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Flat knot 6.1210

Min(phi) over symmetries of the knot is: [-2,-1,1,2,0,3,2,1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1210']
Arrow polynomial of the knot is: -12*K1**2 - 8*K1*K2 + 4*K1 - 4*K2**2 + 6*K2 + 4*K3 + 2*K4 + 9
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1210']
Outer characteristic polynomial of the knot is: t^5+25t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1210']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 1344*K1**4*K2 - 3264*K1**4 + 448*K1**3*K2*K3 + 256*K1**3*K3*K4 + 256*K1**2*K2**3 - 3360*K1**2*K2**2 + 5152*K1**2*K2 - 1248*K1**2*K3**2 - 576*K1**2*K4**2 - 64*K1**2*K5**2 - 3320*K1**2 + 4032*K1*K2*K3 + 2208*K1*K3*K4 + 640*K1*K4*K5 + 112*K1*K5*K6 - 368*K2**4 - 224*K2**2*K3**2 - 32*K2**2*K4**2 + 336*K2**2*K4 - 2768*K2**2 + 448*K2*K3*K5 + 64*K2*K4*K6 - 96*K3**4 - 128*K3**2*K4**2 + 192*K3**2*K6 - 2120*K3**2 + 128*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 1044*K4**2 - 400*K5**2 - 136*K6**2 - 32*K7**2 - 4*K8**2 + 3966
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1210']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3573', 'vk6.3599', 'vk6.3826', 'vk6.3857', 'vk6.6996', 'vk6.7027', 'vk6.7218', 'vk6.7238', 'vk6.15339', 'vk6.15464', 'vk6.33980', 'vk6.34024', 'vk6.34439', 'vk6.48223', 'vk6.48382', 'vk6.49961', 'vk6.49985', 'vk6.53984', 'vk6.54039', 'vk6.54488']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2O3O4U2U5U4O5O6U1U6U3
R3 orbit {'O1O2O3O4U2U5U4O5O6U1U6U3'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3U2U4O5O4O6U5U1U6U3
Gauss code of -K* O1O2O3U2U4O5O4O6U5U1U6U3
Diagrammatic symmetry type r
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -2 2 2 -1 1],[ 2 0 -1 3 2 1 1],[ 2 1 0 2 1 1 0],[-2 -3 -2 0 1 -3 0],[-2 -2 -1 -1 0 -2 0],[ 1 -1 -1 3 2 0 1],[-1 -1 0 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 -1 -2],[-2 0 0 -3 -2],[-1 0 0 -1 0],[ 1 3 1 0 -1],[ 2 2 0 1 0]]
If based matrix primitive False
Phi of primitive based matrix [-2,-1,1,2,0,3,2,1,0,1]
Phi over symmetry [-2,-1,1,2,0,3,2,1,0,1]
Phi of -K [-2,-1,1,2,0,3,2,1,0,1]
Phi of K* [-2,-1,1,2,1,0,2,1,3,0]
Phi of -K* [-2,-1,1,2,1,0,2,1,3,0]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 17z+35
Enhanced Jones-Krushkal polynomial 17w^2z+35w
Inner characteristic polynomial t^4+15t^2+4
Outer characteristic polynomial t^5+25t^3+14t
Flat arrow polynomial -12*K1**2 - 8*K1*K2 + 4*K1 - 4*K2**2 + 6*K2 + 4*K3 + 2*K4 + 9
2-strand cable arrow polynomial -384*K1**4*K2**2 + 1344*K1**4*K2 - 3264*K1**4 + 448*K1**3*K2*K3 + 256*K1**3*K3*K4 + 256*K1**2*K2**3 - 3360*K1**2*K2**2 + 5152*K1**2*K2 - 1248*K1**2*K3**2 - 576*K1**2*K4**2 - 64*K1**2*K5**2 - 3320*K1**2 + 4032*K1*K2*K3 + 2208*K1*K3*K4 + 640*K1*K4*K5 + 112*K1*K5*K6 - 368*K2**4 - 224*K2**2*K3**2 - 32*K2**2*K4**2 + 336*K2**2*K4 - 2768*K2**2 + 448*K2*K3*K5 + 64*K2*K4*K6 - 96*K3**4 - 128*K3**2*K4**2 + 192*K3**2*K6 - 2120*K3**2 + 128*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 1044*K4**2 - 400*K5**2 - 136*K6**2 - 32*K7**2 - 4*K8**2 + 3966
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
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