920*K1**2*K2**4 + 960*K1**2*K2**3 - 2208*K1**2*K2**2 + 1056*K1**2*K2 - 896*K1**2 + 768*K1*K2**5*K3 + 1728*K1*K2**3*K3 + 1920*K1*K2*K3 + 176*K1*K3*K4 + 80*K1*K4*K5 - 128*K2**6*K4**2 + 384*K2**6*K4 - 1184*K2**6 + 128*K2**5*K4*K6 - 768*K2**4*K3**2 - 320*K2**4*K4**2 + 512*K2**4*K4 - 32*K2**4*K6**2 - 352*K2**4 + 192*K2**3*K3*K5 + 64*K2**3*K4*K6 - 480*K2**2*K3**2 - 176*K2**2*K4**2 + 560*K2**2*K4 - 168*K2**2 + 80*K2*K3*K5 + 64*K2*K4*K6 + 16*K3**2*K6 - 648*K3**2 - 296*K4**2 - 56*K5**2 - 24*K6**2 + 966
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{2, 6}, {5}, {4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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