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

Flat knot 6.799

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,0,2,3,3,-1,0,1,2,0,0,0,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.799']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']
Outer characteristic polynomial of the knot is: t^7+49t^5+78t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.799']
2-strand cable arrow polynomial of the knot is: -1088*K1**4*K2**2 + 2080*K1**4*K2 - 3504*K1**4 + 384*K1**3*K2*K3 - 352*K1**3*K3 - 768*K1**2*K2**4 + 4000*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 10192*K1**2*K2**2 - 480*K1**2*K2*K4 + 10496*K1**2*K2 - 16*K1**2*K3**2 - 4700*K1**2 + 928*K1*K2**3*K3 - 1952*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6184*K1*K2*K3 + 176*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 2320*K2**4 - 288*K2**2*K3**2 - 8*K2**2*K4**2 + 1488*K2**2*K4 - 2664*K2**2 + 56*K2*K3*K5 - 900*K3**2 - 124*K4**2 + 3626
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.799']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73278', 'vk6.73419', 'vk6.74017', 'vk6.74559', 'vk6.75190', 'vk6.75420', 'vk6.76035', 'vk6.76765', 'vk6.78147', 'vk6.78382', 'vk6.78994', 'vk6.79551', 'vk6.79976', 'vk6.80129', 'vk6.80516', 'vk6.80984', 'vk6.81871', 'vk6.82155', 'vk6.82181', 'vk6.82587', 'vk6.83576', 'vk6.83760', 'vk6.84034', 'vk6.84601', 'vk6.84929', 'vk6.85586', 'vk6.85707', 'vk6.85944', 'vk6.86725', 'vk6.87676', 'vk6.88928', 'vk6.89976']
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 O1O2O3O4U5O6U4U1U2O5U3U6
R3 orbit {'O1O2O3O4U5O6U4U1U2O5U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U2O6U3U4U1O5U6
Gauss code of K* O1O2O3U4O5O6U2U3U5U1O4U6
Gauss code of -K* O1O2O3U4O5U3U6U1U2O4O6U5
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 1 0 -2 3],[ 2 0 1 1 0 0 3],[ 0 -1 0 0 0 -1 2],[-1 -1 0 0 1 -2 1],[ 0 0 0 -1 0 0 0],[ 2 0 1 2 0 0 3],[-3 -3 -2 -1 0 -3 0]]
Primitive based matrix [[ 0 3 1 0 0 -2 -2],[-3 0 -1 0 -2 -3 -3],[-1 1 0 1 0 -1 -2],[ 0 0 -1 0 0 0 0],[ 0 2 0 0 0 -1 -1],[ 2 3 1 0 1 0 0],[ 2 3 2 0 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,0,0,2,2,1,0,2,3,3,-1,0,1,2,0,0,0,1,1,0]
Phi over symmetry [-3,-1,0,0,2,2,1,0,2,3,3,-1,0,1,2,0,0,0,1,1,0]
Phi of -K [-2,-2,0,0,1,3,0,1,2,1,2,1,2,2,2,0,1,1,2,3,1]
Phi of K* [-3,-1,0,0,2,2,1,1,3,2,2,1,2,1,2,0,1,1,2,2,0]
Phi of -K* [-2,-2,0,0,1,3,0,0,1,1,3,0,1,2,3,0,-1,0,0,2,1]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+31t^4+44t^2
Outer characteristic polynomial t^7+49t^5+78t^3+3t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial -1088*K1**4*K2**2 + 2080*K1**4*K2 - 3504*K1**4 + 384*K1**3*K2*K3 - 352*K1**3*K3 - 768*K1**2*K2**4 + 4000*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 10192*K1**2*K2**2 - 480*K1**2*K2*K4 + 10496*K1**2*K2 - 16*K1**2*K3**2 - 4700*K1**2 + 928*K1*K2**3*K3 - 1952*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6184*K1*K2*K3 + 176*K1*K3*K4 - 32*K2**6 + 32*K2**4*K4 - 2320*K2**4 - 288*K2**2*K3**2 - 8*K2**2*K4**2 + 1488*K2**2*K4 - 2664*K2**2 + 56*K2*K3*K5 - 900*K3**2 - 124*K4**2 + 3626
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}]]
If K is slice False
Contact