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Flat knot 6.123

Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,-1,1,3,2,3,1,3,2,3,-1,0,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.123']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 - 4*K2**2 + 2*K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.84', '6.123', '6.244', '6.864']
Outer characteristic polynomial of the knot is: t^7+78t^5+41t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.123']
2-strand cable arrow polynomial of the knot is: -2496*K1**4 + 448*K1**3*K2*K3 + 128*K1**3*K3*K4 - 448*K1**3*K3 + 192*K1**2*K2**2*K4 - 1024*K1**2*K2**2 - 1152*K1**2*K2*K4 + 4128*K1**2*K2 - 1344*K1**2*K3**2 - 64*K1**2*K3*K5 - 1824*K1**2*K4**2 - 64*K1**2*K4*K6 - 3816*K1**2 - 128*K1*K2**2*K3 - 384*K1*K2*K3*K4 + 3920*K1*K2*K3 + 4944*K1*K3*K4 + 1888*K1*K4*K5 + 80*K1*K5*K6 - 32*K2**4 - 80*K2**2*K4**2 + 848*K2**2*K4 - 2668*K2**2 + 256*K2*K3*K5 + 160*K2*K4*K6 - 64*K3**4 - 224*K3**2*K4**2 + 32*K3**2*K6 - 2552*K3**2 + 128*K3*K4*K7 - 176*K4**4 + 96*K4**2*K8 - 2184*K4**2 - 552*K5**2 - 84*K6**2 - 8*K7**2 - 4*K8**2 + 4122
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.123']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11649', 'vk6.12000', 'vk6.12991', 'vk6.13264', 'vk6.20451', 'vk6.21805', 'vk6.27835', 'vk6.29343', 'vk6.31451', 'vk6.32626', 'vk6.32970', 'vk6.39259', 'vk6.47570', 'vk6.52366', 'vk6.53248', 'vk6.57312', 'vk6.62001', 'vk6.64321', 'vk6.64501', 'vk6.66903']
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 O1O2O3O4O5O6U3U6U1U5U4U2
R3 orbit {'O1O2O3O4O5O6U3U6U1U5U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U3U2U6U1U4
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U5U3U2U6U1U4
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 -3 1 -3 2 2 1],[ 3 0 3 -1 3 2 1],[-1 -3 0 -3 1 1 1],[ 3 1 3 0 3 2 1],[-2 -3 -1 -3 0 0 0],[-2 -2 -1 -2 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 2 2 1 1 -3 -3],[-2 0 0 0 -1 -2 -2],[-2 0 0 0 -1 -3 -3],[-1 0 0 0 -1 -1 -1],[-1 1 1 1 0 -3 -3],[ 3 2 3 1 3 0 1],[ 3 2 3 1 3 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,-1,3,3,0,0,1,2,2,0,1,3,3,1,1,1,3,3,-1]
Phi over symmetry [-3,-3,1,1,2,2,-1,1,3,2,3,1,3,2,3,-1,0,0,1,1,0]
Phi of -K [-3,-3,1,1,2,2,-1,1,3,2,3,1,3,2,3,-1,0,0,1,1,0]
Phi of K* [-2,-2,-1,-1,3,3,0,0,1,2,2,0,1,3,3,1,1,1,3,3,-1]
Phi of -K* [-3,-3,1,1,2,2,-1,1,3,2,3,1,3,2,3,-1,0,0,1,1,0]
Symmetry type of based matrix +
u-polynomial 2t^3-2t^2-2t
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+50t^4+15t^2
Outer characteristic polynomial t^7+78t^5+41t^3+4t
Flat arrow polynomial -4*K1*K2 + 2*K1 - 4*K2**2 + 2*K3 + 2*K4 + 3
2-strand cable arrow polynomial -2496*K1**4 + 448*K1**3*K2*K3 + 128*K1**3*K3*K4 - 448*K1**3*K3 + 192*K1**2*K2**2*K4 - 1024*K1**2*K2**2 - 1152*K1**2*K2*K4 + 4128*K1**2*K2 - 1344*K1**2*K3**2 - 64*K1**2*K3*K5 - 1824*K1**2*K4**2 - 64*K1**2*K4*K6 - 3816*K1**2 - 128*K1*K2**2*K3 - 384*K1*K2*K3*K4 + 3920*K1*K2*K3 + 4944*K1*K3*K4 + 1888*K1*K4*K5 + 80*K1*K5*K6 - 32*K2**4 - 80*K2**2*K4**2 + 848*K2**2*K4 - 2668*K2**2 + 256*K2*K3*K5 + 160*K2*K4*K6 - 64*K3**4 - 224*K3**2*K4**2 + 32*K3**2*K6 - 2552*K3**2 + 128*K3*K4*K7 - 176*K4**4 + 96*K4**2*K8 - 2184*K4**2 - 552*K5**2 - 84*K6**2 - 8*K7**2 - 4*K8**2 + 4122
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {4, 5}, {1, 3}], [{6}, {1, 5}, {3, 4}, {2}]]
If K is slice False
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