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

Flat knot 6.262

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,0,2,4,3,4,1,2,1,2,2,2,3,2,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.262']
Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.83', '6.151', '6.160', '6.190', '6.247', '6.262', '6.491', '6.514']
Outer characteristic polynomial of the knot is: t^7+102t^5+69t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.262']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2 - 1424*K1**4 + 640*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1248*K1**3*K3 - 1360*K1**2*K2**2 - 704*K1**2*K2*K4 + 4288*K1**2*K2 - 1104*K1**2*K3**2 - 160*K1**2*K4**2 - 3296*K1**2 + 64*K1*K2**3*K3 - 192*K1*K2**2*K3 + 128*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 384*K1*K2*K3*K4 + 3984*K1*K2*K3 - 64*K1*K3**2*K5 + 1648*K1*K3*K4 + 416*K1*K4*K5 + 16*K1*K5*K6 - 40*K2**4 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 592*K2**2*K4 - 2412*K2**2 + 320*K2*K3*K5 + 40*K2*K4*K6 + 8*K2*K5*K7 - 128*K3**4 - 80*K3**2*K4**2 + 104*K3**2*K6 - 1492*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 750*K4**2 - 232*K5**2 - 44*K6**2 - 12*K7**2 - 2*K8**2 + 2662
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.262']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16993', 'vk6.17235', 'vk6.20209', 'vk6.21493', 'vk6.23400', 'vk6.23709', 'vk6.27395', 'vk6.29017', 'vk6.35460', 'vk6.35903', 'vk6.38816', 'vk6.40997', 'vk6.42897', 'vk6.43197', 'vk6.45569', 'vk6.47346', 'vk6.55156', 'vk6.55402', 'vk6.57051', 'vk6.58158', 'vk6.59535', 'vk6.59879', 'vk6.61555', 'vk6.62731', 'vk6.64966', 'vk6.65172', 'vk6.66670', 'vk6.67505', 'vk6.68259', 'vk6.68414', 'vk6.69319', 'vk6.70075']
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 O1O2O3O4O5U1U3O6U2U6U5U4
R3 orbit {'O1O2O3O4O5U1U3O6U2U6U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U1U6U4O6U3U5
Gauss code of K* O1O2O3O4U5U1U6U4U3O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U2U1U5U4U6
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 -4 -2 -1 3 3 1],[ 4 0 2 1 4 3 1],[ 2 -2 0 0 4 3 1],[ 1 -1 0 0 2 1 0],[-3 -4 -4 -2 0 0 0],[-3 -3 -3 -1 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 3 1 -1 -2 -4],[-3 0 0 0 -1 -3 -3],[-3 0 0 0 -2 -4 -4],[-1 0 0 0 0 -1 -1],[ 1 1 2 0 0 0 -1],[ 2 3 4 1 0 0 -2],[ 4 3 4 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-1,1,2,4,0,0,1,3,3,0,2,4,4,0,1,1,0,1,2]
Phi over symmetry [-4,-2,-1,1,3,3,0,2,4,3,4,1,2,1,2,2,2,3,2,2,0]
Phi of -K [-4,-2,-1,1,3,3,0,2,4,3,4,1,2,1,2,2,2,3,2,2,0]
Phi of K* [-3,-3,-1,1,2,4,0,2,2,1,3,2,3,2,4,2,2,4,1,2,0]
Phi of -K* [-4,-2,-1,1,3,3,2,1,1,3,4,0,1,3,4,0,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial 5z^2+22z+25
Enhanced Jones-Krushkal polynomial 5w^3z^2+22w^2z+25w
Inner characteristic polynomial t^6+62t^4+20t^2
Outer characteristic polynomial t^7+102t^5+69t^3+4t
Flat arrow polynomial -2*K1**2 - 4*K1*K2 + 2*K1 - 2*K2**2 + K2 + 2*K3 + K4 + 3
2-strand cable arrow polynomial 256*K1**4*K2 - 1424*K1**4 + 640*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1248*K1**3*K3 - 1360*K1**2*K2**2 - 704*K1**2*K2*K4 + 4288*K1**2*K2 - 1104*K1**2*K3**2 - 160*K1**2*K4**2 - 3296*K1**2 + 64*K1*K2**3*K3 - 192*K1*K2**2*K3 + 128*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 384*K1*K2*K3*K4 + 3984*K1*K2*K3 - 64*K1*K3**2*K5 + 1648*K1*K3*K4 + 416*K1*K4*K5 + 16*K1*K5*K6 - 40*K2**4 - 128*K2**2*K3**2 - 48*K2**2*K4**2 + 592*K2**2*K4 - 2412*K2**2 + 320*K2*K3*K5 + 40*K2*K4*K6 + 8*K2*K5*K7 - 128*K3**4 - 80*K3**2*K4**2 + 104*K3**2*K6 - 1492*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 750*K4**2 - 232*K5**2 - 44*K6**2 - 12*K7**2 - 2*K8**2 + 2662
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}]]
If K is slice False
Contact