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

Flat knot 6.1811

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,2,2,1,1,0,1,1,0,2,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1811']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.927', '6.1364', '6.1367', '6.1540', '6.1675', '6.1779', '6.1811', '6.1876', '6.2075']
Outer characteristic polynomial of the knot is: t^7+33t^5+128t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1811']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 2048*K1**4*K2 - 1824*K1**4 - 512*K1**3*K2**2*K3 + 704*K1**3*K2*K3 - 1056*K1**3*K3 - 704*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 4480*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 11456*K1**2*K2**2 - 1152*K1**2*K2*K4 + 9208*K1**2*K2 - 128*K1**2*K3**2 - 5452*K1**2 + 2368*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 2720*K1*K2**2*K3 - 288*K1*K2**2*K5 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9280*K1*K2*K3 + 912*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 352*K2**4*K4 - 3600*K2**4 - 32*K2**3*K6 - 1488*K2**2*K3**2 - 416*K2**2*K4**2 + 2912*K2**2*K4 - 2788*K2**2 + 632*K2*K3*K5 + 96*K2*K4*K6 - 2100*K3**2 - 676*K4**2 - 80*K5**2 - 12*K6**2 + 4226
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1811']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16914', 'vk6.17158', 'vk6.20512', 'vk6.21896', 'vk6.23302', 'vk6.23603', 'vk6.27953', 'vk6.29432', 'vk6.35328', 'vk6.35762', 'vk6.39365', 'vk6.41545', 'vk6.42824', 'vk6.43108', 'vk6.45932', 'vk6.47619', 'vk6.55073', 'vk6.55324', 'vk6.57377', 'vk6.58536', 'vk6.59462', 'vk6.59755', 'vk6.62030', 'vk6.63026', 'vk6.64910', 'vk6.65125', 'vk6.66926', 'vk6.67777', 'vk6.68213', 'vk6.68359', 'vk6.69530', 'vk6.70236']
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 O1O2O3U4U5O6O5U1U2O4U6U3
R3 orbit {'O1O2O3U4U5O6O5U1U2O4U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4O5U2U3O6O4U6U5
Gauss code of K* O1O2U3O4O5U1U2U5O3O6U4U6
Gauss code of -K* O1O2U3O4O5U6U2O6O3U1U4U5
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 0 2 -1 1 0],[ 2 0 1 2 1 2 0],[ 0 -1 0 1 0 0 -1],[-2 -2 -1 0 -1 -1 -2],[ 1 -1 0 1 0 2 1],[-1 -2 0 1 -2 0 0],[ 0 0 1 2 -1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -2 -1 -2],[-1 1 0 0 0 -2 -2],[ 0 1 0 0 -1 0 -1],[ 0 2 0 1 0 -1 0],[ 1 1 2 0 1 0 -1],[ 2 2 2 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,1,1,2,1,2,0,0,2,2,1,0,1,1,0,1]
Phi over symmetry [-2,-1,0,0,1,2,0,0,1,2,2,1,1,0,1,1,0,2,1,1,0]
Phi of -K [-2,-1,0,0,1,2,0,1,2,1,2,1,0,0,2,1,1,1,1,0,0]
Phi of K* [-2,-1,0,0,1,2,0,0,1,2,2,1,1,0,1,1,0,2,1,1,0]
Phi of -K* [-2,-1,0,0,1,2,1,0,1,2,2,1,0,2,1,1,0,2,0,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+23t^4+80t^2+4
Outer characteristic polynomial t^7+33t^5+128t^3+14t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 2048*K1**4*K2 - 1824*K1**4 - 512*K1**3*K2**2*K3 + 704*K1**3*K2*K3 - 1056*K1**3*K3 - 704*K1**2*K2**4 - 256*K1**2*K2**3*K4 + 4480*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 11456*K1**2*K2**2 - 1152*K1**2*K2*K4 + 9208*K1**2*K2 - 128*K1**2*K3**2 - 5452*K1**2 + 2368*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 2720*K1*K2**2*K3 - 288*K1*K2**2*K5 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9280*K1*K2*K3 + 912*K1*K3*K4 + 64*K1*K4*K5 - 64*K2**6 + 352*K2**4*K4 - 3600*K2**4 - 32*K2**3*K6 - 1488*K2**2*K3**2 - 416*K2**2*K4**2 + 2912*K2**2*K4 - 2788*K2**2 + 632*K2*K3*K5 + 96*K2*K4*K6 - 2100*K3**2 - 676*K4**2 - 80*K5**2 - 12*K6**2 + 4226
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
Contact