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

Flat knot 6.423

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,3,2,4,3,1,1,1,1,1,0,2,-1,0,3]
Flat knots (up to 7 crossings) with same phi are :['6.423']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.172', '6.274', '6.286', '6.423', '6.461']
Outer characteristic polynomial of the knot is: t^7+86t^5+84t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.423']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 352*K1**4*K2 - 1600*K1**4 + 128*K1**3*K2**3*K3 + 608*K1**3*K2*K3 - 224*K1**3*K3 - 384*K1**2*K2**4 + 2016*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 6128*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 5984*K1**2*K2 - 320*K1**2*K3**2 - 96*K1**2*K4**2 - 3472*K1**2 - 384*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 1824*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 832*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 + 4936*K1*K2*K3 + 752*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 384*K2**4*K4 - 1800*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1120*K2**2*K3**2 - 296*K2**2*K4**2 + 1016*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 1700*K2**2 + 376*K2*K3*K5 + 32*K2*K4*K6 - 1264*K3**2 - 400*K4**2 - 48*K5**2 - 4*K6**2 + 2806
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.423']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19945', 'vk6.20044', 'vk6.21192', 'vk6.21330', 'vk6.26914', 'vk6.27109', 'vk6.28670', 'vk6.28802', 'vk6.38338', 'vk6.38494', 'vk6.40480', 'vk6.40699', 'vk6.45207', 'vk6.45394', 'vk6.47032', 'vk6.47146', 'vk6.56741', 'vk6.56852', 'vk6.57844', 'vk6.57995', 'vk6.61178', 'vk6.61379', 'vk6.62418', 'vk6.62546', 'vk6.66445', 'vk6.66565', 'vk6.67217', 'vk6.67360', 'vk6.69093', 'vk6.69217', 'vk6.69876', 'vk6.69962']
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 O1O2O3O4O5U6U1O6U3U4U2U5
R3 orbit {'O1O2O3O4O5U6U1O6U3U4U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U4U2U3O6U5U6
Gauss code of K* O1O2O3O4U5U3U1U2U4O6O5U6
Gauss code of -K* O1O2O3O4U5O6O5U1U3U4U2U6
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 0 -1 1 4 -1],[ 3 0 2 0 1 3 3],[ 0 -2 0 -1 1 3 0],[ 1 0 1 0 1 2 1],[-1 -1 -1 -1 0 1 -1],[-4 -3 -3 -2 -1 0 -4],[ 1 -3 0 -1 1 4 0]]
Primitive based matrix [[ 0 4 1 0 -1 -1 -3],[-4 0 -1 -3 -2 -4 -3],[-1 1 0 -1 -1 -1 -1],[ 0 3 1 0 -1 0 -2],[ 1 2 1 1 0 1 0],[ 1 4 1 0 -1 0 -3],[ 3 3 1 2 0 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-1,0,1,1,3,1,3,2,4,3,1,1,1,1,1,0,2,-1,0,3]
Phi over symmetry [-4,-1,0,1,1,3,1,3,2,4,3,1,1,1,1,1,0,2,-1,0,3]
Phi of -K [-3,-1,-1,0,1,4,-1,2,1,3,4,1,1,1,1,0,1,3,0,1,2]
Phi of K* [-4,-1,0,1,1,3,2,1,1,3,4,0,1,1,3,1,0,1,-1,-1,2]
Phi of -K* [-3,-1,-1,0,1,4,0,3,2,1,3,1,1,1,2,0,1,4,1,3,1]
Symmetry type of based matrix c
u-polynomial -t^4+t^3+t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial t^6+58t^4+30t^2+1
Outer characteristic polynomial t^7+86t^5+84t^3+9t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 2*K2 + 3
2-strand cable arrow polynomial -192*K1**4*K2**2 + 352*K1**4*K2 - 1600*K1**4 + 128*K1**3*K2**3*K3 + 608*K1**3*K2*K3 - 224*K1**3*K3 - 384*K1**2*K2**4 + 2016*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 6128*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 5984*K1**2*K2 - 320*K1**2*K3**2 - 96*K1**2*K4**2 - 3472*K1**2 - 384*K1*K2**4*K3 - 256*K1*K2**3*K3*K4 + 1824*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 832*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 448*K1*K2*K3*K4 + 4936*K1*K2*K3 + 752*K1*K3*K4 + 160*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 384*K2**4*K4 - 1800*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1120*K2**2*K3**2 - 296*K2**2*K4**2 + 1016*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 1700*K2**2 + 376*K2*K3*K5 + 32*K2*K4*K6 - 1264*K3**2 - 400*K4**2 - 48*K5**2 - 4*K6**2 + 2806
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
Contact