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Flat knot 6.841

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,-1,0,2,3,4,1,1,3,2,0,1,1,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.841']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.841']
Outer characteristic polynomial of the knot is: t^7+66t^5+105t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.841']
2-strand cable arrow polynomial of the knot is: -656*K1**4 - 32*K1**3*K3 - 768*K1**2*K2**6 + 896*K1**2*K2**5 - 2624*K1**2*K2**4 + 3744*K1**2*K2**3 - 7888*K1**2*K2**2 - 352*K1**2*K2*K4 + 6464*K1**2*K2 - 48*K1**2*K3**2 - 32*K1**2*K4**2 - 4228*K1**2 + 1152*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 3296*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 576*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5760*K1*K2*K3 + 520*K1*K3*K4 + 120*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 448*K2**4*K3**2 - 192*K2**4*K4**2 + 1056*K2**4*K4 - 3440*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1248*K2**2*K3**2 - 560*K2**2*K4**2 + 2360*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 1044*K2**2 - 32*K2*K3**2*K4 + 448*K2*K3*K5 + 104*K2*K4*K6 - 1348*K3**2 - 450*K4**2 - 72*K5**2 - 4*K6**2 + 3136
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.841']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11412', 'vk6.11695', 'vk6.12712', 'vk6.13069', 'vk6.20269', 'vk6.21588', 'vk6.27531', 'vk6.29105', 'vk6.31145', 'vk6.31470', 'vk6.32293', 'vk6.32738', 'vk6.38936', 'vk6.41161', 'vk6.45696', 'vk6.47410', 'vk6.52159', 'vk6.52384', 'vk6.52970', 'vk6.53306', 'vk6.57102', 'vk6.58274', 'vk6.61679', 'vk6.62838', 'vk6.63737', 'vk6.63833', 'vk6.64153', 'vk6.64353', 'vk6.66737', 'vk6.67611', 'vk6.69387', 'vk6.70119']
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 O1O2O3O4U1U2O5O6U5U3U4U6
R3 orbit {'O1O2O3O4U1U2O5O6U5U3U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1U2U6O5O6U3U4
Gauss code of K* O1O2O3O4U5U6U2U3O5O6U1U4
Gauss code of -K* O1O2O3O4U1U4O5O6U2U3U5U6
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 -3 -1 0 2 -1 3],[ 3 0 1 2 3 0 2],[ 1 -1 0 1 2 0 2],[ 0 -2 -1 0 1 0 3],[-2 -3 -2 -1 0 0 2],[ 1 0 0 0 0 0 1],[-3 -2 -2 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 2 0 -1 -1 -3],[-3 0 -2 -3 -1 -2 -2],[-2 2 0 -1 0 -2 -3],[ 0 3 1 0 0 -1 -2],[ 1 1 0 0 0 0 0],[ 1 2 2 1 0 0 -1],[ 3 2 3 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,1,1,3,2,3,1,2,2,1,0,2,3,0,1,2,0,0,1]
Phi over symmetry [-3,-2,0,1,1,3,-1,0,2,3,4,1,1,3,2,0,1,1,0,1,2]
Phi of -K [-3,-1,-1,0,2,3,1,2,1,2,4,0,0,1,2,1,3,3,1,0,-1]
Phi of K* [-3,-2,0,1,1,3,-1,0,2,3,4,1,1,3,2,0,1,1,0,1,2]
Phi of -K* [-3,-1,-1,0,2,3,0,1,2,3,2,0,0,0,1,1,2,2,1,3,2]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial -4w^4z^2+8w^3z^2-4w^3z+21w^2z+19w
Inner characteristic polynomial t^6+42t^4+34t^2
Outer characteristic polynomial t^7+66t^5+105t^3+7t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + 3*K2 + 4
2-strand cable arrow polynomial -656*K1**4 - 32*K1**3*K3 - 768*K1**2*K2**6 + 896*K1**2*K2**5 - 2624*K1**2*K2**4 + 3744*K1**2*K2**3 - 7888*K1**2*K2**2 - 352*K1**2*K2*K4 + 6464*K1**2*K2 - 48*K1**2*K3**2 - 32*K1**2*K4**2 - 4228*K1**2 + 1152*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 3296*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 576*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5760*K1*K2*K3 + 520*K1*K3*K4 + 120*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1088*K2**6 - 448*K2**4*K3**2 - 192*K2**4*K4**2 + 1056*K2**4*K4 - 3440*K2**4 + 160*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1248*K2**2*K3**2 - 560*K2**2*K4**2 + 2360*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 1044*K2**2 - 32*K2*K3**2*K4 + 448*K2*K3*K5 + 104*K2*K4*K6 - 1348*K3**2 - 450*K4**2 - 72*K5**2 - 4*K6**2 + 3136
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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