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Flat knot 6.263

Min(phi) over symmetries of the knot is: [-4,-1,-1,0,3,3,0,2,2,3,4,1,0,1,1,0,2,2,2,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.263']
Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 + 2*K2 + 2*K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.180', '6.263', '6.295', '6.317', '6.350', '6.473', '6.504']
Outer characteristic polynomial of the knot is: t^7+110t^5+86t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.263']
2-strand cable arrow polynomial of the knot is: -784*K1**4 + 320*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 - 1440*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 3216*K1**2*K2 - 2096*K1**2*K3**2 - 96*K1**2*K3*K5 - 64*K1**2*K4**2 - 64*K1**2*K6**2 - 4848*K1**2 + 96*K1*K2**3*K3 - 704*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 7344*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 3128*K1*K3*K4 + 184*K1*K4*K5 + 256*K1*K5*K6 + 80*K1*K6*K7 - 72*K2**4 - 32*K2**3*K6 - 352*K2**2*K3**2 - 48*K2**2*K4**2 + 640*K2**2*K4 - 8*K2**2*K6**2 - 4000*K2**2 - 32*K2*K3**2*K4 + 632*K2*K3*K5 + 320*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 + 120*K3**2*K6 - 3656*K3**2 - 1184*K4**2 - 336*K5**2 - 280*K6**2 - 24*K7**2 - 2*K8**2 + 4656
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.263']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81569', 'vk6.81572', 'vk6.81650', 'vk6.81654', 'vk6.81737', 'vk6.81742', 'vk6.81851', 'vk6.81858', 'vk6.82236', 'vk6.82238', 'vk6.82392', 'vk6.82399', 'vk6.82506', 'vk6.82515', 'vk6.82569', 'vk6.82572', 'vk6.83165', 'vk6.83180', 'vk6.83593', 'vk6.83601', 'vk6.84128', 'vk6.84135', 'vk6.84341', 'vk6.84350', 'vk6.84558', 'vk6.84559', 'vk6.86474', 'vk6.86496', 'vk6.88729', 'vk6.88742', 'vk6.88912', 'vk6.88915']
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 O1O2O3O4O5U1U3O6U4U2U5U6
R3 orbit {'O1O2O3O4O5U1U3O6U4U2U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U1U4U2O6U3U5
Gauss code of K* O1O2O3O4U5U2U6U1U3O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U2U4U5U3U6
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 -4 -1 -1 0 3 3],[ 4 0 3 1 2 4 3],[ 1 -3 0 -1 1 3 3],[ 1 -1 1 0 1 2 2],[ 0 -2 -1 -1 0 1 2],[-3 -4 -3 -2 -1 0 1],[-3 -3 -3 -2 -2 -1 0]]
Primitive based matrix [[ 0 3 3 0 -1 -1 -4],[-3 0 1 -1 -2 -3 -4],[-3 -1 0 -2 -2 -3 -3],[ 0 1 2 0 -1 -1 -2],[ 1 2 2 1 0 1 -1],[ 1 3 3 1 -1 0 -3],[ 4 4 3 2 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,0,1,1,4,-1,1,2,3,4,2,2,3,3,1,1,2,-1,1,3]
Phi over symmetry [-4,-1,-1,0,3,3,0,2,2,3,4,1,0,1,1,0,2,2,2,1,-1]
Phi of -K [-4,-1,-1,0,3,3,0,2,2,3,4,1,0,1,1,0,2,2,2,1,-1]
Phi of K* [-3,-3,0,1,1,4,-1,1,1,2,4,2,1,2,3,0,0,2,-1,0,2]
Phi of -K* [-4,-1,-1,0,3,3,1,3,2,3,4,1,1,2,2,1,3,3,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+2t
Normalized Jones-Krushkal polynomial 4z^2+25z+35
Enhanced Jones-Krushkal polynomial 4w^3z^2+25w^2z+35w
Inner characteristic polynomial t^6+74t^4+30t^2+1
Outer characteristic polynomial t^7+110t^5+86t^3+11t
Flat arrow polynomial -2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 + 2*K2 + 2*K3 + K4 + 2
2-strand cable arrow polynomial -784*K1**4 + 320*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 - 1440*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 3216*K1**2*K2 - 2096*K1**2*K3**2 - 96*K1**2*K3*K5 - 64*K1**2*K4**2 - 64*K1**2*K6**2 - 4848*K1**2 + 96*K1*K2**3*K3 - 704*K1*K2**2*K3 - 96*K1*K2**2*K5 - 64*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 7344*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 3128*K1*K3*K4 + 184*K1*K4*K5 + 256*K1*K5*K6 + 80*K1*K6*K7 - 72*K2**4 - 32*K2**3*K6 - 352*K2**2*K3**2 - 48*K2**2*K4**2 + 640*K2**2*K4 - 8*K2**2*K6**2 - 4000*K2**2 - 32*K2*K3**2*K4 + 632*K2*K3*K5 + 320*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 + 120*K3**2*K6 - 3656*K3**2 - 1184*K4**2 - 336*K5**2 - 280*K6**2 - 24*K7**2 - 2*K8**2 + 4656
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
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