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Flat knot 6.1833

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,0,1,1,1,1,0,1,1,1,-1,-1,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1833']
Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.313', '6.623', '6.1031', '6.1201', '6.1327', '6.1378', '6.1640', '6.1697', '6.1797', '6.1833']
Outer characteristic polynomial of the knot is: t^7+40t^5+131t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1833']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2**3 - 832*K1**4*K2**2 + 1344*K1**4*K2 - 1152*K1**4 - 128*K1**3*K2**2*K3 + 384*K1**3*K2*K3 - 288*K1**3*K3 - 576*K1**2*K2**4 + 2816*K1**2*K2**3 - 9456*K1**2*K2**2 - 256*K1**2*K2*K4 + 7408*K1**2*K2 - 4024*K1**2 + 992*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 96*K1*K2**2*K5 + 6320*K1*K2*K3 + 80*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 1928*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 1072*K2**2*K4 - 1912*K2**2 + 48*K2*K3*K5 - 1128*K3**2 - 102*K4**2 + 2852
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1833']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4713', 'vk6.5026', 'vk6.6236', 'vk6.6690', 'vk6.8210', 'vk6.8644', 'vk6.9588', 'vk6.9919', 'vk6.20301', 'vk6.21634', 'vk6.27597', 'vk6.29149', 'vk6.39019', 'vk6.41267', 'vk6.45787', 'vk6.47464', 'vk6.48753', 'vk6.48952', 'vk6.49552', 'vk6.49768', 'vk6.50763', 'vk6.50965', 'vk6.51240', 'vk6.51449', 'vk6.57164', 'vk6.58352', 'vk6.61790', 'vk6.62909', 'vk6.66781', 'vk6.67657', 'vk6.69429', 'vk6.70151']
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 O1O2O3U1U4O5U6O4O6U2U3U5
R3 orbit {'O1O2O3U1U4O5U6O4O6U2U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1U2O5O6U5O4U6U3
Gauss code of K* O1O2O3U4U1U2O4O5U3O6U5U6
Gauss code of -K* O1O2O3U4U5O4U1O5O6U2U3U6
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 2 2 2],[ 1 -1 0 1 0 1 2],[-1 -2 -1 0 -2 0 0],[ 0 -2 0 2 0 2 0],[-1 -2 -1 0 -2 0 -1],[-1 -2 -2 0 0 1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -2 -2],[-1 -1 0 0 -2 -1 -2],[-1 0 0 0 -2 -1 -2],[ 0 0 2 2 0 0 -2],[ 1 2 1 1 0 0 -1],[ 2 2 2 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,0,2,2,0,2,1,2,2,1,2,0,2,1]
Phi over symmetry [-2,-1,0,1,1,1,0,0,1,1,1,1,0,1,1,1,-1,-1,-1,0,0]
Phi of -K [-2,-1,0,1,1,1,0,0,1,1,1,1,0,1,1,1,-1,-1,-1,0,0]
Phi of K* [-1,-1,-1,0,1,2,-1,0,-1,1,1,0,1,0,1,-1,1,1,1,0,0]
Phi of -K* [-2,-1,0,1,1,1,1,2,2,2,2,0,1,1,2,2,2,0,0,-1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+32t^4+90t^2
Outer characteristic polynomial t^7+40t^5+131t^3+13t
Flat arrow polynomial 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2
2-strand cable arrow polynomial 128*K1**4*K2**3 - 832*K1**4*K2**2 + 1344*K1**4*K2 - 1152*K1**4 - 128*K1**3*K2**2*K3 + 384*K1**3*K2*K3 - 288*K1**3*K3 - 576*K1**2*K2**4 + 2816*K1**2*K2**3 - 9456*K1**2*K2**2 - 256*K1**2*K2*K4 + 7408*K1**2*K2 - 4024*K1**2 + 992*K1*K2**3*K3 - 1248*K1*K2**2*K3 - 96*K1*K2**2*K5 + 6320*K1*K2*K3 + 80*K1*K3*K4 - 64*K2**6 + 128*K2**4*K4 - 1928*K2**4 - 336*K2**2*K3**2 - 48*K2**2*K4**2 + 1072*K2**2*K4 - 1912*K2**2 + 48*K2*K3*K5 - 1128*K3**2 - 102*K4**2 + 2852
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}, {4, 5}, {1, 2}]]
If K is slice False
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