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Flat knot 6.124

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,1,1,3,3,1,1,2,2,1,1,2,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.124']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 8*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + 3*K2 + 3*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.124']
Outer characteristic polynomial of the knot is: t^7+72t^5+57t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.124']
2-strand cable arrow polynomial of the knot is: -128*K1**6 + 704*K1**4*K2 - 2112*K1**4 + 224*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1248*K1**3*K3 - 128*K1**2*K2**4 + 192*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 2304*K1**2*K2**2 + 224*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 448*K1**2*K2*K4 + 6400*K1**2*K2 - 1504*K1**2*K3**2 - 96*K1**2*K3*K5 - 448*K1**2*K4**2 - 96*K1**2*K4*K6 - 4792*K1**2 + 384*K1*K2**3*K3 - 896*K1*K2**2*K3 - 32*K1*K2**2*K5 + 96*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 864*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6184*K1*K2*K3 - 32*K1*K2*K4*K7 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2464*K1*K3*K4 + 680*K1*K4*K5 + 56*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 432*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 896*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 320*K2**2*K4**2 + 1208*K2**2*K4 - 8*K2**2*K6**2 - 3662*K2**2 - 128*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 808*K2*K3*K5 - 32*K2*K4**2*K6 + 280*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 160*K3**4 - 80*K3**2*K4**2 + 168*K3**2*K6 - 2232*K3**2 + 96*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1026*K4**2 - 248*K5**2 - 90*K6**2 - 32*K7**2 - 2*K8**2 + 3978
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.124']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4826', 'vk6.5169', 'vk6.6388', 'vk6.6819', 'vk6.8349', 'vk6.8781', 'vk6.9719', 'vk6.10022', 'vk6.11625', 'vk6.11976', 'vk6.12967', 'vk6.20475', 'vk6.20746', 'vk6.21829', 'vk6.27860', 'vk6.29369', 'vk6.31428', 'vk6.32602', 'vk6.39291', 'vk6.39778', 'vk6.41469', 'vk6.46338', 'vk6.47590', 'vk6.47913', 'vk6.49051', 'vk6.49877', 'vk6.51305', 'vk6.51522', 'vk6.53216', 'vk6.57334', 'vk6.62021', 'vk6.64297']
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 O1O2O3O4O5O6U3U6U4U1U5U2
R3 orbit {'O1O2O3O4O5O6U3U6U4U1U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U2U6U3U1U4
Gauss code of K* O1O2O3O4O5O6U4U6U1U3U5U2
Gauss code of -K* O1O2O3O4O5O6U5U2U4U6U1U3
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 -2 1 -3 0 3 1],[ 2 0 2 -2 1 3 1],[-1 -2 0 -3 0 2 1],[ 3 2 3 0 2 3 1],[ 0 -1 0 -2 0 1 0],[-3 -3 -2 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 1 1 0 -2 -3],[-3 0 0 -2 -1 -3 -3],[-1 0 0 -1 0 -1 -1],[-1 2 1 0 0 -2 -3],[ 0 1 0 0 0 -1 -2],[ 2 3 1 2 1 0 -2],[ 3 3 1 3 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,0,2,3,0,2,1,3,3,1,0,1,1,0,2,3,1,2,2]
Phi over symmetry [-3,-2,0,1,1,3,-1,1,1,3,3,1,1,2,2,1,1,2,-1,0,2]
Phi of -K [-3,-2,0,1,1,3,-1,1,1,3,3,1,1,2,2,1,1,2,-1,0,2]
Phi of K* [-3,-1,-1,0,2,3,0,2,2,2,3,1,1,1,1,1,2,3,1,1,-1]
Phi of -K* [-3,-2,0,1,1,3,2,2,1,3,3,1,1,2,3,0,0,1,-1,0,2]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+48t^4+16t^2
Outer characteristic polynomial t^7+72t^5+57t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 8*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + 3*K2 + 3*K3 + K4 + 5
2-strand cable arrow polynomial -128*K1**6 + 704*K1**4*K2 - 2112*K1**4 + 224*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1248*K1**3*K3 - 128*K1**2*K2**4 + 192*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 2304*K1**2*K2**2 + 224*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 448*K1**2*K2*K4 + 6400*K1**2*K2 - 1504*K1**2*K3**2 - 96*K1**2*K3*K5 - 448*K1**2*K4**2 - 96*K1**2*K4*K6 - 4792*K1**2 + 384*K1*K2**3*K3 - 896*K1*K2**2*K3 - 32*K1*K2**2*K5 + 96*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 864*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6184*K1*K2*K3 - 32*K1*K2*K4*K7 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2464*K1*K3*K4 + 680*K1*K4*K5 + 56*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 432*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 + 64*K2**2*K3**2*K4 - 896*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 320*K2**2*K4**2 + 1208*K2**2*K4 - 8*K2**2*K6**2 - 3662*K2**2 - 128*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 808*K2*K3*K5 - 32*K2*K4**2*K6 + 280*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 - 160*K3**4 - 80*K3**2*K4**2 + 168*K3**2*K6 - 2232*K3**2 + 96*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1026*K4**2 - 248*K5**2 - 90*K6**2 - 32*K7**2 - 2*K8**2 + 3978
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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