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

Flat knot 6.506

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,0,3,1,3,4,1,0,1,1,0,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.506']
Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.87', '6.88', '6.184', '6.302', '6.459', '6.467', '6.506']
Outer characteristic polynomial of the knot is: t^7+81t^5+77t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.506']
2-strand cable arrow polynomial of the knot is: 800*K1**4*K2 - 3792*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1152*K1**3*K3 + 256*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 3584*K1**2*K2**2 - 704*K1**2*K2*K4 + 8200*K1**2*K2 - 1488*K1**2*K3**2 - 256*K1**2*K4**2 - 5276*K1**2 + 64*K1*K2**3*K3 - 704*K1*K2**2*K3 - 160*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 7328*K1*K2*K3 + 2736*K1*K3*K4 + 632*K1*K4*K5 + 8*K1*K5*K6 - 352*K2**4 - 192*K2**2*K3**2 - 112*K2**2*K4**2 + 1344*K2**2*K4 - 4692*K2**2 - 64*K2*K3**2*K4 + 584*K2*K3*K5 + 144*K2*K4*K6 - 96*K3**4 - 48*K3**2*K4**2 + 120*K3**2*K6 - 2856*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1364*K4**2 - 352*K5**2 - 68*K6**2 - 4*K7**2 - 2*K8**2 + 5100
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.506']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16980', 'vk6.16982', 'vk6.17223', 'vk6.17225', 'vk6.20873', 'vk6.20877', 'vk6.22280', 'vk6.22284', 'vk6.23384', 'vk6.23693', 'vk6.23695', 'vk6.28348', 'vk6.35440', 'vk6.35881', 'vk6.35883', 'vk6.39976', 'vk6.39980', 'vk6.42047', 'vk6.43180', 'vk6.43182', 'vk6.46516', 'vk6.46520', 'vk6.55133', 'vk6.55135', 'vk6.55390', 'vk6.57676', 'vk6.57680', 'vk6.58872', 'vk6.59855', 'vk6.59857', 'vk6.68406', 'vk6.69740']
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 O1O2O3O4O5U1U6U3O6U5U4U2
R3 orbit {'O1O2O3O4O5U1U6U3O6U5U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U2U1O6U3U6U5
Gauss code of K* O1O2O3U2O4O5O6U1U6U3U5U4
Gauss code of -K* O1O2O3U4O5O4O6U3U2U5U1U6
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 0 2 2 -1],[ 4 0 4 1 3 2 3],[-1 -4 0 -1 1 1 -2],[ 0 -1 1 0 1 0 0],[-2 -3 -1 -1 0 0 -2],[-2 -2 -1 0 0 0 -2],[ 1 -3 2 0 2 2 0]]
Primitive based matrix [[ 0 2 2 1 0 -1 -4],[-2 0 0 -1 0 -2 -2],[-2 0 0 -1 -1 -2 -3],[-1 1 1 0 -1 -2 -4],[ 0 0 1 1 0 0 -1],[ 1 2 2 2 0 0 -3],[ 4 2 3 4 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,1,4,0,1,0,2,2,1,1,2,3,1,2,4,0,1,3]
Phi over symmetry [-4,-1,0,1,2,2,0,3,1,3,4,1,0,1,1,0,1,2,0,0,0]
Phi of -K [-4,-1,0,1,2,2,0,3,1,3,4,1,0,1,1,0,1,2,0,0,0]
Phi of K* [-2,-2,-1,0,1,4,0,0,1,1,3,0,2,1,4,0,0,1,1,3,0]
Phi of -K* [-4,-1,0,1,2,2,3,1,4,2,3,0,2,2,2,1,0,1,1,1,0]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 6z^2+27z+31
Enhanced Jones-Krushkal polynomial 6w^3z^2+27w^2z+31w
Inner characteristic polynomial t^6+55t^4+35t^2+1
Outer characteristic polynomial t^7+81t^5+77t^3+5t
Flat arrow polynomial -4*K1*K2 + 2*K1 - 2*K2**2 + 2*K3 + K4 + 2
2-strand cable arrow polynomial 800*K1**4*K2 - 3792*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1152*K1**3*K3 + 256*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 3584*K1**2*K2**2 - 704*K1**2*K2*K4 + 8200*K1**2*K2 - 1488*K1**2*K3**2 - 256*K1**2*K4**2 - 5276*K1**2 + 64*K1*K2**3*K3 - 704*K1*K2**2*K3 - 160*K1*K2**2*K5 - 544*K1*K2*K3*K4 + 7328*K1*K2*K3 + 2736*K1*K3*K4 + 632*K1*K4*K5 + 8*K1*K5*K6 - 352*K2**4 - 192*K2**2*K3**2 - 112*K2**2*K4**2 + 1344*K2**2*K4 - 4692*K2**2 - 64*K2*K3**2*K4 + 584*K2*K3*K5 + 144*K2*K4*K6 - 96*K3**4 - 48*K3**2*K4**2 + 120*K3**2*K6 - 2856*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1364*K4**2 - 352*K5**2 - 68*K6**2 - 4*K7**2 - 2*K8**2 + 5100
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
Contact