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

Flat knot 6.560

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,2,3,2,2,1,1,0,1,0,1,1,2,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.560']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 - 2*K2**2 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.130', '6.560']
Outer characteristic polynomial of the knot is: t^7+49t^5+55t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.560']
2-strand cable arrow polynomial of the knot is: -192*K1**2*K2**4 + 832*K1**2*K2**3 - 3632*K1**2*K2**2 - 512*K1**2*K2*K4 + 4928*K1**2*K2 - 64*K1**2*K3**2 - 4448*K1**2 + 320*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1728*K1*K2**2*K3 - 448*K1*K2**2*K5 + 32*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 5608*K1*K2*K3 - 224*K1*K2*K4*K5 + 1384*K1*K3*K4 + 264*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1432*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 848*K2**2*K3**2 + 32*K2**2*K4**3 - 360*K2**2*K4**2 - 32*K2**2*K4*K8 + 2472*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3708*K2**2 - 160*K2*K3**2*K4 + 1224*K2*K3*K5 + 352*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 112*K3**2*K6 - 2056*K3**2 - 8*K4**4 + 8*K4**2*K8 - 1096*K4**2 - 392*K5**2 - 76*K6**2 - 2*K8**2 + 3752
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.560']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4732', 'vk6.5054', 'vk6.6262', 'vk6.6707', 'vk6.8233', 'vk6.8678', 'vk6.9620', 'vk6.9940', 'vk6.20659', 'vk6.22092', 'vk6.28145', 'vk6.29576', 'vk6.39587', 'vk6.41820', 'vk6.46202', 'vk6.47822', 'vk6.48764', 'vk6.48968', 'vk6.49566', 'vk6.49777', 'vk6.50778', 'vk6.50987', 'vk6.51260', 'vk6.51462', 'vk6.57575', 'vk6.58743', 'vk6.62245', 'vk6.63193', 'vk6.67045', 'vk6.67920', 'vk6.69670', 'vk6.70353']
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 O1O2O3O4O5U4U1U5O6U3U2U6
R3 orbit {'O1O2O3O4O5U4U1U5O6U3U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U4U3O6U1U5U2
Gauss code of K* O1O2O3U4O5O6O4U2U6U5U1U3
Gauss code of -K* O1O2O3U1O4O5O6U4U6U3U2U5
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 -3 0 0 -1 2 2],[ 3 0 3 2 0 2 2],[ 0 -3 0 0 -1 1 2],[ 0 -2 0 0 -1 1 1],[ 1 0 1 1 0 1 0],[-2 -2 -1 -1 -1 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 0 -1 -1 -1 -2],[-2 0 0 -1 -2 0 -2],[ 0 1 1 0 0 -1 -2],[ 0 1 2 0 0 -1 -3],[ 1 1 0 1 1 0 0],[ 3 2 2 2 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,0,1,1,1,2,1,2,0,2,0,1,2,1,3,0]
Phi over symmetry [-3,-1,0,0,2,2,0,2,3,2,2,1,1,0,1,0,1,1,2,1,0]
Phi of -K [-3,-1,0,0,2,2,2,0,1,3,3,0,0,2,3,0,1,0,1,1,0]
Phi of K* [-2,-2,0,0,1,3,0,0,1,3,3,1,1,2,3,0,0,0,0,1,2]
Phi of -K* [-3,-1,0,0,2,2,0,2,3,2,2,1,1,0,1,0,1,1,2,1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial 7w^3z^2-4w^3z+28w^2z+21w
Inner characteristic polynomial t^6+31t^4+27t^2+1
Outer characteristic polynomial t^7+49t^5+55t^3+9t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 - 2*K2**2 + K4 + 2
2-strand cable arrow polynomial -192*K1**2*K2**4 + 832*K1**2*K2**3 - 3632*K1**2*K2**2 - 512*K1**2*K2*K4 + 4928*K1**2*K2 - 64*K1**2*K3**2 - 4448*K1**2 + 320*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1728*K1*K2**2*K3 - 448*K1*K2**2*K5 + 32*K1*K2*K3**3 - 128*K1*K2*K3*K4 + 5608*K1*K2*K3 - 224*K1*K2*K4*K5 + 1384*K1*K3*K4 + 264*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1432*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 848*K2**2*K3**2 + 32*K2**2*K4**3 - 360*K2**2*K4**2 - 32*K2**2*K4*K8 + 2472*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3708*K2**2 - 160*K2*K3**2*K4 + 1224*K2*K3*K5 + 352*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 - 64*K3**4 + 112*K3**2*K6 - 2056*K3**2 - 8*K4**4 + 8*K4**2*K8 - 1096*K4**2 - 392*K5**2 - 76*K6**2 - 2*K8**2 + 3752
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
Contact