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

Flat knot 6.930

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,2,2,1,1,1,1,0,0,0,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.930']
Arrow polynomial of the knot is: -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.372', '6.930', '6.1007', '6.1701', '6.1714', '6.1760', '6.1788']
Outer characteristic polynomial of the knot is: t^7+24t^5+31t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.930']
2-strand cable arrow polynomial of the knot is: -832*K1**6 - 576*K1**4*K2**2 + 2144*K1**4*K2 - 5712*K1**4 + 1312*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1536*K1**3*K3 - 5456*K1**2*K2**2 - 640*K1**2*K2*K4 + 11344*K1**2*K2 - 2064*K1**2*K3**2 - 336*K1**2*K4**2 - 5668*K1**2 - 512*K1*K2**2*K3 - 288*K1*K2*K3*K4 + 9000*K1*K2*K3 + 2744*K1*K3*K4 + 416*K1*K4*K5 - 520*K2**4 - 752*K2**2*K3**2 - 256*K2**2*K4**2 + 1072*K2**2*K4 - 5240*K2**2 - 128*K2*K3**2*K4 + 728*K2*K3*K5 + 320*K2*K4*K6 + 64*K3**2*K6 - 3052*K3**2 - 1082*K4**2 - 232*K5**2 - 96*K6**2 + 5800
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.930']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4132', 'vk6.4163', 'vk6.5374', 'vk6.5405', 'vk6.7492', 'vk6.7521', 'vk6.8997', 'vk6.9028', 'vk6.12437', 'vk6.12468', 'vk6.13344', 'vk6.13565', 'vk6.13596', 'vk6.14252', 'vk6.14701', 'vk6.14752', 'vk6.15211', 'vk6.15859', 'vk6.15912', 'vk6.30846', 'vk6.30877', 'vk6.32034', 'vk6.32065', 'vk6.33070', 'vk6.33101', 'vk6.33858', 'vk6.34321', 'vk6.48478', 'vk6.50257', 'vk6.53516', 'vk6.53939', 'vk6.54269']
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 O1O2O3O4U3U5O6O5U4U1U6U2
R3 orbit {'O1O2O3O4U3U5O6O5U4U1U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U4U1O6O5U6U2
Gauss code of K* O1O2O3O4U2U4U5U1O5O6U3U6
Gauss code of -K* O1O2O3O4U5U2O5O6U4U6U1U3
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 1 -1 0 1 1],[ 2 0 2 -1 1 2 1],[-1 -2 0 -1 0 0 0],[ 1 1 1 0 1 1 1],[ 0 -1 0 -1 0 0 0],[-1 -2 0 -1 0 0 1],[-1 -1 0 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 1 0 -1 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 0 -1 -2],[ 0 0 0 0 0 -1 -1],[ 1 1 1 1 1 0 1],[ 2 2 1 2 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,0,1,2,-1,0,0,1,2,0,0,1,1,0,1,2,1,1,-1]
Phi over symmetry [-2,-1,0,1,1,1,-1,1,1,2,2,1,1,1,1,0,0,0,-1,0,0]
Phi of -K [-2,-1,0,1,1,1,2,1,1,1,2,0,1,1,1,1,1,1,0,-1,0]
Phi of K* [-1,-1,-1,0,1,2,-1,0,1,1,2,0,1,1,1,1,1,1,0,1,2]
Phi of -K* [-2,-1,0,1,1,1,-1,1,1,2,2,1,1,1,1,0,0,0,-1,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial 2z^2+23z+39
Enhanced Jones-Krushkal polynomial 2w^3z^2+23w^2z+39w
Inner characteristic polynomial t^6+16t^4+14t^2+1
Outer characteristic polynomial t^7+24t^5+31t^3+5t
Flat arrow polynomial -10*K1**2 - 8*K1*K2 + 4*K1 + 5*K2 + 4*K3 + 6
2-strand cable arrow polynomial -832*K1**6 - 576*K1**4*K2**2 + 2144*K1**4*K2 - 5712*K1**4 + 1312*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1536*K1**3*K3 - 5456*K1**2*K2**2 - 640*K1**2*K2*K4 + 11344*K1**2*K2 - 2064*K1**2*K3**2 - 336*K1**2*K4**2 - 5668*K1**2 - 512*K1*K2**2*K3 - 288*K1*K2*K3*K4 + 9000*K1*K2*K3 + 2744*K1*K3*K4 + 416*K1*K4*K5 - 520*K2**4 - 752*K2**2*K3**2 - 256*K2**2*K4**2 + 1072*K2**2*K4 - 5240*K2**2 - 128*K2*K3**2*K4 + 728*K2*K3*K5 + 320*K2*K4*K6 + 64*K3**2*K6 - 3052*K3**2 - 1082*K4**2 - 232*K5**2 - 96*K6**2 + 5800
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{6}, {1, 5}, {3, 4}, {2}]]
If K is slice False
Contact