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

Flat knot 6.1716

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,1,1,1,2,0,1,0,0,1,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1716']
Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 8*K1*K2 + K1 + 7*K2 + 3*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.474', '6.1684', '6.1716', '6.1749', '6.1781']
Outer characteristic polynomial of the knot is: t^7+25t^5+29t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1716']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 896*K1**4*K2 - 3712*K1**4 + 320*K1**3*K2*K3 - 736*K1**3*K3 + 320*K1**2*K2**3 - 4064*K1**2*K2**2 - 352*K1**2*K2*K4 + 8504*K1**2*K2 - 448*K1**2*K3**2 - 4488*K1**2 + 96*K1*K2**3*K3 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5520*K1*K2*K3 + 800*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 696*K2**4 - 32*K2**3*K6 - 352*K2**2*K3**2 - 128*K2**2*K4**2 + 1112*K2**2*K4 - 3970*K2**2 + 424*K2*K3*K5 + 104*K2*K4*K6 - 1688*K3**2 - 494*K4**2 - 112*K5**2 - 22*K6**2 + 4060
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1716']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17098', 'vk6.17340', 'vk6.20585', 'vk6.21994', 'vk6.23489', 'vk6.23826', 'vk6.28047', 'vk6.29506', 'vk6.35638', 'vk6.36078', 'vk6.39457', 'vk6.41658', 'vk6.43005', 'vk6.43316', 'vk6.46041', 'vk6.47709', 'vk6.55237', 'vk6.55488', 'vk6.57471', 'vk6.58634', 'vk6.59639', 'vk6.59985', 'vk6.62142', 'vk6.63104', 'vk6.65039', 'vk6.65237', 'vk6.66995', 'vk6.67860', 'vk6.68307', 'vk6.68456', 'vk6.69610', 'vk6.70303']
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 O1O2O3U1U4O5O6U3U6O4U2U5
R3 orbit {'O1O2O3U1U4O5O6U3U6O4U2U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O5U6U1O6O4U5U3
Gauss code of K* O1O2U3O4O5U6U4U1O6O3U5U2
Gauss code of -K* O1O2U3O4O5U4U1O3O6U5U2U6
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 0 0 1 1],[ 2 0 2 1 1 2 1],[ 0 -2 0 0 -1 1 1],[ 0 -1 0 0 -1 1 1],[ 0 -1 1 1 0 1 1],[-1 -2 -1 -1 -1 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 0 -2],[-1 0 0 -1 -1 -1 -1],[-1 0 0 -1 -1 -1 -2],[ 0 1 1 0 1 1 -1],[ 0 1 1 -1 0 0 -1],[ 0 1 1 -1 0 0 -2],[ 2 1 2 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,0,2,0,1,1,1,1,1,1,1,2,-1,-1,1,0,1,2]
Phi over symmetry [-2,0,0,0,1,1,0,1,1,1,2,0,1,0,0,1,0,0,0,0,0]
Phi of -K [-2,0,0,0,1,1,0,1,1,1,2,0,1,0,0,1,0,0,0,0,0]
Phi of K* [-1,-1,0,0,0,2,0,0,0,0,1,0,0,0,2,-1,0,0,1,1,1]
Phi of -K* [-2,0,0,0,1,1,1,1,2,1,2,-1,0,1,1,1,1,1,1,1,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+19t^4+12t^2
Outer characteristic polynomial t^7+25t^5+29t^3+4t
Flat arrow polynomial 4*K1**3 - 14*K1**2 - 8*K1*K2 + K1 + 7*K2 + 3*K3 + 8
2-strand cable arrow polynomial -192*K1**4*K2**2 + 896*K1**4*K2 - 3712*K1**4 + 320*K1**3*K2*K3 - 736*K1**3*K3 + 320*K1**2*K2**3 - 4064*K1**2*K2**2 - 352*K1**2*K2*K4 + 8504*K1**2*K2 - 448*K1**2*K3**2 - 4488*K1**2 + 96*K1*K2**3*K3 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5520*K1*K2*K3 + 800*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 696*K2**4 - 32*K2**3*K6 - 352*K2**2*K3**2 - 128*K2**2*K4**2 + 1112*K2**2*K4 - 3970*K2**2 + 424*K2*K3*K5 + 104*K2*K4*K6 - 1688*K3**2 - 494*K4**2 - 112*K5**2 - 22*K6**2 + 4060
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}]]
If K is slice False
Contact