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Flat knot 6.44

Min(phi) over symmetries of the knot is: [-5,-1,0,0,3,3,2,1,4,3,5,0,1,2,2,1,1,1,2,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.44']
Arrow polynomial of the knot is: 16*K1**3 + 4*K1**2*K3 - 8*K1**2 - 12*K1*K2 - 6*K1 - 2*K2*K3 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.44']
Outer characteristic polynomial of the knot is: t^7+120t^5+92t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.44']
2-strand cable arrow polynomial of the knot is: 192*K1**4*K2 - 1344*K1**4 - 128*K1**3*K3 + 384*K1**2*K2**3 - 5568*K1**2*K2**2 - 512*K1**2*K2*K4 + 9104*K1**2*K2 - 7576*K1**2 + 1280*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1920*K1*K2**2*K3 - 320*K1*K2**2*K5 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9328*K1*K2*K3 + 1632*K1*K3*K4 + 144*K1*K4*K5 - 128*K2**6 - 256*K2**4*K3**2 + 192*K2**4*K4 - 32*K2**4*K6**2 - 2896*K2**4 + 128*K2**3*K3*K5 + 128*K2**3*K4*K6 - 192*K2**3*K6 + 256*K2**2*K3**2*K4 - 2048*K2**2*K3**2 - 64*K2**2*K3*K7 - 528*K2**2*K4**2 + 32*K2**2*K4*K6**2 + 4416*K2**2*K4 - 96*K2**2*K6**2 - 6468*K2**2 - 192*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1456*K2*K3*K5 - 64*K2*K4**2*K6 + 816*K2*K4*K6 + 80*K2*K6*K8 - 64*K3**2*K4**2 + 80*K3**2*K6 - 3328*K3**2 + 32*K3*K4*K7 - 8*K4**2*K6**2 - 1804*K4**2 - 248*K5**2 - 260*K6**2 - 16*K8**2 + 6826
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.44']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71599', 'vk6.71722', 'vk6.72140', 'vk6.72334', 'vk6.74049', 'vk6.74616', 'vk6.76803', 'vk6.77214', 'vk6.77522', 'vk6.77668', 'vk6.79048', 'vk6.79613', 'vk6.80568', 'vk6.81018', 'vk6.81344', 'vk6.81391', 'vk6.85411', 'vk6.85490', 'vk6.87983', 'vk6.89321']
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 O1O2O3O4O5O6U1U5U3U6U2U4
R3 orbit {'O1O2O3O4O5O6U1U5U3U6U2U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U5U1U4U2U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U3U5U1U4U2U6
Diagrammatic symmetry type +
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 0 -1 3 0 3],[ 5 0 4 2 5 1 3],[ 0 -4 0 -1 2 -1 2],[ 1 -2 1 0 2 0 2],[-3 -5 -2 -2 0 -1 1],[ 0 -1 1 0 1 0 1],[-3 -3 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 3 0 0 -1 -5],[-3 0 1 -1 -2 -2 -5],[-3 -1 0 -1 -2 -2 -3],[ 0 1 1 0 1 0 -1],[ 0 2 2 -1 0 -1 -4],[ 1 2 2 0 1 0 -2],[ 5 5 3 1 4 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,0,0,1,5,-1,1,2,2,5,1,2,2,3,-1,0,1,1,4,2]
Phi over symmetry [-5,-1,0,0,3,3,2,1,4,3,5,0,1,2,2,1,1,1,2,2,-1]
Phi of -K [-5,-1,0,0,3,3,2,1,4,3,5,0,1,2,2,1,1,1,2,2,-1]
Phi of K* [-3,-3,0,0,1,5,-1,1,2,2,5,1,2,2,3,-1,0,1,1,4,2]
Phi of -K* [-5,-1,0,0,3,3,2,1,4,3,5,0,1,2,2,1,1,1,2,2,-1]
Symmetry type of based matrix +
u-polynomial t^5-2t^3+t
Normalized Jones-Krushkal polynomial 6z^2+26z+29
Enhanced Jones-Krushkal polynomial 6w^3z^2+26w^2z+29w
Inner characteristic polynomial t^6+76t^4+22t^2+1
Outer characteristic polynomial t^7+120t^5+92t^3+7t
Flat arrow polynomial 16*K1**3 + 4*K1**2*K3 - 8*K1**2 - 12*K1*K2 - 6*K1 - 2*K2*K3 + 4*K2 + 5
2-strand cable arrow polynomial 192*K1**4*K2 - 1344*K1**4 - 128*K1**3*K3 + 384*K1**2*K2**3 - 5568*K1**2*K2**2 - 512*K1**2*K2*K4 + 9104*K1**2*K2 - 7576*K1**2 + 1280*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1920*K1*K2**2*K3 - 320*K1*K2**2*K5 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9328*K1*K2*K3 + 1632*K1*K3*K4 + 144*K1*K4*K5 - 128*K2**6 - 256*K2**4*K3**2 + 192*K2**4*K4 - 32*K2**4*K6**2 - 2896*K2**4 + 128*K2**3*K3*K5 + 128*K2**3*K4*K6 - 192*K2**3*K6 + 256*K2**2*K3**2*K4 - 2048*K2**2*K3**2 - 64*K2**2*K3*K7 - 528*K2**2*K4**2 + 32*K2**2*K4*K6**2 + 4416*K2**2*K4 - 96*K2**2*K6**2 - 6468*K2**2 - 192*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1456*K2*K3*K5 - 64*K2*K4**2*K6 + 816*K2*K4*K6 + 80*K2*K6*K8 - 64*K3**2*K4**2 + 80*K3**2*K6 - 3328*K3**2 + 32*K3*K4*K7 - 8*K4**2*K6**2 - 1804*K4**2 - 248*K5**2 - 260*K6**2 - 16*K8**2 + 6826
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}]]
If K is slice False
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