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Flat knot 6.284

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,1,4,2,4,0,2,1,1,2,2,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.284']
Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511']
Outer characteristic polynomial of the knot is: t^7+90t^5+120t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.284']
2-strand cable arrow polynomial of the knot is: -144*K1**4 + 384*K1**3*K2*K3 - 384*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 1024*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 128*K1**2*K2*K4 + 2384*K1**2*K2 - 1296*K1**2*K3**2 - 3224*K1**2 + 416*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 704*K1*K2**2*K3 - 32*K1*K2**2*K5 + 384*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5208*K1*K2*K3 - 96*K1*K3**2*K5 + 1240*K1*K3*K4 + 32*K1*K4*K5 + 24*K1*K5*K6 - 192*K2**4 - 1568*K2**2*K3**2 - 40*K2**2*K4**2 + 656*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3122*K2**2 - 96*K2*K3**2*K4 + 1752*K2*K3*K5 + 48*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 - 192*K3**4 + 160*K3**2*K6 - 2340*K3**2 + 8*K3*K4*K7 + 24*K3*K5*K8 - 458*K4**2 - 528*K5**2 - 46*K6**2 - 12*K7**2 - 18*K8**2 + 3162
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.284']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16924', 'vk6.17167', 'vk6.20228', 'vk6.21523', 'vk6.23314', 'vk6.23610', 'vk6.27434', 'vk6.29044', 'vk6.35348', 'vk6.35774', 'vk6.38848', 'vk6.41038', 'vk6.42833', 'vk6.43114', 'vk6.45605', 'vk6.47363', 'vk6.55078', 'vk6.55330', 'vk6.57064', 'vk6.58192', 'vk6.59467', 'vk6.59760', 'vk6.61588', 'vk6.62767', 'vk6.64914', 'vk6.65127', 'vk6.66686', 'vk6.67529', 'vk6.68215', 'vk6.68360', 'vk6.69334', 'vk6.70086']
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 O1O2O3O4O5U1U5O6U3U2U4U6
R3 orbit {'O1O2O3O4O5U1U5O6U3U2U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U2U4U3O6U1U5
Gauss code of K* O1O2O3O4U5U2U1U3U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U2U4U3U6
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 -1 -1 2 1 3],[ 4 0 3 2 4 1 3],[ 1 -3 0 0 2 0 3],[ 1 -2 0 0 1 0 2],[-2 -4 -2 -1 0 0 1],[-1 -1 0 0 0 0 0],[-3 -3 -3 -2 -1 0 0]]
Primitive based matrix [[ 0 3 2 1 -1 -1 -4],[-3 0 -1 0 -2 -3 -3],[-2 1 0 0 -1 -2 -4],[-1 0 0 0 0 0 -1],[ 1 2 1 0 0 0 -2],[ 1 3 2 0 0 0 -3],[ 4 3 4 1 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,1,4,1,0,2,3,3,0,1,2,4,0,0,1,0,2,3]
Phi over symmetry [-4,-1,-1,1,2,3,0,1,4,2,4,0,2,1,1,2,2,2,1,2,0]
Phi of -K [-4,-1,-1,1,2,3,0,1,4,2,4,0,2,1,1,2,2,2,1,2,0]
Phi of K* [-3,-2,-1,1,1,4,0,2,1,2,4,1,1,2,2,2,2,4,0,0,1]
Phi of -K* [-4,-1,-1,1,2,3,2,3,1,4,3,0,0,1,2,0,2,3,0,0,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2-4w^3z+26w^2z+25w
Inner characteristic polynomial t^6+58t^4+39t^2+1
Outer characteristic polynomial t^7+90t^5+120t^3+13t
Flat arrow polynomial -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial -144*K1**4 + 384*K1**3*K2*K3 - 384*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 1024*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 128*K1**2*K2*K4 + 2384*K1**2*K2 - 1296*K1**2*K3**2 - 3224*K1**2 + 416*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 704*K1*K2**2*K3 - 32*K1*K2**2*K5 + 384*K1*K2*K3**3 - 224*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 5208*K1*K2*K3 - 96*K1*K3**2*K5 + 1240*K1*K3*K4 + 32*K1*K4*K5 + 24*K1*K5*K6 - 192*K2**4 - 1568*K2**2*K3**2 - 40*K2**2*K4**2 + 656*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3122*K2**2 - 96*K2*K3**2*K4 + 1752*K2*K3*K5 + 48*K2*K4*K6 + 24*K2*K5*K7 + 16*K2*K6*K8 - 192*K3**4 + 160*K3**2*K6 - 2340*K3**2 + 8*K3*K4*K7 + 24*K3*K5*K8 - 458*K4**2 - 528*K5**2 - 46*K6**2 - 12*K7**2 - 18*K8**2 + 3162
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
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