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Flat knot 6.56

Min(phi) over symmetries of the knot is: [-5,1,1,1,1,1,1,2,3,4,5,0,0,0,0,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.56']
Arrow polynomial of the knot is: K1 - 2*K2*K3 + K5 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.13', '6.30', '6.33', '6.42', '6.56']
Outer characteristic polynomial of the knot is: t^7+85t^5+50t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.56']
2-strand cable arrow polynomial of the knot is: -200*K1**2 + 240*K1*K4*K5 + 160*K1*K5*K6 - 2*K10**2 + 8*K10*K4*K6 - 50*K2**2 + 104*K2*K4*K6 - 8*K4**2*K6**2 - 172*K4**2 - 200*K5**2 - 132*K6**2 + 250
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.56']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20197', 'vk6.20199', 'vk6.21473', 'vk6.21476', 'vk6.27351', 'vk6.27359', 'vk6.28997', 'vk6.29001', 'vk6.38786', 'vk6.38794', 'vk6.40953', 'vk6.40961', 'vk6.47332', 'vk6.47336', 'vk6.57038', 'vk6.57040', 'vk6.62709', 'vk6.62712', 'vk6.70063', 'vk6.70064']
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 O1O2O3O4O5O6U1U6U5U4U3U2
R3 orbit {'O1O2O3O4O5O6U1U6U5U4U3U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U4U3U2U1U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U5U4U3U2U1U6
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 1 1],[ 5 0 5 4 3 2 1],[-1 -5 0 0 0 0 0],[-1 -4 0 0 0 0 0],[-1 -3 0 0 0 0 0],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 1 1 1 1 1 -5],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[-1 0 0 0 0 0 -3],[-1 0 0 0 0 0 -4],[-1 0 0 0 0 0 -5],[ 5 1 2 3 4 5 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,-1,-1,5,0,0,0,0,1,0,0,0,2,0,0,3,0,4,5]
Phi over symmetry [-5,1,1,1,1,1,1,2,3,4,5,0,0,0,0,0,0,0,0,0,0]
Phi of -K [-5,1,1,1,1,1,1,2,3,4,5,0,0,0,0,0,0,0,0,0,0]
Phi of K* [-1,-1,-1,-1,-1,5,0,0,0,0,1,0,0,0,2,0,0,3,0,4,5]
Phi of -K* [-5,1,1,1,1,1,1,2,3,4,5,0,0,0,0,0,0,0,0,0,0]
Symmetry type of based matrix +
u-polynomial t^5-5t
Normalized Jones-Krushkal polynomial z+3
Enhanced Jones-Krushkal polynomial -16w^5z+16w^4z+4w^3z-3w^2z+3w
Inner characteristic polynomial t^6+55t^4
Outer characteristic polynomial t^7+85t^5+50t^3
Flat arrow polynomial K1 - 2*K2*K3 + K5 + 1
2-strand cable arrow polynomial -200*K1**2 + 240*K1*K4*K5 + 160*K1*K5*K6 - 2*K10**2 + 8*K10*K4*K6 - 50*K2**2 + 104*K2*K4*K6 - 8*K4**2*K6**2 - 172*K4**2 - 200*K5**2 - 132*K6**2 + 250
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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