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

Flat knot 6.1248

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,0,1,2,2,0,0,1,1,0,1,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.1135', '6.1248']
Arrow polynomial of the knot is: -4*K1**2 - 4*K1*K3 + 4*K2 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1248']
Outer characteristic polynomial of the knot is: t^7+33t^5+55t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1127', '6.1135', '6.1248']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 256*K1**4*K2 - 1792*K1**4 + 896*K1**3*K2*K3 - 640*K1**3*K3 + 128*K1**2*K2**3 - 2112*K1**2*K2**2 - 320*K1**2*K2*K4 + 4224*K1**2*K2 - 2272*K1**2*K3**2 - 4480*K1**2 - 320*K1*K2**2*K3 + 128*K1*K2*K3**3 - 64*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 7584*K1*K2*K3 - 64*K1*K2*K4*K5 + 2656*K1*K3*K4 + 144*K1*K4*K5 + 176*K1*K5*K6 - 80*K2**4 - 1024*K2**2*K3**2 + 304*K2**2*K4 - 48*K2**2*K6**2 - 3824*K2**2 - 192*K2*K3**2*K4 + 976*K2*K3*K5 + 240*K2*K4*K6 + 32*K2*K6*K8 - 224*K3**4 + 336*K3**2*K6 - 3352*K3**2 - 920*K4**2 - 328*K5**2 - 216*K6**2 - 4*K8**2 + 4514
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1248']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3565', 'vk6.3585', 'vk6.3808', 'vk6.3839', 'vk6.6982', 'vk6.7013', 'vk6.7200', 'vk6.7230', 'vk6.15359', 'vk6.15484', 'vk6.33986', 'vk6.34036', 'vk6.34445', 'vk6.48209', 'vk6.48367', 'vk6.49952', 'vk6.49971', 'vk6.54004', 'vk6.54053', 'vk6.54502']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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 O1O2O3O4U3U5U4O6O5U1U6U2
R3 orbit {'O1O2O3O4U3U5U4O6O5U1U6U2'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* O1O2O3U4U2O5O4O6U5U6U1U3
Gauss code of -K* O1O2O3U4U2O5O4O6U5U6U1U3
Diagrammatic symmetry type r
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 1 -1 2 0 0],[ 2 0 2 -1 2 1 0],[-1 -2 0 -1 2 -1 -1],[ 1 1 1 0 1 0 0],[-2 -2 -2 -1 0 -2 -1],[ 0 -1 1 0 2 0 0],[ 0 0 1 0 1 0 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 -2 -1 -2 -1 -2],[-1 2 0 -1 -1 -1 -2],[ 0 1 1 0 0 0 0],[ 0 2 1 0 0 0 -1],[ 1 1 1 0 0 0 1],[ 2 2 2 0 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,2,1,2,1,2,1,1,1,2,0,0,0,0,1,-1]
Phi over symmetry [-2,-1,0,0,1,2,-1,0,1,2,2,0,0,1,1,0,1,1,1,2,2]
Phi of -K [-2,-1,0,0,1,2,2,1,2,1,2,1,1,1,2,0,0,0,0,1,-1]
Phi of K* [-2,-1,0,0,1,2,-1,0,1,2,2,0,0,1,1,0,1,1,1,2,2]
Phi of -K* [-2,-1,0,0,1,2,-1,0,1,2,2,0,0,1,1,0,1,1,1,2,2]
Symmetry type of based matrix r
u-polynomial 0
Normalized Jones-Krushkal polynomial 20z+41
Enhanced Jones-Krushkal polynomial 20w^2z+41w
Inner characteristic polynomial t^6+23t^4+23t^2+1
Outer characteristic polynomial t^7+33t^5+55t^3+7t
Flat arrow polynomial -4*K1**2 - 4*K1*K3 + 4*K2 + 2*K4 + 3
2-strand cable arrow polynomial -128*K1**4*K2**2 + 256*K1**4*K2 - 1792*K1**4 + 896*K1**3*K2*K3 - 640*K1**3*K3 + 128*K1**2*K2**3 - 2112*K1**2*K2**2 - 320*K1**2*K2*K4 + 4224*K1**2*K2 - 2272*K1**2*K3**2 - 4480*K1**2 - 320*K1*K2**2*K3 + 128*K1*K2*K3**3 - 64*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 7584*K1*K2*K3 - 64*K1*K2*K4*K5 + 2656*K1*K3*K4 + 144*K1*K4*K5 + 176*K1*K5*K6 - 80*K2**4 - 1024*K2**2*K3**2 + 304*K2**2*K4 - 48*K2**2*K6**2 - 3824*K2**2 - 192*K2*K3**2*K4 + 976*K2*K3*K5 + 240*K2*K4*K6 + 32*K2*K6*K8 - 224*K3**4 + 336*K3**2*K6 - 3352*K3**2 - 920*K4**2 - 328*K5**2 - 216*K6**2 - 4*K8**2 + 4514
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {2, 3}]]
If K is slice False
Contact