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Flat knot 6.289

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,1,1,4,4,3,0,1,2,1,-1,0,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.289']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + K2 + 3*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.289']
Outer characteristic polynomial of the knot is: t^7+66t^5+58t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.289']
2-strand cable arrow polynomial of the knot is: 1248*K1**4*K2 - 3344*K1**4 + 1184*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1920*K1**3*K3 - 128*K1**2*K2**4 + 352*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 4272*K1**2*K2**2 - 1024*K1**2*K2*K4 + 7376*K1**2*K2 - 1872*K1**2*K3**2 - 288*K1**2*K4**2 - 3968*K1**2 + 512*K1*K2**3*K3 - 800*K1*K2**2*K3 - 96*K1*K2**2*K5 + 128*K1*K2*K3**3 - 864*K1*K2*K3*K4 + 6936*K1*K2*K3 - 64*K1*K3**2*K5 + 2512*K1*K3*K4 + 432*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 160*K2**4*K4 - 696*K2**4 + 64*K2**2*K3**2*K4 - 672*K2**2*K3**2 - 320*K2**2*K4**2 + 1448*K2**2*K4 - 3386*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 704*K2*K3*K5 + 192*K2*K4*K6 - 128*K3**4 - 112*K3**2*K4**2 + 112*K3**2*K6 - 2172*K3**2 + 80*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1054*K4**2 - 200*K5**2 - 54*K6**2 - 12*K7**2 - 2*K8**2 + 3742
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.289']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13898', 'vk6.13993', 'vk6.14187', 'vk6.14426', 'vk6.14969', 'vk6.15090', 'vk6.15655', 'vk6.16109', 'vk6.16711', 'vk6.16739', 'vk6.16855', 'vk6.18816', 'vk6.19280', 'vk6.19572', 'vk6.23146', 'vk6.23236', 'vk6.25414', 'vk6.26465', 'vk6.33717', 'vk6.33792', 'vk6.34271', 'vk6.35138', 'vk6.37535', 'vk6.42740', 'vk6.44685', 'vk6.54134', 'vk6.54916', 'vk6.54942', 'vk6.56402', 'vk6.56603', 'vk6.59341', 'vk6.64609']
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 O1O2O3O4O5U1U5O6U3U6U4U2
R3 orbit {'O1O2O3O4O5U1U5O6U3U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U2U6U3O6U1U5
Gauss code of K* O1O2O3O4U5U4U1U3U6O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U5U2U4U1U6
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 -1 2 1 1],[ 4 0 4 2 3 1 1],[-1 -4 0 -2 1 0 1],[ 1 -2 2 0 2 0 1],[-2 -3 -1 -2 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 1 1 1 -1 -4],[-2 0 0 0 -1 -2 -3],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 -1 -1],[-1 1 0 1 0 -2 -4],[ 1 2 0 1 2 0 -2],[ 4 3 1 1 4 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,1,4,0,0,1,2,3,0,0,0,1,1,1,1,2,4,2]
Phi over symmetry [-4,-1,1,1,1,2,1,1,4,4,3,0,1,2,1,-1,0,0,0,1,1]
Phi of -K [-4,-1,1,1,1,2,1,1,4,4,3,0,1,2,1,-1,0,0,0,1,1]
Phi of K* [-2,-1,-1,-1,1,4,0,1,1,1,3,0,1,0,1,0,2,4,1,4,1]
Phi of -K* [-4,-1,1,1,1,2,2,1,1,4,3,0,1,2,2,0,0,0,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial 6w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+42t^4+17t^2+1
Outer characteristic polynomial t^7+66t^5+58t^3+6t
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + K2 + 3*K3 + K4 + 3
2-strand cable arrow polynomial 1248*K1**4*K2 - 3344*K1**4 + 1184*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1920*K1**3*K3 - 128*K1**2*K2**4 + 352*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 4272*K1**2*K2**2 - 1024*K1**2*K2*K4 + 7376*K1**2*K2 - 1872*K1**2*K3**2 - 288*K1**2*K4**2 - 3968*K1**2 + 512*K1*K2**3*K3 - 800*K1*K2**2*K3 - 96*K1*K2**2*K5 + 128*K1*K2*K3**3 - 864*K1*K2*K3*K4 + 6936*K1*K2*K3 - 64*K1*K3**2*K5 + 2512*K1*K3*K4 + 432*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 + 160*K2**4*K4 - 696*K2**4 + 64*K2**2*K3**2*K4 - 672*K2**2*K3**2 - 320*K2**2*K4**2 + 1448*K2**2*K4 - 3386*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 704*K2*K3*K5 + 192*K2*K4*K6 - 128*K3**4 - 112*K3**2*K4**2 + 112*K3**2*K6 - 2172*K3**2 + 80*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1054*K4**2 - 200*K5**2 - 54*K6**2 - 12*K7**2 - 2*K8**2 + 3742
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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