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Flat knot 6.1714

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1714']
Arrow polynomial of the knot is: -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.372', '6.930', '6.1007', '6.1701', '6.1714', '6.1760', '6.1788']
Outer characteristic polynomial of the knot is: t^7+28t^5+23t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1607', '6.1714']
2-strand cable arrow polynomial of the knot is: -448*K1**6 - 192*K1**4*K2**2 + 1824*K1**4*K2 - 5776*K1**4 + 384*K1**3*K2*K3 - 960*K1**3*K3 + 32*K1**2*K2**2*K4 - 4768*K1**2*K2**2 - 192*K1**2*K2*K4 + 11504*K1**2*K2 - 1072*K1**2*K3**2 - 160*K1**2*K4**2 - 6004*K1**2 - 576*K1*K2**2*K3 - 256*K1*K2*K3*K4 + 7320*K1*K2*K3 + 1760*K1*K3*K4 + 224*K1*K4*K5 - 456*K2**4 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 1112*K2**2*K4 - 5608*K2**2 + 472*K2*K3*K5 + 128*K2*K4*K6 - 2580*K3**2 - 874*K4**2 - 152*K5**2 - 32*K6**2 + 5856
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1714']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20020', 'vk6.20069', 'vk6.21292', 'vk6.21349', 'vk6.27067', 'vk6.27134', 'vk6.28772', 'vk6.28821', 'vk6.38460', 'vk6.38535', 'vk6.40649', 'vk6.40730', 'vk6.45340', 'vk6.45435', 'vk6.47109', 'vk6.47175', 'vk6.56835', 'vk6.56874', 'vk6.57969', 'vk6.58010', 'vk6.61349', 'vk6.61404', 'vk6.62525', 'vk6.62559', 'vk6.66547', 'vk6.66582', 'vk6.67336', 'vk6.67371', 'vk6.69189', 'vk6.69234', 'vk6.69940', 'vk6.69973']
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 O1O2O3U1U4O5O6U2U6O4U3U5
R3 orbit {'O1O2O3U1U4O5O6U2U6O4U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1O5U6U2O6O4U5U3
Gauss code of K* O1O2U3O4O5U6U1U4O6O3U5U2
Gauss code of -K* O1O2U3O4O5U4U1O3O6U2U5U6
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 -2 -1 1 0 1 1],[ 2 0 1 2 1 2 1],[ 1 -1 0 1 0 2 1],[-1 -2 -1 0 -1 0 0],[ 0 -1 0 1 0 1 1],[-1 -2 -2 0 -1 0 0],[-1 -1 -1 0 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 -1 -1 -1],[-1 0 0 0 -1 -1 -2],[-1 0 0 0 -1 -2 -2],[ 0 1 1 1 0 0 -1],[ 1 1 1 2 0 0 -1],[ 2 1 2 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,0,0,1,1,1,0,1,1,2,1,2,2,0,1,1]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,0,0,0,0,0,0]
Phi of -K [-2,-1,0,1,1,1,0,1,1,1,2,1,0,1,1,0,0,0,0,0,0]
Phi of K* [-1,-1,-1,0,1,2,0,0,0,0,1,0,0,1,1,0,1,2,1,1,0]
Phi of -K* [-2,-1,0,1,1,1,1,1,1,2,2,0,1,1,2,1,1,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial z^2+22z+41
Enhanced Jones-Krushkal polynomial w^3z^2+22w^2z+41w
Inner characteristic polynomial t^6+20t^4+10t^2+1
Outer characteristic polynomial t^7+28t^5+23t^3+4t
Flat arrow polynomial -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6
2-strand cable arrow polynomial -448*K1**6 - 192*K1**4*K2**2 + 1824*K1**4*K2 - 5776*K1**4 + 384*K1**3*K2*K3 - 960*K1**3*K3 + 32*K1**2*K2**2*K4 - 4768*K1**2*K2**2 - 192*K1**2*K2*K4 + 11504*K1**2*K2 - 1072*K1**2*K3**2 - 160*K1**2*K4**2 - 6004*K1**2 - 576*K1*K2**2*K3 - 256*K1*K2*K3*K4 + 7320*K1*K2*K3 + 1760*K1*K3*K4 + 224*K1*K4*K5 - 456*K2**4 - 384*K2**2*K3**2 - 128*K2**2*K4**2 + 1112*K2**2*K4 - 5608*K2**2 + 472*K2*K3*K5 + 128*K2*K4*K6 - 2580*K3**2 - 874*K4**2 - 152*K5**2 - 32*K6**2 + 5856
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}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice False
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