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Flat knot 6.1093

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,0,1,3,3,3,1,0,0,1,0,1,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1093']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.211', '6.557', '6.676', '6.685', '6.750', '6.751', '6.856', '6.919', '6.1093', '6.1371']
Outer characteristic polynomial of the knot is: t^7+32t^5+49t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1093']
2-strand cable arrow polynomial of the knot is: -704*K1**6 - 448*K1**4*K2**2 + 2432*K1**4*K2 - 5632*K1**4 + 928*K1**3*K2*K3 - 1056*K1**3*K3 - 192*K1**2*K2**4 + 1152*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6800*K1**2*K2**2 - 1120*K1**2*K2*K4 + 10160*K1**2*K2 - 992*K1**2*K3**2 - 96*K1**2*K3*K5 - 112*K1**2*K4**2 - 4380*K1**2 + 480*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 352*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7888*K1*K2*K3 + 2184*K1*K3*K4 + 456*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 816*K2**4 - 32*K2**3*K6 - 272*K2**2*K3**2 - 88*K2**2*K4**2 + 1544*K2**2*K4 - 4604*K2**2 + 616*K2*K3*K5 + 120*K2*K4*K6 + 32*K3**2*K6 - 2560*K3**2 - 1128*K4**2 - 308*K5**2 - 52*K6**2 + 4990
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1093']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4135', 'vk6.4168', 'vk6.5377', 'vk6.5410', 'vk6.7503', 'vk6.7530', 'vk6.9008', 'vk6.9041', 'vk6.12432', 'vk6.12465', 'vk6.13337', 'vk6.13556', 'vk6.13589', 'vk6.14265', 'vk6.14712', 'vk6.14725', 'vk6.15204', 'vk6.15872', 'vk6.15883', 'vk6.30833', 'vk6.30866', 'vk6.32021', 'vk6.32054', 'vk6.33061', 'vk6.33094', 'vk6.33851', 'vk6.34314', 'vk6.48493', 'vk6.50278', 'vk6.53525', 'vk6.53948', 'vk6.54260']
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 O1O2O3O4U1U4O5U3U5O6U2U6
R3 orbit {'O1O2O3O4U1U4O5U3U5O6U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3O5U6U2O6U1U4
Gauss code of K* O1O2U3O4O3U5O6O5U1U6U4U2
Gauss code of -K* O1O2U1O3O4U3O5O6U5U4U2U6
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 -3 0 0 1 1 1],[ 3 0 3 2 1 1 1],[ 0 -3 0 -1 0 1 1],[ 0 -2 1 0 0 1 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 -1 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 1 1 1 0 0 -3],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 -1 -1],[-1 0 0 0 -1 -1 -1],[ 0 0 0 1 0 1 -2],[ 0 0 1 1 -1 0 -3],[ 3 1 1 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,0,3,0,0,0,0,1,0,0,1,1,1,1,1,-1,2,3]
Phi over symmetry [-3,0,0,1,1,1,0,1,3,3,3,1,0,0,1,0,1,1,0,0,0]
Phi of -K [-3,0,0,1,1,1,0,1,3,3,3,1,0,0,1,0,1,1,0,0,0]
Phi of K* [-1,-1,-1,0,0,3,0,0,0,0,3,0,0,1,3,1,1,3,-1,0,1]
Phi of -K* [-3,0,0,1,1,1,2,3,1,1,1,1,0,0,1,0,1,1,0,0,0]
Symmetry type of based matrix c
u-polynomial t^3-3t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+20t^4+15t^2+1
Outer characteristic polynomial t^7+32t^5+49t^3+7t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 6*K1*K2 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -704*K1**6 - 448*K1**4*K2**2 + 2432*K1**4*K2 - 5632*K1**4 + 928*K1**3*K2*K3 - 1056*K1**3*K3 - 192*K1**2*K2**4 + 1152*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 6800*K1**2*K2**2 - 1120*K1**2*K2*K4 + 10160*K1**2*K2 - 992*K1**2*K3**2 - 96*K1**2*K3*K5 - 112*K1**2*K4**2 - 4380*K1**2 + 480*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 352*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7888*K1*K2*K3 + 2184*K1*K3*K4 + 456*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 + 96*K2**4*K4 - 816*K2**4 - 32*K2**3*K6 - 272*K2**2*K3**2 - 88*K2**2*K4**2 + 1544*K2**2*K4 - 4604*K2**2 + 616*K2*K3*K5 + 120*K2*K4*K6 + 32*K3**2*K6 - 2560*K3**2 - 1128*K4**2 - 308*K5**2 - 52*K6**2 + 4990
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
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