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Flat knot 6.30

Min(phi) over symmetries of the knot is: [-5,-1,-1,-1,4,4,1,2,3,4,5,0,0,1,2,0,2,3,3,4,0]
Flat knots (up to 7 crossings) with same phi are :['6.30']
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+158t^5+136t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.30']
2-strand cable arrow polynomial of the knot is: -576*K1**2*K4**2 - 488*K1**2 + 336*K1*K3*K4 + 64*K1*K4**3*K5 + 1104*K1*K4*K5 + 96*K1*K5*K6 - 2*K10**2 + 8*K10*K4*K6 - 18*K2**2 + 40*K2*K4*K6 - 32*K3**2 - 32*K4**4 - 64*K4**2*K5**2 - 8*K4**2*K6**2 - 428*K4**2 + 16*K4*K5*K9 - 456*K5**2 - 68*K6**2 + 506
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.30']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81848', 'vk6.81898', 'vk6.82075', 'vk6.82094', 'vk6.82781', 'vk6.82790', 'vk6.82853', 'vk6.82952', 'vk6.83296', 'vk6.83394', 'vk6.83454', 'vk6.84550', 'vk6.84646', 'vk6.84775', 'vk6.84784', 'vk6.86270', 'vk6.86849', 'vk6.88464', 'vk6.88547', 'vk6.90026']
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 O1O2O3O4O5O6U1U4U3U2U6U5
R3 orbit {'O1O2O3O4O5O6U1U4U3U2U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U2U1U5U4U3U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U2U1U5U4U3U6
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 4 4],[ 5 0 3 2 1 5 4],[ 1 -3 0 0 0 4 3],[ 1 -2 0 0 0 3 2],[ 1 -1 0 0 0 2 1],[-4 -5 -4 -3 -2 0 0],[-4 -4 -3 -2 -1 0 0]]
Primitive based matrix [[ 0 4 4 -1 -1 -1 -5],[-4 0 0 -1 -2 -3 -4],[-4 0 0 -2 -3 -4 -5],[ 1 1 2 0 0 0 -1],[ 1 2 3 0 0 0 -2],[ 1 3 4 0 0 0 -3],[ 5 4 5 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-4,1,1,1,5,0,1,2,3,4,2,3,4,5,0,0,1,0,2,3]
Phi over symmetry [-5,-1,-1,-1,4,4,1,2,3,4,5,0,0,1,2,0,2,3,3,4,0]
Phi of -K [-5,-1,-1,-1,4,4,1,2,3,4,5,0,0,1,2,0,2,3,3,4,0]
Phi of K* [-4,-4,1,1,1,5,0,1,2,3,4,2,3,4,5,0,0,1,0,2,3]
Phi of -K* [-5,-1,-1,-1,4,4,1,2,3,4,5,0,0,1,2,0,2,3,3,4,0]
Symmetry type of based matrix +
u-polynomial t^5-2t^4+3t
Normalized Jones-Krushkal polynomial z+3
Enhanced Jones-Krushkal polynomial -8w^5z+8w^4z-6w^3z+7w^2z+3w
Inner characteristic polynomial t^6+98t^4+26t^2
Outer characteristic polynomial t^7+158t^5+136t^3
Flat arrow polynomial K1 - 2*K2*K3 + K5 + 1
2-strand cable arrow polynomial -576*K1**2*K4**2 - 488*K1**2 + 336*K1*K3*K4 + 64*K1*K4**3*K5 + 1104*K1*K4*K5 + 96*K1*K5*K6 - 2*K10**2 + 8*K10*K4*K6 - 18*K2**2 + 40*K2*K4*K6 - 32*K3**2 - 32*K4**4 - 64*K4**2*K5**2 - 8*K4**2*K6**2 - 428*K4**2 + 16*K4*K5*K9 - 456*K5**2 - 68*K6**2 + 506
Genus of based matrix 1
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}]]
If K is slice False
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