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Flat knot 6.507

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,1,4,2,4,0,1,0,1,1,1,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.507']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 + 3*K2 + 2*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.167', '6.507']
Outer characteristic polynomial of the knot is: t^7+100t^5+37t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.507']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 1312*K1**4*K2 - 1968*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 1024*K1**3*K2*K3 - 896*K1**3*K3 - 320*K1**2*K2**4 + 320*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 5744*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 8496*K1**2*K2 - 1264*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 6744*K1**2 + 672*K1*K2**3*K3 - 1728*K1*K2**2*K3 - 448*K1*K2**2*K5 + 96*K1*K2*K3**3 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9296*K1*K2*K3 - 32*K1*K3**2*K5 + 2000*K1*K3*K4 + 264*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 848*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 - 848*K2**2*K3**2 - 152*K2**2*K4**2 + 2016*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 5888*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1240*K2*K3*K5 + 128*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 96*K3**4 + 152*K3**2*K6 - 3420*K3**2 + 8*K3*K4*K7 - 1090*K4**2 - 408*K5**2 - 64*K6**2 - 4*K7**2 - 2*K8**2 + 5866
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.507']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11444', 'vk6.11740', 'vk6.12757', 'vk6.13101', 'vk6.20317', 'vk6.21658', 'vk6.27621', 'vk6.29165', 'vk6.31201', 'vk6.31541', 'vk6.32369', 'vk6.32784', 'vk6.39045', 'vk6.41305', 'vk6.45801', 'vk6.47476', 'vk6.52197', 'vk6.52458', 'vk6.53027', 'vk6.53348', 'vk6.57188', 'vk6.58399', 'vk6.61802', 'vk6.62923', 'vk6.63769', 'vk6.63880', 'vk6.64196', 'vk6.64383', 'vk6.66801', 'vk6.67669', 'vk6.69441', 'vk6.70163']
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 O1O2O3O4O5U1U6U4O6U2U5U3
R3 orbit {'O1O2O3O4O5U1U6U4O6U2U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U1U4O6U2U6U5
Gauss code of K* O1O2O3U2O4O5O6U1U4U6U3U5
Gauss code of -K* O1O2O3U4O5O4O6U2U5U1U3U6
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 -1 2 1 3 -1],[ 4 0 2 4 1 3 3],[ 1 -2 0 2 1 2 0],[-2 -4 -2 0 0 1 -3],[-1 -1 -1 0 0 0 -1],[-3 -3 -2 -1 0 0 -3],[ 1 -3 0 3 1 3 0]]
Primitive based matrix [[ 0 3 2 1 -1 -1 -4],[-3 0 -1 0 -2 -3 -3],[-2 1 0 0 -2 -3 -4],[-1 0 0 0 -1 -1 -1],[ 1 2 2 1 0 0 -2],[ 1 3 3 1 0 0 -3],[ 4 3 4 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,1,4,1,0,2,3,3,0,2,3,4,1,1,1,0,2,3]
Phi over symmetry [-4,-1,-1,1,2,3,0,1,4,2,4,0,1,0,1,1,1,2,1,2,0]
Phi of -K [-4,-1,-1,1,2,3,0,1,4,2,4,0,1,0,1,1,1,2,1,2,0]
Phi of K* [-3,-2,-1,1,1,4,0,2,1,2,4,1,0,1,2,1,1,4,0,0,1]
Phi of -K* [-4,-1,-1,1,2,3,2,3,1,4,3,0,1,2,2,1,3,3,0,0,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+68t^4+12t^2
Outer characteristic polynomial t^7+100t^5+37t^3+3t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 + 3*K2 + 2*K3 + K4 + 3
2-strand cable arrow polynomial -256*K1**4*K2**2 + 1312*K1**4*K2 - 1968*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 1024*K1**3*K2*K3 - 896*K1**3*K3 - 320*K1**2*K2**4 + 320*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 5744*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 384*K1**2*K2*K4 + 8496*K1**2*K2 - 1264*K1**2*K3**2 - 32*K1**2*K3*K5 - 32*K1**2*K4**2 - 6744*K1**2 + 672*K1*K2**3*K3 - 1728*K1*K2**2*K3 - 448*K1*K2**2*K5 + 96*K1*K2*K3**3 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9296*K1*K2*K3 - 32*K1*K3**2*K5 + 2000*K1*K3*K4 + 264*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 848*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 - 848*K2**2*K3**2 - 152*K2**2*K4**2 + 2016*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 5888*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1240*K2*K3*K5 + 128*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 96*K3**4 + 152*K3**2*K6 - 3420*K3**2 + 8*K3*K4*K7 - 1090*K4**2 - 408*K5**2 - 64*K6**2 - 4*K7**2 - 2*K8**2 + 5866
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}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
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