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

Flat knot 6.84

Min(phi) over symmetries of the knot is: [-4,-3,0,0,3,4,0,2,3,3,5,1,2,2,3,0,1,2,2,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.84']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 - 4*K2**2 + 2*K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.84', '6.123', '6.244', '6.864']
Outer characteristic polynomial of the knot is: t^7+129t^5+59t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.84']
2-strand cable arrow polynomial of the knot is: -224*K1**2*K3**2 - 224*K1**2*K4**2 - 832*K1**2 + 608*K1*K2*K3 + 1216*K1*K3*K4 + 304*K1*K4*K5 + 16*K1*K7*K8 - 16*K2**2*K4**2 + 192*K2**2*K4 - 492*K2**2 + 96*K2*K3*K5 + 16*K2*K4*K6 + 16*K2*K5*K7 - 32*K3**4 - 64*K3**2*K4**2 + 32*K3**2*K6 - 792*K3**2 + 64*K3*K4*K7 - 16*K4**4 + 16*K4**2*K8 - 688*K4**2 - 152*K5**2 - 12*K6**2 - 32*K7**2 - 12*K8**2 + 1010
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.84']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73335', 'vk6.73496', 'vk6.75258', 'vk6.75504', 'vk6.78221', 'vk6.78460', 'vk6.80046', 'vk6.80194', 'vk6.81589', 'vk6.81675', 'vk6.82097', 'vk6.82189', 'vk6.82275', 'vk6.82670', 'vk6.84163', 'vk6.84655', 'vk6.84738', 'vk6.84970', 'vk6.85745', 'vk6.85974', 'vk6.87323', 'vk6.87682', 'vk6.88138', 'vk6.89717', 'vk6.90043', 'vk6.90081']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4O5O6U2U5U1U3U6U4
R3 orbit {'O1O2O3O4O5U1U6U3U2U5O6U4', 'O1O2O3O4O5O6U2U5U1U3U6U4'}
R3 orbit length 2
Gauss code of -K O1O2O3O4O5O6U3U1U4U6U2U5
Gauss code of K* O1O2O3O4O5O6U3U1U4U6U2U5
Gauss code of -K* Same
Diagrammatic symmetry type -
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -4 0 3 0 4],[ 3 0 -1 2 4 1 4],[ 4 1 0 2 4 1 3],[ 0 -2 -2 0 2 0 2],[-3 -4 -4 -2 0 -1 1],[ 0 -1 -1 0 1 0 1],[-4 -4 -3 -2 -1 -1 0]]
Primitive based matrix [[ 0 4 3 0 0 -3 -4],[-4 0 -1 -1 -2 -4 -3],[-3 1 0 -1 -2 -4 -4],[ 0 1 1 0 0 -1 -1],[ 0 2 2 0 0 -2 -2],[ 3 4 4 1 2 0 -1],[ 4 3 4 1 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,0,0,3,4,1,1,2,4,3,1,2,4,4,0,1,1,2,2,1]
Phi over symmetry [-4,-3,0,0,3,4,0,2,3,3,5,1,2,2,3,0,1,2,2,3,0]
Phi of -K [-4,-3,0,0,3,4,0,2,3,3,5,1,2,2,3,0,1,2,2,3,0]
Phi of K* [-4,-3,0,0,3,4,0,2,3,3,5,1,2,2,3,0,1,2,2,3,0]
Phi of -K* [-4,-3,0,0,3,4,1,1,2,4,3,1,2,4,4,0,1,1,2,2,1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial -4w^3z+11w^2z+15w
Inner characteristic polynomial t^6+79t^4+19t^2
Outer characteristic polynomial t^7+129t^5+59t^3
Flat arrow polynomial -4*K1*K2 + 2*K1 - 4*K2**2 + 2*K3 + 2*K4 + 3
2-strand cable arrow polynomial -224*K1**2*K3**2 - 224*K1**2*K4**2 - 832*K1**2 + 608*K1*K2*K3 + 1216*K1*K3*K4 + 304*K1*K4*K5 + 16*K1*K7*K8 - 16*K2**2*K4**2 + 192*K2**2*K4 - 492*K2**2 + 96*K2*K3*K5 + 16*K2*K4*K6 + 16*K2*K5*K7 - 32*K3**4 - 64*K3**2*K4**2 + 32*K3**2*K6 - 792*K3**2 + 64*K3*K4*K7 - 16*K4**4 + 16*K4**2*K8 - 688*K4**2 - 152*K5**2 - 12*K6**2 - 32*K7**2 - 12*K8**2 + 1010
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}]]
If K is slice True
Contact