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

Flat knot 6.436

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,0,0,2,1,3,-1,3,2,4,1,1,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.436']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.436', '6.1909']
Outer characteristic polynomial of the knot is: t^7+88t^5+335t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.436']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 320*K1**4*K2 - 2144*K1**4 + 160*K1**3*K2*K3 - 96*K1**3*K3 + 256*K1**2*K2**5 - 1536*K1**2*K2**4 + 2304*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 8048*K1**2*K2**2 - 384*K1**2*K2*K4 + 9160*K1**2*K2 - 128*K1**2*K3**2 - 5400*K1**2 + 128*K1*K2**5*K3 + 1984*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 7656*K1*K2*K3 + 904*K1*K3*K4 + 176*K1*K4*K5 + 16*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1056*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1312*K2**4*K4 - 3384*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 992*K2**2*K3**2 - 688*K2**2*K4**2 + 3248*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 3226*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 648*K2*K3*K5 + 184*K2*K4*K6 + 24*K2*K5*K7 - 2128*K3**2 - 872*K4**2 - 208*K5**2 - 30*K6**2 + 4742
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.436']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17086', 'vk6.17329', 'vk6.20244', 'vk6.21550', 'vk6.23472', 'vk6.23811', 'vk6.27467', 'vk6.29063', 'vk6.35613', 'vk6.36061', 'vk6.38881', 'vk6.41083', 'vk6.42986', 'vk6.43298', 'vk6.45646', 'vk6.47382', 'vk6.55231', 'vk6.55483', 'vk6.57077', 'vk6.58231', 'vk6.59631', 'vk6.59978', 'vk6.61615', 'vk6.62794', 'vk6.65030', 'vk6.65232', 'vk6.66710', 'vk6.67568', 'vk6.68300', 'vk6.68450', 'vk6.69360', 'vk6.70103']
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 O1O2O3O4O5U6U2O6U1U4U5U3
R3 orbit {'O1O2O3O4O5U6U2O6U1U4U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U1U2U5O6U4U6
Gauss code of K* O1O2O3O4U1U5U4U2U3O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U2U3U1U6U4
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 -2 2 1 3 -1],[ 3 0 1 4 2 3 2],[ 2 -1 0 2 0 1 2],[-2 -4 -2 0 -1 1 -2],[-1 -2 0 1 0 1 -1],[-3 -3 -1 -1 -1 0 -3],[ 1 -2 -2 2 1 3 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 -1 -1 -3 -1 -3],[-2 1 0 -1 -2 -2 -4],[-1 1 1 0 -1 0 -2],[ 1 3 2 1 0 -2 -2],[ 2 1 2 0 2 0 -1],[ 3 3 4 2 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,1,1,3,1,3,1,2,2,4,1,0,2,2,2,1]
Phi over symmetry [-3,-2,-1,1,2,3,0,0,2,1,3,-1,3,2,4,1,1,1,0,1,0]
Phi of -K [-3,-2,-1,1,2,3,0,0,2,1,3,-1,3,2,4,1,1,1,0,1,0]
Phi of K* [-3,-2,-1,1,2,3,0,1,1,4,3,0,1,2,1,1,3,2,-1,0,0]
Phi of -K* [-3,-2,-1,1,2,3,1,2,2,4,3,2,0,2,1,1,2,3,1,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+24z+29
Enhanced Jones-Krushkal polynomial -2w^4z^2+7w^3z^2+24w^2z+29w
Inner characteristic polynomial t^6+60t^4+185t^2
Outer characteristic polynomial t^7+88t^5+335t^3+9t
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 8*K1**2*K2 - 2*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -192*K1**4*K2**2 + 320*K1**4*K2 - 2144*K1**4 + 160*K1**3*K2*K3 - 96*K1**3*K3 + 256*K1**2*K2**5 - 1536*K1**2*K2**4 + 2304*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 8048*K1**2*K2**2 - 384*K1**2*K2*K4 + 9160*K1**2*K2 - 128*K1**2*K3**2 - 5400*K1**2 + 128*K1*K2**5*K3 + 1984*K1*K2**3*K3 + 320*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 448*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 7656*K1*K2*K3 + 904*K1*K3*K4 + 176*K1*K4*K5 + 16*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1056*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 1312*K2**4*K4 - 3384*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 224*K2**3*K6 - 992*K2**2*K3**2 - 688*K2**2*K4**2 + 3248*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 3226*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 648*K2*K3*K5 + 184*K2*K4*K6 + 24*K2*K5*K7 - 2128*K3**2 - 872*K4**2 - 208*K5**2 - 30*K6**2 + 4742
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
Contact