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

Flat knot 6.1339

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,2,0,2,2,1,1,2,1,1,1,3,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1339']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.395', '6.430', '6.440', '6.548', '6.551', '6.774', '6.832', '6.887', '6.908', '6.911', '6.1205', '6.1332', '6.1339', '6.1341', '6.1346', '6.1382', '6.1488', '6.1651', '6.1655', '6.1686']
Outer characteristic polynomial of the knot is: t^7+40t^5+79t^3+15t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1339']
2-strand cable arrow polynomial of the knot is: -1472*K1**2*K2**4 + 1312*K1**2*K2**3 - 6000*K1**2*K2**2 - 96*K1**2*K2*K4 + 4152*K1**2*K2 - 2488*K1**2 + 2016*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 640*K1*K2**2*K3 - 192*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4872*K1*K2*K3 + 152*K1*K3*K4 - 288*K2**6 + 448*K2**4*K4 - 1792*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 208*K2**2*K4**2 + 1152*K2**2*K4 - 1006*K2**2 + 144*K2*K3*K5 + 24*K2*K4*K6 - 1072*K3**2 - 204*K4**2 - 2*K6**2 + 1906
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1339']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16784', 'vk6.16788', 'vk6.16817', 'vk6.16821', 'vk6.18163', 'vk6.18165', 'vk6.18500', 'vk6.18502', 'vk6.23200', 'vk6.23204', 'vk6.24624', 'vk6.25039', 'vk6.25041', 'vk6.35215', 'vk6.35246', 'vk6.36760', 'vk6.37184', 'vk6.37186', 'vk6.42696', 'vk6.42700', 'vk6.44339', 'vk6.44341', 'vk6.54975', 'vk6.55010', 'vk6.55969', 'vk6.55971', 'vk6.59365', 'vk6.59369', 'vk6.60505', 'vk6.65634', 'vk6.68166', 'vk6.68170']
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 O1O2O3U1O4O5U4U2U3O6U5U6
R3 orbit {'O1O2O3U1O4O5U4U2U3O6U5U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U1U2U6O5O6U3
Gauss code of K* O1O2O3U4O5O4U6U2U3O6U1U5
Gauss code of -K* O1O2O3U4U3O5U1U2U5O6O4U6
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 1 -1 2 1],[ 2 0 1 2 0 2 0],[ 1 -1 0 1 0 3 1],[-1 -2 -1 0 0 2 1],[ 1 0 0 0 0 1 1],[-2 -2 -3 -2 -1 0 1],[-1 0 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 1 -2 -1 -3 -2],[-1 -1 0 -1 -1 -1 0],[-1 2 1 0 0 -1 -2],[ 1 1 1 0 0 0 0],[ 1 3 1 1 0 0 -1],[ 2 2 0 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,-1,2,1,3,2,1,1,1,0,0,1,2,0,0,1]
Phi over symmetry [-2,-1,-1,1,1,2,-1,2,0,2,2,1,1,2,1,1,1,3,0,0,1]
Phi of -K [-2,-1,-1,1,1,2,0,1,1,3,2,0,1,1,0,2,1,2,-1,-1,2]
Phi of K* [-2,-1,-1,1,1,2,-1,2,0,2,2,1,1,2,1,1,1,3,0,0,1]
Phi of -K* [-2,-1,-1,1,1,2,0,1,0,2,2,0,1,0,1,1,1,3,-1,-1,2]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 2z^2+7z+7
Enhanced Jones-Krushkal polynomial -4w^4z^2+6w^3z^2-16w^3z+23w^2z+7w
Inner characteristic polynomial t^6+28t^4+27t^2+1
Outer characteristic polynomial t^7+40t^5+79t^3+15t
Flat arrow polynomial 4*K1**3 - 4*K1*K2 - K1 + K3 + 1
2-strand cable arrow polynomial -1472*K1**2*K2**4 + 1312*K1**2*K2**3 - 6000*K1**2*K2**2 - 96*K1**2*K2*K4 + 4152*K1**2*K2 - 2488*K1**2 + 2016*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 640*K1*K2**2*K3 - 192*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 4872*K1*K2*K3 + 152*K1*K3*K4 - 288*K2**6 + 448*K2**4*K4 - 1792*K2**4 - 32*K2**3*K6 - 688*K2**2*K3**2 - 208*K2**2*K4**2 + 1152*K2**2*K4 - 1006*K2**2 + 144*K2*K3*K5 + 24*K2*K4*K6 - 1072*K3**2 - 204*K4**2 - 2*K6**2 + 1906
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
Contact