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Flat knot 6.770

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,2,1,3,3,0,0,1,1,1,2,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.770']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 10*K1*K2 - K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.770', '6.1299', '6.1366']
Outer characteristic polynomial of the knot is: t^7+58t^5+28t^3+2t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.770']
2-strand cable arrow polynomial of the knot is: 3904*K1**4*K2 - 7808*K1**4 + 1696*K1**3*K2*K3 - 1792*K1**3*K3 - 384*K1**2*K2**4 + 1728*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 10656*K1**2*K2**2 - 1888*K1**2*K2*K4 + 10872*K1**2*K2 - 1312*K1**2*K3**2 - 96*K1**2*K4**2 - 2472*K1**2 + 1152*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 640*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 9296*K1*K2*K3 + 2176*K1*K3*K4 + 280*K1*K4*K5 - 192*K2**6 + 384*K2**4*K4 - 1808*K2**4 - 96*K2**3*K6 - 800*K2**2*K3**2 - 264*K2**2*K4**2 + 2264*K2**2*K4 - 3538*K2**2 + 824*K2*K3*K5 + 168*K2*K4*K6 - 2200*K3**2 - 1016*K4**2 - 248*K5**2 - 30*K6**2 + 4126
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.770']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13925', 'vk6.14022', 'vk6.14193', 'vk6.14432', 'vk6.14992', 'vk6.15115', 'vk6.15665', 'vk6.16119', 'vk6.16705', 'vk6.16730', 'vk6.16853', 'vk6.18797', 'vk6.19272', 'vk6.19564', 'vk6.23139', 'vk6.23232', 'vk6.25391', 'vk6.26457', 'vk6.33736', 'vk6.33813', 'vk6.34288', 'vk6.35132', 'vk6.37524', 'vk6.42743', 'vk6.44681', 'vk6.54128', 'vk6.54922', 'vk6.54949', 'vk6.56391', 'vk6.56607', 'vk6.59346', 'vk6.64599']
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 O1O2O3O4U3O5U1U6U4O6U5U2
R3 orbit {'O1O2O3O4U3O5U1U6U4O6U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5O6U1U6U4O5U2
Gauss code of K* O1O2O3U2O4O5U1U5U6U3O6U4
Gauss code of -K* O1O2O3U4O5U1U5U6U3O6O4U2
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 -3 1 -1 2 2 -1],[ 3 0 3 0 2 2 2],[-1 -3 0 -1 1 1 -2],[ 1 0 1 0 1 1 0],[-2 -2 -1 -1 0 0 -2],[-2 -2 -1 -1 0 0 -2],[ 1 -2 2 0 2 2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 -1 -1 -2 -2],[-2 0 0 -1 -1 -2 -2],[-1 1 1 0 -1 -2 -3],[ 1 1 1 1 0 0 0],[ 1 2 2 2 0 0 -2],[ 3 2 2 3 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,1,1,2,2,1,1,2,2,1,2,3,0,0,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,2,1,3,3,0,0,1,1,1,2,2,0,0,0]
Phi of -K [-3,-1,-1,1,2,2,0,2,1,3,3,0,0,1,1,1,2,2,0,0,0]
Phi of K* [-2,-2,-1,1,1,3,0,0,1,2,3,0,1,2,3,0,1,1,0,0,2]
Phi of -K* [-3,-1,-1,1,2,2,0,2,3,2,2,0,1,1,1,2,2,2,1,1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+38t^4+14t^2
Outer characteristic polynomial t^7+58t^5+28t^3+2t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 10*K1*K2 - K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial 3904*K1**4*K2 - 7808*K1**4 + 1696*K1**3*K2*K3 - 1792*K1**3*K3 - 384*K1**2*K2**4 + 1728*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 10656*K1**2*K2**2 - 1888*K1**2*K2*K4 + 10872*K1**2*K2 - 1312*K1**2*K3**2 - 96*K1**2*K4**2 - 2472*K1**2 + 1152*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 640*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 9296*K1*K2*K3 + 2176*K1*K3*K4 + 280*K1*K4*K5 - 192*K2**6 + 384*K2**4*K4 - 1808*K2**4 - 96*K2**3*K6 - 800*K2**2*K3**2 - 264*K2**2*K4**2 + 2264*K2**2*K4 - 3538*K2**2 + 824*K2*K3*K5 + 168*K2*K4*K6 - 2200*K3**2 - 1016*K4**2 - 248*K5**2 - 30*K6**2 + 4126
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}], [{2, 6}, {5}, {4}, {1, 3}]]
If K is slice False
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