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Flat knot 6.893

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,1,1,2,4,0,1,0,2,0,1,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.893']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.893']
Outer characteristic polynomial of the knot is: t^7+53t^5+78t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.893']
2-strand cable arrow polynomial of the knot is: 128*K1**4*K2**3 - 640*K1**4*K2**2 + 1056*K1**4*K2 - 1504*K1**4 - 256*K1**3*K2**2*K3 + 576*K1**3*K2*K3 - 320*K1**3*K3 - 512*K1**2*K2**4 + 2208*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 7808*K1**2*K2**2 - 704*K1**2*K2*K4 + 6904*K1**2*K2 - 160*K1**2*K3**2 - 4716*K1**2 + 2016*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8208*K1*K2*K3 - 64*K1*K2*K4*K5 + 960*K1*K3*K4 + 80*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 - 192*K2**4*K3**2 - 64*K2**4*K4**2 + 352*K2**4*K4 - 2792*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 2304*K2**2*K3**2 - 32*K2**2*K3*K7 - 536*K2**2*K4**2 + 2424*K2**2*K4 - 112*K2**2*K5**2 - 48*K2**2*K6**2 - 3030*K2**2 - 128*K2*K3**2*K4 + 1344*K2*K3*K5 + 352*K2*K4*K6 + 8*K2*K5*K7 + 16*K3**2*K6 - 2380*K3**2 - 778*K4**2 - 216*K5**2 - 66*K6**2 + 4248
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.893']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3613', 'vk6.3684', 'vk6.3877', 'vk6.3998', 'vk6.7035', 'vk6.7072', 'vk6.7249', 'vk6.7368', 'vk6.17703', 'vk6.17750', 'vk6.24250', 'vk6.24309', 'vk6.36553', 'vk6.36628', 'vk6.43659', 'vk6.43764', 'vk6.48241', 'vk6.48308', 'vk6.48393', 'vk6.48422', 'vk6.49997', 'vk6.50026', 'vk6.50111', 'vk6.50142', 'vk6.55743', 'vk6.55798', 'vk6.60315', 'vk6.60396', 'vk6.65443', 'vk6.65470', 'vk6.68571', 'vk6.68598']
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 O1O2O3O4U2U3O5O6U5U1U6U4
R3 orbit {'O1O2O3O4U2U3O5O6U5U1U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U5U4U6O5O6U2U3
Gauss code of K* O1O2O3O4U2U5U6U4O5O6U1U3
Gauss code of -K* O1O2O3O4U2U4O5O6U1U5U6U3
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 -2 0 3 -1 2],[ 2 0 -1 1 4 0 2],[ 2 1 0 1 2 0 0],[ 0 -1 -1 0 1 0 0],[-3 -4 -2 -1 0 -1 1],[ 1 0 0 0 1 0 1],[-2 -2 0 0 -1 -1 0]]
Primitive based matrix [[ 0 3 2 0 -1 -2 -2],[-3 0 1 -1 -1 -2 -4],[-2 -1 0 0 -1 0 -2],[ 0 1 0 0 0 -1 -1],[ 1 1 1 0 0 0 0],[ 2 2 0 1 0 0 1],[ 2 4 2 1 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,0,1,2,2,-1,1,1,2,4,0,1,0,2,0,1,1,0,0,-1]
Phi over symmetry [-3,-2,0,1,2,2,-1,1,1,2,4,0,1,0,2,0,1,1,0,0,-1]
Phi of -K [-2,-2,-1,0,2,3,-1,1,1,4,3,1,1,2,1,1,2,3,2,2,2]
Phi of K* [-3,-2,0,1,2,2,2,2,3,1,3,2,2,2,4,1,1,1,1,1,-1]
Phi of -K* [-2,-2,-1,0,2,3,-1,0,1,2,4,0,1,0,2,0,1,1,0,1,-1]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+31t^4+39t^2+1
Outer characteristic polynomial t^7+53t^5+78t^3+14t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial 128*K1**4*K2**3 - 640*K1**4*K2**2 + 1056*K1**4*K2 - 1504*K1**4 - 256*K1**3*K2**2*K3 + 576*K1**3*K2*K3 - 320*K1**3*K3 - 512*K1**2*K2**4 + 2208*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 7808*K1**2*K2**2 - 704*K1**2*K2*K4 + 6904*K1**2*K2 - 160*K1**2*K3**2 - 4716*K1**2 + 2016*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8208*K1*K2*K3 - 64*K1*K2*K4*K5 + 960*K1*K3*K4 + 80*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 - 192*K2**4*K3**2 - 64*K2**4*K4**2 + 352*K2**4*K4 - 2792*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 2304*K2**2*K3**2 - 32*K2**2*K3*K7 - 536*K2**2*K4**2 + 2424*K2**2*K4 - 112*K2**2*K5**2 - 48*K2**2*K6**2 - 3030*K2**2 - 128*K2*K3**2*K4 + 1344*K2*K3*K5 + 352*K2*K4*K6 + 8*K2*K5*K7 + 16*K3**2*K6 - 2380*K3**2 - 778*K4**2 - 216*K5**2 - 66*K6**2 + 4248
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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