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Flat knot 6.250

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,1,1,2,3,4,0,1,2,2,1,2,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.250']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 4*K1**2*K2 + 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.250', '6.309']
Outer characteristic polynomial of the knot is: t^7+122t^5+102t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.250']
2-strand cable arrow polynomial of the knot is: -384*K1**4 - 1024*K1**2*K2**6 + 3328*K1**2*K2**5 - 6400*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 5056*K1**2*K2**3 - 5536*K1**2*K2**2 - 96*K1**2*K2*K4 + 3736*K1**2*K2 - 32*K1**2*K3**2 - 1932*K1**2 + 1408*K1*K2**5*K3 - 1536*K1*K2**4*K3 - 128*K1*K2**4*K5 + 3168*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 96*K1*K2**2*K5 + 2608*K1*K2*K3 + 96*K1*K3*K4 - 128*K2**8 + 128*K2**6*K4 - 3008*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 1632*K2**4*K4 - 1032*K2**4 + 32*K2**3*K3*K5 - 368*K2**2*K3**2 - 80*K2**2*K4**2 + 1208*K2**2*K4 + 104*K2**2 + 72*K2*K3*K5 - 368*K3**2 - 138*K4**2 - 4*K5**2 + 1344
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.250']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81816', 'vk6.81818', 'vk6.82033', 'vk6.82037', 'vk6.82527', 'vk6.82530', 'vk6.82769', 'vk6.82771', 'vk6.82966', 'vk6.82968', 'vk6.83044', 'vk6.83046', 'vk6.83515', 'vk6.83517', 'vk6.83891', 'vk6.83894', 'vk6.84510', 'vk6.84512', 'vk6.84878', 'vk6.84880', 'vk6.85790', 'vk6.85796', 'vk6.86047', 'vk6.86051', 'vk6.86380', 'vk6.86385', 'vk6.86831', 'vk6.86832', 'vk6.88799', 'vk6.88809', 'vk6.89859', 'vk6.89864']
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 O1O2O3O4O5U1U2O6U3U4U5U6
R3 orbit {'O1O2O3O4O5U1U2O6U3U4U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U1U2U3O6U4U5
Gauss code of K* O1O2O3O4U5U6U1U2U3O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U2U3U4U5U6
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 -4 -2 -1 1 3 3],[ 4 0 1 2 3 4 3],[ 2 -1 0 1 2 3 3],[ 1 -2 -1 0 1 2 3],[-1 -3 -2 -1 0 1 2],[-3 -4 -3 -2 -1 0 1],[-3 -3 -3 -3 -2 -1 0]]
Primitive based matrix [[ 0 3 3 1 -1 -2 -4],[-3 0 1 -1 -2 -3 -4],[-3 -1 0 -2 -3 -3 -3],[-1 1 2 0 -1 -2 -3],[ 1 2 3 1 0 -1 -2],[ 2 3 3 2 1 0 -1],[ 4 4 3 3 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-1,1,2,4,-1,1,2,3,4,2,3,3,3,1,2,3,1,2,1]
Phi over symmetry [-4,-2,-1,1,3,3,1,1,2,3,4,0,1,2,2,1,2,1,1,0,-1]
Phi of -K [-4,-2,-1,1,3,3,1,1,2,3,4,0,1,2,2,1,2,1,1,0,-1]
Phi of K* [-3,-3,-1,1,2,4,-1,0,1,2,4,1,2,2,3,1,1,2,0,1,1]
Phi of -K* [-4,-2,-1,1,3,3,1,2,3,3,4,1,2,3,3,1,3,2,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial z^2+6z+9
Enhanced Jones-Krushkal polynomial -6w^4z^2+7w^3z^2-10w^3z+16w^2z+9w
Inner characteristic polynomial t^6+82t^4+35t^2
Outer characteristic polynomial t^7+122t^5+102t^3+6t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 4*K1**2*K2 + 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2
2-strand cable arrow polynomial -384*K1**4 - 1024*K1**2*K2**6 + 3328*K1**2*K2**5 - 6400*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 5056*K1**2*K2**3 - 5536*K1**2*K2**2 - 96*K1**2*K2*K4 + 3736*K1**2*K2 - 32*K1**2*K3**2 - 1932*K1**2 + 1408*K1*K2**5*K3 - 1536*K1*K2**4*K3 - 128*K1*K2**4*K5 + 3168*K1*K2**3*K3 - 1088*K1*K2**2*K3 - 96*K1*K2**2*K5 + 2608*K1*K2*K3 + 96*K1*K3*K4 - 128*K2**8 + 128*K2**6*K4 - 3008*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 1632*K2**4*K4 - 1032*K2**4 + 32*K2**3*K3*K5 - 368*K2**2*K3**2 - 80*K2**2*K4**2 + 1208*K2**2*K4 + 104*K2**2 + 72*K2*K3*K5 - 368*K3**2 - 138*K4**2 - 4*K5**2 + 1344
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
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