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Flat knot 6.247

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,0,1,3,4,3,0,1,1,1,0,0,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.247']
Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.83', '6.151', '6.160', '6.190', '6.247', '6.262', '6.491', '6.514']
Outer characteristic polynomial of the knot is: t^7+64t^5+33t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.247']
2-strand cable arrow polynomial of the knot is: -1296*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 224*K1**3*K3 + 96*K1**2*K2**2*K4 - 1328*K1**2*K2**2 - 352*K1**2*K2*K4 + 2952*K1**2*K2 - 432*K1**2*K3**2 - 96*K1**2*K3*K5 - 160*K1**2*K4**2 - 1672*K1**2 - 480*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 2384*K1*K2*K3 + 1224*K1*K3*K4 + 360*K1*K4*K5 + 24*K1*K5*K6 - 72*K2**4 - 96*K2**2*K3**2 - 80*K2**2*K4**2 + 616*K2**2*K4 - 1620*K2**2 - 32*K2*K3*K4*K5 + 304*K2*K3*K5 + 80*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**2*K4**2 + 8*K3**2*K6 - 996*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 638*K4**2 - 200*K5**2 - 28*K6**2 - 4*K7**2 - 2*K8**2 + 1742
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.247']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13385', 'vk6.13468', 'vk6.13657', 'vk6.13765', 'vk6.13923', 'vk6.14020', 'vk6.14195', 'vk6.14199', 'vk6.14434', 'vk6.14442', 'vk6.14994', 'vk6.15117', 'vk6.15663', 'vk6.16117', 'vk6.16123', 'vk6.16738', 'vk6.16783', 'vk6.23147', 'vk6.23194', 'vk6.25387', 'vk6.25664', 'vk6.33136', 'vk6.33185', 'vk6.33734', 'vk6.33811', 'vk6.35137', 'vk6.35181', 'vk6.35208', 'vk6.37517', 'vk6.37765', 'vk6.42672', 'vk6.42689', 'vk6.42739', 'vk6.42777', 'vk6.44689', 'vk6.44733', 'vk6.53569', 'vk6.54941', 'vk6.56600', 'vk6.64601']
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 O1O2O3O4O5U4O6U1U6U5U3U2
R3 orbit {'O1O2O3O4O5U4O6U1U6U5U3U2', 'O1O2O3O4U3O5O6U1U6U4U5U2'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5U4U3U1U6U5O6U2
Gauss code of K* O1O2O3O4O5U1U5U4U6U3O6U2
Gauss code of -K* O1O2O3O4O5U4O6U3U6U2U1U5
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 -1 2 1],[ 4 0 4 3 0 3 1],[-1 -4 0 0 -1 1 0],[-1 -3 0 0 -1 1 0],[ 1 0 1 1 0 1 0],[-2 -3 -1 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 1 -1 -4],[-2 0 0 -1 -1 -1 -3],[-1 0 0 0 0 0 -1],[-1 1 0 0 0 -1 -3],[-1 1 0 0 0 -1 -4],[ 1 1 0 1 1 0 0],[ 4 3 1 3 4 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,-1,1,4,0,1,1,1,3,0,0,0,1,0,1,3,1,4,0]
Phi over symmetry [-4,-1,1,1,1,2,0,1,3,4,3,0,1,1,1,0,0,0,0,1,1]
Phi of -K [-4,-1,1,1,1,2,3,1,2,4,3,1,1,2,2,0,0,0,0,0,1]
Phi of K* [-2,-1,-1,-1,1,4,0,0,1,2,3,0,0,1,1,0,1,2,2,4,3]
Phi of -K* [-4,-1,1,1,1,2,0,1,3,4,3,0,1,1,1,0,0,0,0,1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^2-2t
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial 4w^3z^2+17w^2z+19w
Inner characteristic polynomial t^6+40t^4+8t^2
Outer characteristic polynomial t^7+64t^5+33t^3+3t
Flat arrow polynomial -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
2-strand cable arrow polynomial -1296*K1**4 + 224*K1**3*K2*K3 + 64*K1**3*K3*K4 - 224*K1**3*K3 + 96*K1**2*K2**2*K4 - 1328*K1**2*K2**2 - 352*K1**2*K2*K4 + 2952*K1**2*K2 - 432*K1**2*K3**2 - 96*K1**2*K3*K5 - 160*K1**2*K4**2 - 1672*K1**2 - 480*K1*K2**2*K3 - 64*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 2384*K1*K2*K3 + 1224*K1*K3*K4 + 360*K1*K4*K5 + 24*K1*K5*K6 - 72*K2**4 - 96*K2**2*K3**2 - 80*K2**2*K4**2 + 616*K2**2*K4 - 1620*K2**2 - 32*K2*K3*K4*K5 + 304*K2*K3*K5 + 80*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**2*K4**2 + 8*K3**2*K6 - 996*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 638*K4**2 - 200*K5**2 - 28*K6**2 - 4*K7**2 - 2*K8**2 + 1742
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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