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

Flat knot 6.1296

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,2,1,0,0,0,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.1296']
Arrow polynomial of the knot is: 8*K1**3 - 6*K1**2 - 8*K1*K2 - 2*K1 + 3*K2 + 2*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.561', '6.1296']
Outer characteristic polynomial of the knot is: t^7+41t^5+54t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1296']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 960*K1**4*K2**2 + 2720*K1**4*K2 - 4800*K1**4 + 1088*K1**3*K2*K3 - 1312*K1**3*K3 - 704*K1**2*K2**4 + 3424*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 10768*K1**2*K2**2 - 992*K1**2*K2*K4 + 11624*K1**2*K2 - 704*K1**2*K3**2 - 48*K1**2*K4**2 - 4544*K1**2 + 1408*K1*K2**3*K3 - 2240*K1*K2**2*K3 - 224*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 8360*K1*K2*K3 + 864*K1*K3*K4 + 80*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2344*K2**4 - 64*K2**3*K6 - 720*K2**2*K3**2 - 128*K2**2*K4**2 + 1872*K2**2*K4 - 3212*K2**2 + 336*K2*K3*K5 + 48*K2*K4*K6 - 1572*K3**2 - 342*K4**2 - 28*K5**2 - 4*K6**2 + 4044
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1296']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4700', 'vk6.5003', 'vk6.6194', 'vk6.6665', 'vk6.8191', 'vk6.8609', 'vk6.9569', 'vk6.9908', 'vk6.17382', 'vk6.20919', 'vk6.20976', 'vk6.22331', 'vk6.22398', 'vk6.23553', 'vk6.23890', 'vk6.28395', 'vk6.36142', 'vk6.40057', 'vk6.40169', 'vk6.42110', 'vk6.43057', 'vk6.43361', 'vk6.46585', 'vk6.46674', 'vk6.48732', 'vk6.49528', 'vk6.49731', 'vk6.51432', 'vk6.55540', 'vk6.58927', 'vk6.65286', 'vk6.69771']
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 O1O2O3U2O4O5U6U1O6U4U5U3
R3 orbit {'O1O2O3U2O4O5U6U1O6U4U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3U1U4U5O6U3U6O4O5U2
Gauss code of K* O1O2O3U4U5U3O5U1U2O6O4U6
Gauss code of -K* O1O2O3U4O5O4U2U3O6U1U6U5
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 0 2 -1],[ 2 0 0 3 0 1 2],[ 1 0 0 1 0 0 1],[-2 -3 -1 0 -1 1 -2],[ 0 0 0 1 0 1 0],[-2 -1 0 -1 -1 0 -2],[ 1 -2 -1 2 0 2 0]]
Primitive based matrix [[ 0 2 2 0 -1 -1 -2],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -1 0 -2 -1],[ 0 1 1 0 0 0 0],[ 1 1 0 0 0 1 0],[ 1 2 2 0 -1 0 -2],[ 2 3 1 0 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,2,1,0,0,0,-1,0,2]
Phi over symmetry [-2,-2,0,1,1,2,-1,1,1,2,3,1,0,2,1,0,0,0,-1,0,2]
Phi of -K [-2,-1,-1,0,2,2,-1,1,2,1,3,1,1,1,1,1,2,3,1,1,-1]
Phi of K* [-2,-2,0,1,1,2,-1,1,1,3,3,1,1,2,1,1,1,2,-1,-1,1]
Phi of -K* [-2,-1,-1,0,2,2,0,2,0,1,3,1,0,0,1,0,2,2,1,1,-1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+27t^4+23t^2
Outer characteristic polynomial t^7+41t^5+54t^3+6t
Flat arrow polynomial 8*K1**3 - 6*K1**2 - 8*K1*K2 - 2*K1 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -256*K1**6 - 960*K1**4*K2**2 + 2720*K1**4*K2 - 4800*K1**4 + 1088*K1**3*K2*K3 - 1312*K1**3*K3 - 704*K1**2*K2**4 + 3424*K1**2*K2**3 + 448*K1**2*K2**2*K4 - 10768*K1**2*K2**2 - 992*K1**2*K2*K4 + 11624*K1**2*K2 - 704*K1**2*K3**2 - 48*K1**2*K4**2 - 4544*K1**2 + 1408*K1*K2**3*K3 - 2240*K1*K2**2*K3 - 224*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 8360*K1*K2*K3 + 864*K1*K3*K4 + 80*K1*K4*K5 - 192*K2**6 + 320*K2**4*K4 - 2344*K2**4 - 64*K2**3*K6 - 720*K2**2*K3**2 - 128*K2**2*K4**2 + 1872*K2**2*K4 - 3212*K2**2 + 336*K2*K3*K5 + 48*K2*K4*K6 - 1572*K3**2 - 342*K4**2 - 28*K5**2 - 4*K6**2 + 4044
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
Contact