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Flat knot 6.764

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,1,3,3,0,0,1,1,1,2,2,-1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.764']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.552', '6.652', '6.764', '6.776', '6.784', '6.839', '6.903', '6.1010', '6.1166']
Outer characteristic polynomial of the knot is: t^7+54t^5+39t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.764']
2-strand cable arrow polynomial of the knot is: -1536*K1**4*K2**2 + 2912*K1**4*K2 - 4336*K1**4 - 384*K1**3*K2**2*K3 + 1472*K1**3*K2*K3 - 768*K1**3*K3 + 384*K1**2*K2**5 - 1344*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3072*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 9296*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 896*K1**2*K2*K4 + 8664*K1**2*K2 - 1232*K1**2*K3**2 - 3480*K1**2 + 2272*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 - 448*K1*K2**2*K5 + 192*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7024*K1*K2*K3 + 1056*K1*K3*K4 - 288*K2**6 + 352*K2**4*K4 - 1664*K2**4 - 32*K2**3*K6 - 880*K2**2*K3**2 - 152*K2**2*K4**2 + 1048*K2**2*K4 - 2460*K2**2 - 64*K2*K3**2*K4 + 344*K2*K3*K5 + 64*K2*K4*K6 - 32*K3**4 + 32*K3**2*K6 - 1424*K3**2 - 292*K4**2 - 24*K5**2 - 12*K6**2 + 3434
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.764']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16956', 'vk6.16957', 'vk6.17198', 'vk6.17200', 'vk6.20848', 'vk6.20851', 'vk6.22251', 'vk6.22253', 'vk6.23355', 'vk6.23649', 'vk6.23651', 'vk6.28311', 'vk6.35404', 'vk6.35825', 'vk6.35827', 'vk6.39927', 'vk6.39931', 'vk6.42021', 'vk6.43156', 'vk6.43158', 'vk6.46477', 'vk6.46481', 'vk6.55116', 'vk6.55117', 'vk6.55378', 'vk6.57662', 'vk6.57665', 'vk6.58846', 'vk6.59823', 'vk6.59825', 'vk6.68397', 'vk6.69717']
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 O1O2O3O4U3O5U1U2U4O6U5U6
R3 orbit {'O1O2O3O4U3O5U1U2U4O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U6O5U1U3U4O6U2
Gauss code of K* O1O2O3U4O5O4U1U2U6U3O6U5
Gauss code of -K* O1O2O3U4O5U1U5U2U3O6O4U6
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 -1 2 2 1],[ 3 0 1 0 3 3 1],[ 1 -1 0 0 2 2 1],[ 1 0 0 0 1 1 0],[-2 -3 -2 -1 0 1 1],[-2 -3 -2 -1 -1 0 1],[-1 -1 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 1 1 -1 -2 -3],[-2 -1 0 1 -1 -2 -3],[-1 -1 -1 0 0 -1 -1],[ 1 1 1 0 0 0 0],[ 1 2 2 1 0 0 -1],[ 3 3 3 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,-1,-1,1,2,3,-1,1,2,3,0,1,1,0,0,1]
Phi over symmetry [-3,-1,-1,1,2,2,0,1,1,3,3,0,0,1,1,1,2,2,-1,-1,-1]
Phi of -K [-3,-1,-1,1,2,2,1,2,3,2,2,0,1,1,1,2,2,2,2,2,-1]
Phi of K* [-2,-2,-1,1,1,3,-1,2,1,2,2,2,1,2,2,1,2,3,0,1,2]
Phi of -K* [-3,-1,-1,1,2,2,0,1,1,3,3,0,0,1,1,1,2,2,-1,-1,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 6z^2+26z+29
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+26w^2z+29w
Inner characteristic polynomial t^6+34t^4+9t^2
Outer characteristic polynomial t^7+54t^5+39t^3+6t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 6*K1*K2 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -1536*K1**4*K2**2 + 2912*K1**4*K2 - 4336*K1**4 - 384*K1**3*K2**2*K3 + 1472*K1**3*K2*K3 - 768*K1**3*K3 + 384*K1**2*K2**5 - 1344*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3072*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 + 448*K1**2*K2**2*K4 - 9296*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 896*K1**2*K2*K4 + 8664*K1**2*K2 - 1232*K1**2*K3**2 - 3480*K1**2 + 2272*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 1184*K1*K2**2*K3 - 448*K1*K2**2*K5 + 192*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7024*K1*K2*K3 + 1056*K1*K3*K4 - 288*K2**6 + 352*K2**4*K4 - 1664*K2**4 - 32*K2**3*K6 - 880*K2**2*K3**2 - 152*K2**2*K4**2 + 1048*K2**2*K4 - 2460*K2**2 - 64*K2*K3**2*K4 + 344*K2*K3*K5 + 64*K2*K4*K6 - 32*K3**4 + 32*K3**2*K6 - 1424*K3**2 - 292*K4**2 - 24*K5**2 - 12*K6**2 + 3434
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}]]
If K is slice False
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