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

Flat knot 6.484

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,2,4,4,3,1,1,2,1,2,2,2,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.484']
Arrow polynomial of the knot is: -2*K1**2 - 6*K1*K2 + 3*K1 - 2*K2**2 + K2 + 3*K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.186', '6.484']
Outer characteristic polynomial of the knot is: t^7+80t^5+77t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.484']
2-strand cable arrow polynomial of the knot is: -1120*K1**4 + 992*K1**3*K2*K3 + 64*K1**3*K3*K4 - 480*K1**3*K3 - 1696*K1**2*K2**2 - 672*K1**2*K2*K4 + 2896*K1**2*K2 - 2272*K1**2*K3**2 - 288*K1**2*K4**2 - 3952*K1**2 + 96*K1*K2**3*K3 - 288*K1*K2**2*K3 + 256*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 128*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 6688*K1*K2*K3 - 64*K1*K2*K4*K7 + 32*K1*K3**3*K4 + 3400*K1*K3*K4 + 288*K1*K4*K5 + 48*K1*K5*K6 + 16*K1*K6*K7 - 72*K2**4 - 528*K2**2*K3**2 - 120*K2**2*K4**2 + 600*K2**2*K4 - 3266*K2**2 - 288*K2*K3**2*K4 + 392*K2*K3*K5 + 248*K2*K4*K6 + 8*K2*K5*K7 - 256*K3**4 - 144*K3**2*K4**2 + 288*K3**2*K6 - 3176*K3**2 + 88*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1286*K4**2 - 148*K5**2 - 134*K6**2 - 20*K7**2 - 2*K8**2 + 4030
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.484']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11020', 'vk6.11100', 'vk6.12188', 'vk6.12297', 'vk6.18188', 'vk6.18523', 'vk6.24644', 'vk6.25073', 'vk6.30595', 'vk6.30692', 'vk6.31863', 'vk6.31911', 'vk6.36782', 'vk6.37230', 'vk6.44025', 'vk6.44365', 'vk6.51819', 'vk6.51888', 'vk6.52689', 'vk6.52785', 'vk6.55995', 'vk6.56267', 'vk6.60531', 'vk6.60872', 'vk6.63505', 'vk6.63551', 'vk6.63985', 'vk6.64031', 'vk6.65659', 'vk6.65940', 'vk6.68707', 'vk6.68915']
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 O1O2O3O4O5U1U3U5O6U2U6U4
R3 orbit {'O1O2O3O4O5U1U3U5O6U2U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U4O6U1U3U5
Gauss code of K* O1O2O3U4O5O4O6U1U5U2U6U3
Gauss code of -K* O1O2O3U2O4O5O6U4U1U5U3U6
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 -1 -1 3 2 1],[ 4 0 3 1 4 2 1],[ 1 -3 0 -1 3 1 1],[ 1 -1 1 0 2 1 0],[-3 -4 -3 -2 0 0 0],[-2 -2 -1 -1 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 2 1 -1 -1 -4],[-3 0 0 0 -2 -3 -4],[-2 0 0 0 -1 -1 -2],[-1 0 0 0 0 -1 -1],[ 1 2 1 0 0 1 -1],[ 1 3 1 1 -1 0 -3],[ 4 4 2 1 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,1,4,0,0,2,3,4,0,1,1,2,0,1,1,-1,1,3]
Phi over symmetry [-4,-1,-1,1,2,3,0,2,4,4,3,1,1,2,1,2,2,2,1,2,1]
Phi of -K [-4,-1,-1,1,2,3,0,2,4,4,3,1,1,2,1,2,2,2,1,2,1]
Phi of K* [-3,-2,-1,1,1,4,1,2,1,2,3,1,2,2,4,1,2,4,-1,0,2]
Phi of -K* [-4,-1,-1,1,2,3,1,3,1,2,4,1,0,1,2,1,1,3,0,0,0]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+48t^4+22t^2+1
Outer characteristic polynomial t^7+80t^5+77t^3+11t
Flat arrow polynomial -2*K1**2 - 6*K1*K2 + 3*K1 - 2*K2**2 + K2 + 3*K3 + K4 + 3
2-strand cable arrow polynomial -1120*K1**4 + 992*K1**3*K2*K3 + 64*K1**3*K3*K4 - 480*K1**3*K3 - 1696*K1**2*K2**2 - 672*K1**2*K2*K4 + 2896*K1**2*K2 - 2272*K1**2*K3**2 - 288*K1**2*K4**2 - 3952*K1**2 + 96*K1*K2**3*K3 - 288*K1*K2**2*K3 + 256*K1*K2*K3**3 + 96*K1*K2*K3*K4**2 - 128*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 6688*K1*K2*K3 - 64*K1*K2*K4*K7 + 32*K1*K3**3*K4 + 3400*K1*K3*K4 + 288*K1*K4*K5 + 48*K1*K5*K6 + 16*K1*K6*K7 - 72*K2**4 - 528*K2**2*K3**2 - 120*K2**2*K4**2 + 600*K2**2*K4 - 3266*K2**2 - 288*K2*K3**2*K4 + 392*K2*K3*K5 + 248*K2*K4*K6 + 8*K2*K5*K7 - 256*K3**4 - 144*K3**2*K4**2 + 288*K3**2*K6 - 3176*K3**2 + 88*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1286*K4**2 - 148*K5**2 - 134*K6**2 - 20*K7**2 - 2*K8**2 + 4030
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {2, 3}]]
If K is slice False
Contact