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Flat knot 6.9

Min(phi) over symmetries of the knot is: [-5,-3,1,1,3,3,1,2,4,3,5,1,3,2,4,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.9']
Arrow polynomial of the knot is: -8*K1**3*K2 + 20*K1**3 + 4*K1**2*K3 - 4*K1**2 - 8*K1*K2 - 8*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.9']
Outer characteristic polynomial of the knot is: t^7+145t^5+239t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.9']
2-strand cable arrow polynomial of the knot is: -256*K1**4 - 512*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2624*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 6816*K1**2*K2**2 - 320*K1**2*K2*K4 + 5120*K1**2*K2 - 3072*K1**2 + 256*K1*K2**4*K3*K4 - 512*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 2944*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1728*K1*K2**2*K3 - 320*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4512*K1*K2*K3 + 240*K1*K3*K4 + 16*K1*K4*K5 - 128*K2**6*K4**2 + 640*K2**6*K4 - 3936*K2**6 + 128*K2**5*K4*K6 - 256*K2**5*K6 - 512*K2**4*K3**2 - 832*K2**4*K4**2 + 5056*K2**4*K4 - 32*K2**4*K6**2 - 6928*K2**4 + 320*K2**3*K3*K5 + 384*K2**3*K4*K6 - 640*K2**3*K6 - 1504*K2**2*K3**2 - 1248*K2**2*K4**2 + 4368*K2**2*K4 - 32*K2**2*K6**2 + 1352*K2**2 + 320*K2*K3*K5 + 208*K2*K4*K6 - 848*K3**2 - 516*K4**2 - 16*K5**2 - 8*K6**2 + 2418
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.9']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81095', 'vk6.81125', 'vk6.81129', 'vk6.81200', 'vk6.81203', 'vk6.81246', 'vk6.81248', 'vk6.82056', 'vk6.82552', 'vk6.83018', 'vk6.83486', 'vk6.83488', 'vk6.83952', 'vk6.83996', 'vk6.86300', 'vk6.86302', 'vk6.88525', 'vk6.88529', 'vk6.88847', 'vk6.89889']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4O5O6U1U2U5U6U3U4
R3 orbit {'O1O2O3O4O5O6U1U2U5U6U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U4U1U2U5U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U3U4U1U2U5U6
Diagrammatic symmetry type +
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 -3 1 3 1 3],[ 5 0 1 4 5 2 3],[ 3 -1 0 3 4 1 2],[-1 -4 -3 0 1 -1 1],[-3 -5 -4 -1 0 -1 1],[-1 -2 -1 1 1 0 1],[-3 -3 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 3 1 1 -3 -5],[-3 0 1 -1 -1 -4 -5],[-3 -1 0 -1 -1 -2 -3],[-1 1 1 0 1 -1 -2],[-1 1 1 -1 0 -3 -4],[ 3 4 2 1 3 0 -1],[ 5 5 3 2 4 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-1,-1,3,5,-1,1,1,4,5,1,1,2,3,-1,1,2,3,4,1]
Phi over symmetry [-5,-3,1,1,3,3,1,2,4,3,5,1,3,2,4,1,1,1,1,1,-1]
Phi of -K [-5,-3,1,1,3,3,1,2,4,3,5,1,3,2,4,1,1,1,1,1,-1]
Phi of K* [-3,-3,-1,-1,3,5,-1,1,1,4,5,1,1,2,3,-1,1,2,3,4,1]
Phi of -K* [-5,-3,1,1,3,3,1,2,4,3,5,1,3,2,4,1,1,1,1,1,-1]
Symmetry type of based matrix +
u-polynomial t^5-t^3-2t
Normalized Jones-Krushkal polynomial z^2+6z+9
Enhanced Jones-Krushkal polynomial -6w^4z^2+7w^3z^2-16w^3z+22w^2z+9w
Inner characteristic polynomial t^6+91t^4+39t^2+1
Outer characteristic polynomial t^7+145t^5+239t^3+19t
Flat arrow polynomial -8*K1**3*K2 + 20*K1**3 + 4*K1**2*K3 - 4*K1**2 - 8*K1*K2 - 8*K1 + 2*K2 + 3
2-strand cable arrow polynomial -256*K1**4 - 512*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 2624*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 6816*K1**2*K2**2 - 320*K1**2*K2*K4 + 5120*K1**2*K2 - 3072*K1**2 + 256*K1*K2**4*K3*K4 - 512*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 2944*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1728*K1*K2**2*K3 - 320*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4512*K1*K2*K3 + 240*K1*K3*K4 + 16*K1*K4*K5 - 128*K2**6*K4**2 + 640*K2**6*K4 - 3936*K2**6 + 128*K2**5*K4*K6 - 256*K2**5*K6 - 512*K2**4*K3**2 - 832*K2**4*K4**2 + 5056*K2**4*K4 - 32*K2**4*K6**2 - 6928*K2**4 + 320*K2**3*K3*K5 + 384*K2**3*K4*K6 - 640*K2**3*K6 - 1504*K2**2*K3**2 - 1248*K2**2*K4**2 + 4368*K2**2*K4 - 32*K2**2*K6**2 + 1352*K2**2 + 320*K2*K3*K5 + 208*K2*K4*K6 - 848*K3**2 - 516*K4**2 - 16*K5**2 - 8*K6**2 + 2418
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}]]
If K is slice False
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