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

Flat knot 6.1541

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,0,2,2,0,1,0,1,0,2,1,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1541']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1209', '6.1245', '6.1509', '6.1541', '6.1704', '6.1778', '6.1914']
Outer characteristic polynomial of the knot is: t^7+50t^5+200t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1541']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 1072*K1**4 + 128*K1**3*K2**3*K3 + 1408*K1**3*K2*K3 - 704*K1**3*K3 - 320*K1**2*K2**4 + 224*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 7168*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 9592*K1**2*K2 - 1200*K1**2*K3**2 - 7520*K1**2 + 1568*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 640*K1*K2**2*K5 + 224*K1*K2*K3**3 - 384*K1*K2*K3*K4 + 10656*K1*K2*K3 + 1472*K1*K3*K4 + 112*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1312*K2**4 - 32*K2**3*K6 - 1184*K2**2*K3**2 - 64*K2**2*K4**2 + 1976*K2**2*K4 - 5690*K2**2 + 920*K2*K3*K5 + 48*K2*K4*K6 - 32*K3**4 - 3196*K3**2 - 664*K4**2 - 164*K5**2 - 6*K6**2 + 5710
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1541']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11432', 'vk6.11728', 'vk6.12745', 'vk6.13089', 'vk6.20329', 'vk6.21670', 'vk6.27633', 'vk6.29177', 'vk6.31185', 'vk6.31527', 'vk6.32353', 'vk6.32770', 'vk6.39061', 'vk6.41319', 'vk6.45817', 'vk6.47488', 'vk6.52185', 'vk6.52442', 'vk6.53015', 'vk6.53332', 'vk6.57200', 'vk6.58415', 'vk6.61814', 'vk6.62939', 'vk6.63757', 'vk6.63868', 'vk6.64184', 'vk6.64371', 'vk6.66813', 'vk6.67681', 'vk6.69453', 'vk6.70175']
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 O1O2O3U4O5U1U6O4O6U2U5U3
R3 orbit {'O1O2O3U4O5U1U6O4O6U2U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U2O5O6U5U3O4U6
Gauss code of K* O1O2O3U4U1U3O5U2O4O6U5U6
Gauss code of -K* O1O2O3U4U5O4O6U2O5U1U3U6
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 2 -1 1 1],[ 2 0 0 2 2 1 3],[ 1 0 0 2 0 0 2],[-2 -2 -2 0 -2 -1 -1],[ 1 -2 0 2 0 2 1],[-1 -1 0 1 -2 0 -1],[-1 -3 -2 1 -1 1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 -1 -1 -2 -2 -2],[-1 1 0 1 -1 -2 -3],[-1 1 -1 0 -2 0 -1],[ 1 2 1 2 0 0 -2],[ 1 2 2 0 0 0 0],[ 2 2 3 1 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,1,1,2,2,2,-1,1,2,3,2,0,1,0,2,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,0,2,2,0,1,0,1,0,2,1,-1,0,0]
Phi of -K [-2,-1,-1,1,1,2,-1,1,0,2,2,0,1,0,1,0,2,1,-1,0,0]
Phi of K* [-2,-1,-1,1,1,2,0,0,1,1,2,-1,0,2,2,1,0,0,0,-1,1]
Phi of -K* [-2,-1,-1,1,1,2,0,2,1,3,2,0,0,2,2,2,1,2,-1,1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+38t^4+142t^2+4
Outer characteristic polynomial t^7+50t^5+200t^3+14t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 8*K1*K2 + K1 + 4*K2 + 3*K3 + 5
2-strand cable arrow polynomial -256*K1**4*K2**2 + 96*K1**4*K2 - 1072*K1**4 + 128*K1**3*K2**3*K3 + 1408*K1**3*K2*K3 - 704*K1**3*K3 - 320*K1**2*K2**4 + 224*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 7168*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 832*K1**2*K2*K4 + 9592*K1**2*K2 - 1200*K1**2*K3**2 - 7520*K1**2 + 1568*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 640*K1*K2**2*K5 + 224*K1*K2*K3**3 - 384*K1*K2*K3*K4 + 10656*K1*K2*K3 + 1472*K1*K3*K4 + 112*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1312*K2**4 - 32*K2**3*K6 - 1184*K2**2*K3**2 - 64*K2**2*K4**2 + 1976*K2**2*K4 - 5690*K2**2 + 920*K2*K3*K5 + 48*K2*K4*K6 - 32*K3**4 - 3196*K3**2 - 664*K4**2 - 164*K5**2 - 6*K6**2 + 5710
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice True
Contact