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

Flat knot 6.1540

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,2,1,2,1,1,0,1,0,0,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1540']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.927', '6.1364', '6.1367', '6.1540', '6.1675', '6.1779', '6.1811', '6.1876', '6.2075']
Outer characteristic polynomial of the knot is: t^7+27t^5+40t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1540']
2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 2112*K1**4*K2 - 1888*K1**4 - 512*K1**3*K2**2*K3 + 576*K1**3*K2*K3 - 960*K1**3*K3 - 512*K1**2*K2**4 + 4160*K1**2*K2**3 - 9744*K1**2*K2**2 - 1056*K1**2*K2*K4 + 8432*K1**2*K2 - 96*K1**2*K3**2 - 5268*K1**2 + 1984*K1*K2**3*K3 - 2560*K1*K2**2*K3 - 128*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 8104*K1*K2*K3 + 928*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 3200*K2**4 - 64*K2**3*K6 - 1024*K2**2*K3**2 - 128*K2**2*K4**2 + 2664*K2**2*K4 - 2876*K2**2 + 376*K2*K3*K5 + 80*K2*K4*K6 - 1948*K3**2 - 664*K4**2 - 72*K5**2 - 12*K6**2 + 4070
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1540']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16332', 'vk6.16373', 'vk6.18073', 'vk6.18409', 'vk6.22663', 'vk6.22744', 'vk6.24520', 'vk6.24940', 'vk6.34611', 'vk6.34688', 'vk6.36661', 'vk6.37083', 'vk6.42306', 'vk6.42335', 'vk6.43939', 'vk6.44255', 'vk6.54595', 'vk6.54634', 'vk6.55893', 'vk6.56181', 'vk6.59074', 'vk6.59117', 'vk6.60417', 'vk6.60776', 'vk6.64630', 'vk6.64668', 'vk6.65531', 'vk6.65845', 'vk6.67989', 'vk6.68011', 'vk6.68613', 'vk6.68829']
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 O1O2O3U4O5U1U2O4O6U5U6U3
R3 orbit {'O1O2O3U4O5U1U2O4O6U5U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O4O6U2U3O5U6
Gauss code of K* O1O2O3U4U5U3O6U1O4O5U6U2
Gauss code of -K* O1O2O3U2U4O5O6U3O4U1U5U6
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 2 -1 0 1],[ 2 0 1 2 1 1 1],[ 0 -1 0 1 0 0 1],[-2 -2 -1 0 -1 -2 1],[ 1 -1 0 1 0 0 0],[ 0 -1 0 2 0 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix [[ 0 2 1 0 0 -1 -2],[-2 0 1 -1 -2 -1 -2],[-1 -1 0 -1 -1 0 -1],[ 0 1 1 0 0 0 -1],[ 0 2 1 0 0 0 -1],[ 1 1 0 0 0 0 -1],[ 2 2 1 1 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,0,1,2,-1,1,2,1,2,1,1,0,1,0,0,1,0,1,1]
Phi over symmetry [-2,-1,0,0,1,2,-1,1,2,1,2,1,1,0,1,0,0,1,0,1,1]
Phi of -K [-2,-1,0,0,1,2,0,1,1,2,2,1,1,2,2,0,0,0,0,1,2]
Phi of K* [-2,-1,0,0,1,2,2,0,1,2,2,0,0,2,2,0,1,1,1,1,0]
Phi of -K* [-2,-1,0,0,1,2,1,1,1,1,2,0,0,0,1,0,1,1,1,2,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+17t^4+12t^2+1
Outer characteristic polynomial t^7+27t^5+40t^3+8t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial 256*K1**4*K2**3 - 1024*K1**4*K2**2 + 2112*K1**4*K2 - 1888*K1**4 - 512*K1**3*K2**2*K3 + 576*K1**3*K2*K3 - 960*K1**3*K3 - 512*K1**2*K2**4 + 4160*K1**2*K2**3 - 9744*K1**2*K2**2 - 1056*K1**2*K2*K4 + 8432*K1**2*K2 - 96*K1**2*K3**2 - 5268*K1**2 + 1984*K1*K2**3*K3 - 2560*K1*K2**2*K3 - 128*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 8104*K1*K2*K3 + 928*K1*K3*K4 + 56*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 3200*K2**4 - 64*K2**3*K6 - 1024*K2**2*K3**2 - 128*K2**2*K4**2 + 2664*K2**2*K4 - 2876*K2**2 + 376*K2*K3*K5 + 80*K2*K4*K6 - 1948*K3**2 - 664*K4**2 - 72*K5**2 - 12*K6**2 + 4070
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
Contact