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Flat knot 6.1209

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,1,1,1,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1209']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1209', '6.1245', '6.1509', '6.1541', '6.1704', '6.1778', '6.1914']
Outer characteristic polynomial of the knot is: t^7+40t^5+32t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1209']
2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 256*K1**4*K2 - 848*K1**4 + 160*K1**3*K2*K3 - 352*K1**3*K3 + 192*K1**2*K2**3 - 1840*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 128*K1**2*K2*K4 + 3800*K1**2*K2 - 304*K1**2*K3**2 - 16*K1**2*K4**2 - 32*K1**2*K5**2 - 3112*K1**2 + 64*K1*K2**3*K3 - 512*K1*K2**2*K3 - 192*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 3328*K1*K2*K3 - 32*K1*K2*K4*K5 + 664*K1*K3*K4 + 216*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 352*K2**4 - 32*K2**3*K6 - 176*K2**2*K3**2 - 64*K2**2*K4**2 + 800*K2**2*K4 - 2458*K2**2 + 392*K2*K3*K5 + 56*K2*K4*K6 - 1232*K3**2 - 460*K4**2 - 184*K5**2 - 22*K6**2 + 2498
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1209']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4434', 'vk6.4529', 'vk6.5816', 'vk6.5943', 'vk6.7881', 'vk6.7992', 'vk6.9303', 'vk6.9422', 'vk6.10169', 'vk6.10240', 'vk6.10387', 'vk6.17877', 'vk6.17940', 'vk6.18281', 'vk6.18618', 'vk6.24384', 'vk6.24684', 'vk6.25169', 'vk6.30064', 'vk6.30125', 'vk6.30906', 'vk6.31029', 'vk6.32094', 'vk6.32213', 'vk6.36891', 'vk6.37270', 'vk6.37351', 'vk6.43819', 'vk6.44108', 'vk6.44433', 'vk6.50528', 'vk6.50613', 'vk6.51133', 'vk6.51998', 'vk6.52093', 'vk6.55828', 'vk6.56071', 'vk6.60554', 'vk6.60894', 'vk6.65963']
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 O1O2O3O4U2U5U3O5O6U4U1U6
R3 orbit {'O1O2O3U1O4U5U3O5O6U2U4U6', 'O1O2O3O4U2U5U3O5O6U4U1U6'}
R3 orbit length 2
Gauss code of -K O1O2O3O4U5U4U1O5O6U2U6U3
Gauss code of K* O1O2O3U2U4O5O6O4U6U1U3U5
Gauss code of -K* O1O2O3U1U4O5O4O6U3U5U6U2
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 -1 -2 1 1 -1 2],[ 1 0 -2 1 2 0 2],[ 2 2 0 1 2 1 1],[-1 -1 -1 0 0 -1 1],[-1 -2 -2 0 0 -1 1],[ 1 0 -1 1 1 0 2],[-2 -2 -1 -1 -1 -2 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -1 -2 -2 -1],[-1 1 0 0 -1 -1 -1],[-1 1 0 0 -1 -2 -2],[ 1 2 1 1 0 0 -1],[ 1 2 1 2 0 0 -2],[ 2 1 1 2 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,1,1,2,2,1,0,1,1,1,1,2,2,0,1,2]
Phi over symmetry [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,1,1,1,1,1,0,0,0]
Phi of -K [-2,-1,-1,1,1,2,-1,0,1,2,3,0,0,1,1,1,1,1,0,0,0]
Phi of K* [-2,-1,-1,1,1,2,0,0,1,1,3,0,0,1,1,1,1,2,0,-1,0]
Phi of -K* [-2,-1,-1,1,1,2,1,2,1,2,1,0,1,1,2,1,2,2,0,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+14z+25
Enhanced Jones-Krushkal polynomial w^3z^2+14w^2z+25w
Inner characteristic polynomial t^6+28t^4+18t^2
Outer characteristic polynomial t^7+40t^5+32t^3+3t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
2-strand cable arrow polynomial -64*K1**4*K2**2 + 256*K1**4*K2 - 848*K1**4 + 160*K1**3*K2*K3 - 352*K1**3*K3 + 192*K1**2*K2**3 - 1840*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 128*K1**2*K2*K4 + 3800*K1**2*K2 - 304*K1**2*K3**2 - 16*K1**2*K4**2 - 32*K1**2*K5**2 - 3112*K1**2 + 64*K1*K2**3*K3 - 512*K1*K2**2*K3 - 192*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 3328*K1*K2*K3 - 32*K1*K2*K4*K5 + 664*K1*K3*K4 + 216*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 352*K2**4 - 32*K2**3*K6 - 176*K2**2*K3**2 - 64*K2**2*K4**2 + 800*K2**2*K4 - 2458*K2**2 + 392*K2*K3*K5 + 56*K2*K4*K6 - 1232*K3**2 - 460*K4**2 - 184*K5**2 - 22*K6**2 + 2498
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}]]
If K is slice False
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