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

Flat knot 6.185

Min(phi) over symmetries of the knot is: [-4,-3,-1,1,3,4,0,2,3,3,5,1,2,2,3,1,2,3,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.185']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 16*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 6*K2 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.185']
Outer characteristic polynomial of the knot is: t^7+136t^5+58t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.185']
2-strand cable arrow polynomial of the knot is: -3552*K1**4 + 1984*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1216*K1**3*K3 - 256*K1**2*K2**4 + 1088*K1**2*K2**3 - 1152*K1**2*K2**2*K3**2 - 11712*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 1664*K1**2*K2*K4 + 14800*K1**2*K2 - 1440*K1**2*K3**2 - 64*K1**2*K3*K5 - 96*K1**2*K4**2 - 8544*K1**2 + 256*K1*K2**3*K3**3 + 4480*K1*K2**3*K3 + 704*K1*K2**2*K3*K4 - 2432*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 960*K1*K2**2*K5 + 384*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 12832*K1*K2*K3 - 64*K1*K2*K4*K5 + 1888*K1*K3*K4 + 176*K1*K4*K5 - 64*K2**6 - 512*K2**4*K3**2 - 64*K2**4*K4**2 + 256*K2**4*K4 - 3360*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 128*K2**2*K3**4 - 2624*K2**2*K3**2 - 496*K2**2*K4**2 + 2960*K2**2*K4 - 64*K2**2*K5**2 - 16*K2**2*K6**2 - 5200*K2**2 + 1056*K2*K3*K5 + 160*K2*K4*K6 - 64*K3**4 - 3096*K3**2 - 724*K4**2 - 72*K5**2 - 8*K6**2 + 6554
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.185']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81564', 'vk6.81645', 'vk6.81825', 'vk6.82044', 'vk6.82227', 'vk6.82339', 'vk6.82535', 'vk6.82992', 'vk6.83142', 'vk6.83562', 'vk6.83931', 'vk6.84091', 'vk6.84528', 'vk6.84891', 'vk6.85916', 'vk6.86394', 'vk6.86456', 'vk6.88827', 'vk6.89769', 'vk6.89885']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is -.
The reverse -K is
The 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 O1O2O3O4O5U2O6U1U3U4U6U5
R3 orbit {'O1O2O3O4O5U2O6U1U3U4U6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U6U2U3U5O6U4
Gauss code of K* O1O2O3O4O5U1U6U2U3U5O6U4
Gauss code of -K* Same
Diagrammatic symmetry type -
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 -3 -1 1 4 3],[ 4 0 0 2 3 5 3],[ 3 0 0 1 2 3 2],[ 1 -2 -1 0 1 3 2],[-1 -3 -2 -1 0 2 1],[-4 -5 -3 -3 -2 0 0],[-3 -3 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 4 3 1 -1 -3 -4],[-4 0 0 -2 -3 -3 -5],[-3 0 0 -1 -2 -2 -3],[-1 2 1 0 -1 -2 -3],[ 1 3 2 1 0 -1 -2],[ 3 3 2 2 1 0 0],[ 4 5 3 3 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-3,-1,1,3,4,0,2,3,3,5,1,2,2,3,1,2,3,1,2,0]
Phi over symmetry [-4,-3,-1,1,3,4,0,2,3,3,5,1,2,2,3,1,2,3,1,2,0]
Phi of -K [-4,-3,-1,1,3,4,1,1,2,4,3,1,2,4,4,1,2,2,1,1,1]
Phi of K* [-4,-3,-1,1,3,4,1,1,2,4,3,1,2,4,4,1,2,2,1,1,1]
Phi of -K* [-4,-3,-1,1,3,4,0,2,3,3,5,1,2,2,3,1,2,3,1,2,0]
Symmetry type of based matrix -
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+84t^4+12t^2
Outer characteristic polynomial t^7+136t^5+58t^3+4t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 16*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 6*K2 + 7
2-strand cable arrow polynomial -3552*K1**4 + 1984*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1216*K1**3*K3 - 256*K1**2*K2**4 + 1088*K1**2*K2**3 - 1152*K1**2*K2**2*K3**2 - 11712*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 1664*K1**2*K2*K4 + 14800*K1**2*K2 - 1440*K1**2*K3**2 - 64*K1**2*K3*K5 - 96*K1**2*K4**2 - 8544*K1**2 + 256*K1*K2**3*K3**3 + 4480*K1*K2**3*K3 + 704*K1*K2**2*K3*K4 - 2432*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 960*K1*K2**2*K5 + 384*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 12832*K1*K2*K3 - 64*K1*K2*K4*K5 + 1888*K1*K3*K4 + 176*K1*K4*K5 - 64*K2**6 - 512*K2**4*K3**2 - 64*K2**4*K4**2 + 256*K2**4*K4 - 3360*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 128*K2**2*K3**4 - 2624*K2**2*K3**2 - 496*K2**2*K4**2 + 2960*K2**2*K4 - 64*K2**2*K5**2 - 16*K2**2*K6**2 - 5200*K2**2 + 1056*K2*K3*K5 + 160*K2*K4*K6 - 64*K3**4 - 3096*K3**2 - 724*K4**2 - 72*K5**2 - 8*K6**2 + 6554
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}]]
If K is slice True
Contact