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

Flat knot 6.132

Min(phi) over symmetries of the knot is: [-2,-2,0,0,2,2,-2,0,1,2,2,1,2,2,2,0,0,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.132']
Arrow polynomial of the knot is: -16*K1**4 + 8*K1**2*K2 + 8*K1**2 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.113', '6.132', '6.220', '6.933', '6.1250', '6.1905']
Outer characteristic polynomial of the knot is: t^7+52t^5+132t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.132']
2-strand cable arrow polynomial of the knot is: -2816*K2**8 + 1536*K2**6*K4 - 2304*K2**6 - 64*K2**4*K4**2 + 2048*K2**4*K4 + 256*K2**4 - 128*K2**2*K4**2 + 800*K2**2*K4 + 672*K2**2 - 64*K4**2 + 62
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.132']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70392', 'vk6.70395', 'vk6.70401', 'vk6.70405', 'vk6.70408', 'vk6.70411', 'vk6.70626', 'vk6.70795', 'vk6.70872', 'vk6.71034', 'vk6.71146', 'vk6.71265', 'vk6.89193', 'vk6.90108']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is a.
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4O5O6U4U5U6U1U2U3
R3 orbit {'O1O2O3O4O5O6U4U5U6U1U2U3'}
R3 orbit length 1
Gauss code of -K Same
Gauss code of K* Same
Gauss code of -K* Same
Diagrammatic symmetry type a
Flat genus of the diagram 2
If K is checkerboard colorable True
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 2 -2 0 2],[ 2 0 1 2 -2 0 2],[ 0 -1 0 1 -2 0 2],[-2 -2 -1 0 -2 0 2],[ 2 2 2 2 0 1 2],[ 0 0 0 0 -1 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix [[ 0 2 2 0 0 -2 -2],[-2 0 2 0 -1 -2 -2],[-2 -2 0 -1 -2 -2 -2],[ 0 0 1 0 0 0 -1],[ 0 1 2 0 0 -1 -2],[ 2 2 2 0 1 0 -2],[ 2 2 2 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,2,2,-2,0,1,2,2,1,2,2,2,0,0,1,1,2,2]
Phi over symmetry [-2,-2,0,0,2,2,-2,0,1,2,2,1,2,2,2,0,0,1,1,2,2]
Phi of -K [-2,-2,0,0,2,2,-2,0,1,2,2,1,2,2,2,0,0,1,1,2,2]
Phi of K* [-2,-2,0,0,2,2,-2,0,1,2,2,1,2,2,2,0,0,1,1,2,2]
Phi of -K* [-2,-2,0,0,2,2,-2,0,1,2,2,1,2,2,2,0,0,1,1,2,2]
Symmetry type of based matrix a
u-polynomial 0
Normalized Jones-Krushkal polynomial z^2+2z+1
Enhanced Jones-Krushkal polynomial -8w^4z^2+9w^3z^2+2w^2z+1
Inner characteristic polynomial t^6+36t^4+36t^2
Outer characteristic polynomial t^7+52t^5+132t^3
Flat arrow polynomial -16*K1**4 + 8*K1**2*K2 + 8*K1**2 + 1
2-strand cable arrow polynomial -2816*K2**8 + 1536*K2**6*K4 - 2304*K2**6 - 64*K2**4*K4**2 + 2048*K2**4*K4 + 256*K2**4 - 128*K2**2*K4**2 + 800*K2**2*K4 + 672*K2**2 - 64*K4**2 + 62
Genus of based matrix 0
Fillings of based matrix [[{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {5}, {1, 3}, {2}]]
If K is slice True
Contact