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

Flat knot 6.34

Min(phi) over symmetries of the knot is: [-5,-1,-1,1,2,4,1,3,2,5,4,1,1,2,2,1,2,3,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.34']
Arrow polynomial of the knot is: 4*K1**2*K2 + 4*K1**2*K3 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K1*K4 + 2*K1 + 3*K2 + 2*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.34']
Outer characteristic polynomial of the knot is: t^7+132t^5+61t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.34']
2-strand cable arrow polynomial of the knot is: -144*K1**4 + 256*K1**3*K2*K3 + 32*K1**3*K3*K4 - 320*K1**3*K3 + 64*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 3408*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 544*K1**2*K2*K4 + 4320*K1**2*K2 - 496*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 4120*K1**2 + 1984*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 1280*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 + 64*K1*K2**2*K5*K6 - 416*K1*K2**2*K5 + 32*K1*K2**2*K6*K7 + 32*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 544*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6192*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 1200*K1*K3*K4 + 208*K1*K4*K5 + 64*K1*K5*K6 + 16*K1*K6*K7 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 32*K2**4*K6**2 - 1840*K2**4 + 352*K2**3*K3*K5 + 96*K2**3*K4*K6 + 32*K2**3*K5*K7 + 32*K2**3*K6*K8 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 1872*K2**2*K3**2 - 96*K2**2*K3*K7 - 424*K2**2*K4**2 - 32*K2**2*K4*K8 + 1968*K2**2*K4 - 160*K2**2*K5**2 - 104*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 2720*K2**2 - 32*K2*K3**2*K4 + 1224*K2*K3*K5 + 336*K2*K4*K6 + 112*K2*K5*K7 + 32*K2*K6*K8 - 32*K3**2*K4**2 + 8*K3**2*K6 - 2076*K3**2 - 686*K4**2 - 252*K5**2 - 80*K6**2 - 16*K7**2 - 4*K8**2 + 3336
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.34']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71566', 'vk6.71675', 'vk6.72089', 'vk6.72304', 'vk6.74039', 'vk6.74600', 'vk6.76087', 'vk6.76797', 'vk6.77182', 'vk6.77279', 'vk6.77476', 'vk6.77643', 'vk6.79027', 'vk6.79603', 'vk6.80562', 'vk6.81012', 'vk6.81101', 'vk6.81141', 'vk6.81162', 'vk6.81209', 'vk6.81307', 'vk6.81452', 'vk6.82257', 'vk6.83497', 'vk6.83829', 'vk6.83970', 'vk6.85391', 'vk6.86320', 'vk6.87101', 'vk6.88022', 'vk6.88320', 'vk6.88953']
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 O1O2O3O4O5O6U1U4U5U2U6U3
R3 orbit {'O1O2O3O4O5O6U1U4U5U2U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U1U5U2U3U6
Gauss code of K* O1O2O3O4O5O6U1U4U6U2U3U5
Gauss code of -K* O1O2O3O4O5O6U2U4U5U1U3U6
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 -5 -1 2 -1 1 4],[ 5 0 3 5 1 2 4],[ 1 -3 0 2 -1 1 3],[-2 -5 -2 0 -2 0 2],[ 1 -1 1 2 0 1 2],[-1 -2 -1 0 -1 0 1],[-4 -4 -3 -2 -2 -1 0]]
Primitive based matrix [[ 0 4 2 1 -1 -1 -5],[-4 0 -2 -1 -2 -3 -4],[-2 2 0 0 -2 -2 -5],[-1 1 0 0 -1 -1 -2],[ 1 2 2 1 0 1 -1],[ 1 3 2 1 -1 0 -3],[ 5 4 5 2 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,1,1,5,2,1,2,3,4,0,2,2,5,1,1,2,-1,1,3]
Phi over symmetry [-5,-1,-1,1,2,4,1,3,2,5,4,1,1,2,2,1,2,3,0,1,2]
Phi of -K [-5,-1,-1,1,2,4,1,3,4,2,5,1,1,1,2,1,1,3,1,2,0]
Phi of K* [-4,-2,-1,1,1,5,0,2,2,3,5,1,1,1,2,1,1,4,-1,1,3]
Phi of -K* [-5,-1,-1,1,2,4,1,3,2,5,4,1,1,2,2,1,2,3,0,1,2]
Symmetry type of based matrix c
u-polynomial t^5-t^4-t^2+t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2+23w^2z+23w
Inner characteristic polynomial t^6+84t^4+8t^2
Outer characteristic polynomial t^7+132t^5+61t^3+4t
Flat arrow polynomial 4*K1**2*K2 + 4*K1**2*K3 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 - 2*K1*K4 + 2*K1 + 3*K2 + 2*K3 + 4
2-strand cable arrow polynomial -144*K1**4 + 256*K1**3*K2*K3 + 32*K1**3*K3*K4 - 320*K1**3*K3 + 64*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 3408*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 544*K1**2*K2*K4 + 4320*K1**2*K2 - 496*K1**2*K3**2 - 64*K1**2*K3*K5 - 32*K1**2*K4**2 - 4120*K1**2 + 1984*K1*K2**3*K3 + 352*K1*K2**2*K3*K4 - 1280*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 + 64*K1*K2**2*K5*K6 - 416*K1*K2**2*K5 + 32*K1*K2**2*K6*K7 + 32*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 544*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6192*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 1200*K1*K3*K4 + 208*K1*K4*K5 + 64*K1*K5*K6 + 16*K1*K6*K7 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 32*K2**4*K6**2 - 1840*K2**4 + 352*K2**3*K3*K5 + 96*K2**3*K4*K6 + 32*K2**3*K5*K7 + 32*K2**3*K6*K8 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 1872*K2**2*K3**2 - 96*K2**2*K3*K7 - 424*K2**2*K4**2 - 32*K2**2*K4*K8 + 1968*K2**2*K4 - 160*K2**2*K5**2 - 104*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 2720*K2**2 - 32*K2*K3**2*K4 + 1224*K2*K3*K5 + 336*K2*K4*K6 + 112*K2*K5*K7 + 32*K2*K6*K8 - 32*K3**2*K4**2 + 8*K3**2*K6 - 2076*K3**2 - 686*K4**2 - 252*K5**2 - 80*K6**2 - 16*K7**2 - 4*K8**2 + 3336
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}]]
If K is slice False
Contact