Table of flat knot invariants
Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo
Glossary Reference List

Flat knot 6.1669

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,1,1,1,1,1,-1,0,1,1,0,0,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1669', '6.1775']
Arrow polynomial of the knot is: -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.665', '6.1301', '6.1514', '6.1646', '6.1669', '6.1709', '6.1710', '6.1744', '6.1776']
Outer characteristic polynomial of the knot is: t^7+17t^5+25t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1669', '6.1775']
2-strand cable arrow polynomial of the knot is: -640*K1**6 - 256*K1**4*K2**2 + 2144*K1**4*K2 - 6784*K1**4 + 416*K1**3*K2*K3 - 1312*K1**3*K3 - 5744*K1**2*K2**2 - 384*K1**2*K2*K4 + 13384*K1**2*K2 - 928*K1**2*K3**2 - 160*K1**2*K4**2 - 6408*K1**2 - 640*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 8320*K1*K2*K3 + 1592*K1*K3*K4 + 168*K1*K4*K5 - 280*K2**4 - 208*K2**2*K3**2 - 48*K2**2*K4**2 + 816*K2**2*K4 - 5956*K2**2 + 184*K2*K3*K5 + 32*K2*K4*K6 - 2636*K3**2 - 670*K4**2 - 76*K5**2 - 4*K6**2 + 6108
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1669']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4773', 'vk6.4785', 'vk6.5110', 'vk6.5122', 'vk6.6339', 'vk6.6768', 'vk6.6780', 'vk6.8292', 'vk6.8312', 'vk6.8745', 'vk6.9662', 'vk6.9682', 'vk6.9973', 'vk6.9993', 'vk6.21009', 'vk6.21026', 'vk6.22433', 'vk6.22448', 'vk6.28464', 'vk6.40237', 'vk6.40254', 'vk6.42167', 'vk6.46739', 'vk6.46756', 'vk6.48812', 'vk6.49030', 'vk6.49042', 'vk6.49846', 'vk6.49858', 'vk6.51504', 'vk6.58957', 'vk6.69799']
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 O1O2O3U2O4U3U5U1O6O5U6U4
R3 orbit {'O1O2O3U2O4U3U5U1O6O5U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O6O5U3U6U1O4U2
Gauss code of K* O1O2O3U4U2O4O5U3U6U1O6U5
Gauss code of -K* O1O2O3U4O5U3U5U1O4O6U2U6
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 0 -1 0 2 0 -1],[ 0 0 -1 1 1 0 -1],[ 1 1 0 1 1 1 0],[ 0 -1 -1 0 1 0 0],[-2 -1 -1 -1 0 -1 -1],[ 0 0 -1 0 1 0 -1],[ 1 1 0 0 1 1 0]]
Primitive based matrix [[ 0 2 0 0 0 -1 -1],[-2 0 -1 -1 -1 -1 -1],[ 0 1 0 1 0 -1 -1],[ 0 1 -1 0 0 0 -1],[ 0 1 0 0 0 -1 -1],[ 1 1 1 0 1 0 0],[ 1 1 1 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,0,0,0,1,1,1,1,1,1,1,-1,0,1,1,0,0,1,1,1,0]
Phi over symmetry [-2,0,0,0,1,1,1,1,1,1,1,-1,0,1,1,0,0,1,1,1,0]
Phi of -K [-1,-1,0,0,0,2,0,0,0,0,2,0,0,1,2,0,-1,1,0,1,1]
Phi of K* [-2,0,0,0,1,1,1,1,1,2,2,-1,0,0,1,0,0,0,0,0,0]
Phi of -K* [-1,-1,0,0,0,2,0,0,1,1,1,1,1,1,1,-1,0,1,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 21z+43
Enhanced Jones-Krushkal polynomial 21w^2z+43w
Inner characteristic polynomial t^6+11t^4+12t^2
Outer characteristic polynomial t^7+17t^5+25t^3+4t
Flat arrow polynomial -14*K1**2 - 4*K1*K2 + 2*K1 + 7*K2 + 2*K3 + 8
2-strand cable arrow polynomial -640*K1**6 - 256*K1**4*K2**2 + 2144*K1**4*K2 - 6784*K1**4 + 416*K1**3*K2*K3 - 1312*K1**3*K3 - 5744*K1**2*K2**2 - 384*K1**2*K2*K4 + 13384*K1**2*K2 - 928*K1**2*K3**2 - 160*K1**2*K4**2 - 6408*K1**2 - 640*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 8320*K1*K2*K3 + 1592*K1*K3*K4 + 168*K1*K4*K5 - 280*K2**4 - 208*K2**2*K3**2 - 48*K2**2*K4**2 + 816*K2**2*K4 - 5956*K2**2 + 184*K2*K3*K5 + 32*K2*K4*K6 - 2636*K3**2 - 670*K4**2 - 76*K5**2 - 4*K6**2 + 6108
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
Contact