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Flat knot 6.1778

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,-1,1,1,1,1,1,1,1,1,0,1,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1778']
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+16t^5+36t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1778']
2-strand cable arrow polynomial of the knot is: -384*K1**6 - 320*K1**4*K2**2 + 1952*K1**4*K2 - 5568*K1**4 + 864*K1**3*K2*K3 + 32*K1**3*K3*K4 - 864*K1**3*K3 - 192*K1**2*K2**4 + 640*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6784*K1**2*K2**2 - 512*K1**2*K2*K4 + 11752*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 5368*K1**2 + 352*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 128*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 7776*K1*K2*K3 + 1640*K1*K3*K4 + 312*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 720*K2**4 - 32*K2**3*K6 - 288*K2**2*K3**2 - 128*K2**2*K4**2 + 1488*K2**2*K4 - 5210*K2**2 + 392*K2*K3*K5 + 104*K2*K4*K6 - 2340*K3**2 - 836*K4**2 - 148*K5**2 - 22*K6**2 + 5290
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1778']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4462', 'vk6.4557', 'vk6.5844', 'vk6.5971', 'vk6.7896', 'vk6.8012', 'vk6.9323', 'vk6.9442', 'vk6.13400', 'vk6.13497', 'vk6.13684', 'vk6.14074', 'vk6.15051', 'vk6.15171', 'vk6.17790', 'vk6.17823', 'vk6.18832', 'vk6.19425', 'vk6.19718', 'vk6.24333', 'vk6.25429', 'vk6.25462', 'vk6.26601', 'vk6.33250', 'vk6.33311', 'vk6.37551', 'vk6.44878', 'vk6.48665', 'vk6.50555', 'vk6.53650', 'vk6.55823', 'vk6.65495']
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 O1O2O3U4U3O5O4U6U2O6U1U5
R3 orbit {'O1O2O3U4U3O5O4U6U2O6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U3O5U2U5O6O4U1U6
Gauss code of K* O1O2U1O3O4U3U2U5O6O5U4U6
Gauss code of -K* O1O2U3O4O3U5U1O6O5U6U4U2
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 0 1 0 1 -1],[ 1 0 1 1 0 1 0],[ 0 -1 0 1 -1 -1 0],[-1 -1 -1 0 -1 -1 -1],[ 0 0 1 1 0 1 -1],[-1 -1 1 1 -1 0 -1],[ 1 0 0 1 1 1 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 1 -1 -1 -1],[-1 -1 0 -1 -1 -1 -1],[ 0 -1 1 0 -1 0 -1],[ 0 1 1 1 0 -1 0],[ 1 1 1 0 1 0 0],[ 1 1 1 1 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,-1,1,1,1,1,1,1,1,1,0,1,1,0,0]
Phi over symmetry [-1,-1,0,0,1,1,-1,-1,1,1,1,1,1,1,1,1,0,1,1,0,0]
Phi of -K [-1,-1,0,0,1,1,0,0,1,1,1,1,0,1,1,-1,0,0,0,2,1]
Phi of K* [-1,-1,0,0,1,1,-1,0,0,1,1,0,2,1,1,1,0,1,1,0,0]
Phi of -K* [-1,-1,0,0,1,1,0,0,1,1,1,1,0,1,1,-1,-1,1,1,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+22z+37
Enhanced Jones-Krushkal polynomial 2w^3z^2+22w^2z+37w
Inner characteristic polynomial t^6+12t^4+16t^2+1
Outer characteristic polynomial t^7+16t^5+36t^3+7t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
2-strand cable arrow polynomial -384*K1**6 - 320*K1**4*K2**2 + 1952*K1**4*K2 - 5568*K1**4 + 864*K1**3*K2*K3 + 32*K1**3*K3*K4 - 864*K1**3*K3 - 192*K1**2*K2**4 + 640*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6784*K1**2*K2**2 - 512*K1**2*K2*K4 + 11752*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K3*K5 - 128*K1**2*K4**2 - 5368*K1**2 + 352*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 128*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 7776*K1*K2*K3 + 1640*K1*K3*K4 + 312*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 720*K2**4 - 32*K2**3*K6 - 288*K2**2*K3**2 - 128*K2**2*K4**2 + 1488*K2**2*K4 - 5210*K2**2 + 392*K2*K3*K5 + 104*K2*K4*K6 - 2340*K3**2 - 836*K4**2 - 148*K5**2 - 22*K6**2 + 5290
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {4}, {1, 3}, {2}], [{6}, {3, 5}, {2, 4}, {1}]]
If K is slice False
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