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

Flat knot 6.604

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,3,3,2,0,1,1,1,0,1,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.604']
Arrow polynomial of the knot is: -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.574', '6.593', '6.604', '6.649', '6.650', '6.673', '6.690', '6.783', '6.973', '6.985', '6.1033', '6.1035']
Outer characteristic polynomial of the knot is: t^7+52t^5+45t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.604']
2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 1408*K1**4*K2 - 4896*K1**4 + 192*K1**3*K2*K3 - 192*K1**3*K3 + 1248*K1**2*K2**3 - 8576*K1**2*K2**2 - 320*K1**2*K2*K4 + 10984*K1**2*K2 - 352*K1**2*K3**2 - 96*K1**2*K3*K5 - 4284*K1**2 - 1696*K1*K2**2*K3 + 7392*K1*K2*K3 + 1016*K1*K3*K4 + 152*K1*K4*K5 - 1816*K2**4 - 176*K2**2*K3**2 - 8*K2**2*K4**2 + 1944*K2**2*K4 - 3902*K2**2 + 184*K2*K3*K5 + 8*K2*K4*K6 - 1760*K3**2 - 634*K4**2 - 100*K5**2 - 2*K6**2 + 4408
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.604']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13955', 'vk6.13958', 'vk6.14051', 'vk6.14052', 'vk6.15025', 'vk6.15026', 'vk6.15145', 'vk6.15148', 'vk6.17447', 'vk6.17466', 'vk6.17476', 'vk6.23959', 'vk6.23965', 'vk6.23990', 'vk6.23996', 'vk6.33759', 'vk6.33835', 'vk6.33836', 'vk6.34300', 'vk6.36250', 'vk6.36258', 'vk6.43412', 'vk6.53887', 'vk6.53890', 'vk6.53922', 'vk6.54435', 'vk6.55593', 'vk6.55597', 'vk6.60081', 'vk6.60094', 'vk6.60104', 'vk6.65306']
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 O1O2O3O4U2O5U4O6U3U6U1U5
R3 orbit {'O1O2O3O4U2O5U4O6U3U6U1U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U6U2O6U1O5U3
Gauss code of K* O1O2O3O4U3U5U1U6O5U4O6U2
Gauss code of -K* O1O2O3O4U3O5U1O6U5U4U6U2
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 -1 -2 -1 0 3 1],[ 1 0 -2 0 1 3 1],[ 2 2 0 2 1 2 1],[ 1 0 -2 0 0 3 1],[ 0 -1 -1 0 0 1 0],[-3 -3 -2 -3 -1 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix [[ 0 3 1 0 -1 -1 -2],[-3 0 0 -1 -3 -3 -2],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 0 -1 -1],[ 1 3 1 0 0 0 -2],[ 1 3 1 1 0 0 -2],[ 2 2 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,1,1,2,0,1,3,3,2,0,1,1,1,0,1,1,0,2,2]
Phi over symmetry [-3,-1,0,1,1,2,0,1,3,3,2,0,1,1,1,0,1,1,0,2,2]
Phi of -K [-2,-1,-1,0,1,3,-1,-1,1,2,3,0,0,1,1,1,1,1,1,2,2]
Phi of K* [-3,-1,0,1,1,2,2,2,1,1,3,1,1,1,2,0,1,1,0,-1,-1]
Phi of -K* [-2,-1,-1,0,1,3,2,2,1,1,2,0,0,1,3,1,1,3,0,1,0]
Symmetry type of based matrix c
u-polynomial -t^3+t^2+t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+36t^4+22t^2+1
Outer characteristic polynomial t^7+52t^5+45t^3+7t
Flat arrow polynomial -10*K1**2 - 2*K1*K2 + K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial -128*K1**4*K2**2 + 1408*K1**4*K2 - 4896*K1**4 + 192*K1**3*K2*K3 - 192*K1**3*K3 + 1248*K1**2*K2**3 - 8576*K1**2*K2**2 - 320*K1**2*K2*K4 + 10984*K1**2*K2 - 352*K1**2*K3**2 - 96*K1**2*K3*K5 - 4284*K1**2 - 1696*K1*K2**2*K3 + 7392*K1*K2*K3 + 1016*K1*K3*K4 + 152*K1*K4*K5 - 1816*K2**4 - 176*K2**2*K3**2 - 8*K2**2*K4**2 + 1944*K2**2*K4 - 3902*K2**2 + 184*K2*K3*K5 + 8*K2*K4*K6 - 1760*K3**2 - 634*K4**2 - 100*K5**2 - 2*K6**2 + 4408
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}]]
If K is slice False
Contact