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

Flat knot 6.512

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,0,2,1,3,5,0,0,0,1,0,1,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.512']
Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.512']
Outer characteristic polynomial of the knot is: t^7+89t^5+76t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.512']
2-strand cable arrow polynomial of the knot is: -256*K1**6 - 832*K1**4*K2**2 + 2560*K1**4*K2 - 4656*K1**4 - 384*K1**3*K2**2*K3 + 1472*K1**3*K2*K3 - 960*K1**3*K3 - 448*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2784*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 512*K1**2*K2**2*K4 - 11024*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 1632*K1**2*K2*K4 + 11472*K1**2*K2 - 848*K1**2*K3**2 - 32*K1**2*K3*K5 - 144*K1**2*K4**2 - 5580*K1**2 - 128*K1*K2**3*K3*K4 + 2368*K1*K2**3*K3 + 800*K1*K2**2*K3*K4 - 1856*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 672*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 9552*K1*K2*K3 - 96*K1*K2*K4*K5 + 1784*K1*K3*K4 + 280*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 512*K2**4*K4 - 2728*K2**4 + 288*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1616*K2**2*K3**2 - 704*K2**2*K4**2 + 2496*K2**2*K4 - 176*K2**2*K5**2 - 8*K2**2*K6**2 - 3984*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 968*K2*K3*K5 + 208*K2*K4*K6 + 24*K2*K5*K7 + 8*K3**2*K6 - 2456*K3**2 - 964*K4**2 - 252*K5**2 - 24*K6**2 + 5322
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.512']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4699', 'vk6.5004', 'vk6.6193', 'vk6.6666', 'vk6.8190', 'vk6.8610', 'vk6.9568', 'vk6.9909', 'vk6.17397', 'vk6.20914', 'vk6.20974', 'vk6.22324', 'vk6.22396', 'vk6.23562', 'vk6.23901', 'vk6.28394', 'vk6.36165', 'vk6.40044', 'vk6.40167', 'vk6.42095', 'vk6.43074', 'vk6.43380', 'vk6.46576', 'vk6.46676', 'vk6.48731', 'vk6.49527', 'vk6.49732', 'vk6.51433', 'vk6.55555', 'vk6.58904', 'vk6.65293', 'vk6.69762']
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 O1O2O3O4O5U1U6U5O6U3U4U2
R3 orbit {'O1O2O3O4O5U1U6U5O6U3U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U2U3O6U1U6U5
Gauss code of K* O1O2O3U2O4O5O6U1U6U4U5U3
Gauss code of -K* O1O2O3U4O5O4O6U5U2U3U1U6
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 2 3 1 3],[-1 -4 0 -1 1 1 -2],[ 0 -2 1 0 1 1 -1],[-2 -3 -1 -1 0 1 -3],[-2 -1 -1 -1 -1 0 -2],[ 1 -3 2 1 3 2 0]]
Primitive based matrix [[ 0 2 2 1 0 -1 -4],[-2 0 1 -1 -1 -3 -3],[-2 -1 0 -1 -1 -2 -1],[-1 1 1 0 -1 -2 -4],[ 0 1 1 1 0 -1 -2],[ 1 3 2 2 1 0 -3],[ 4 3 1 4 2 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,1,4,-1,1,1,3,3,1,1,2,1,1,2,4,1,2,3]
Phi over symmetry [-4,-1,0,1,2,2,0,2,1,3,5,0,0,0,1,0,1,1,0,0,-1]
Phi of -K [-4,-1,0,1,2,2,0,2,1,3,5,0,0,0,1,0,1,1,0,0,-1]
Phi of K* [-2,-2,-1,0,1,4,-1,0,1,1,5,0,1,0,3,0,0,1,0,2,0]
Phi of -K* [-4,-1,0,1,2,2,3,2,4,1,3,1,2,2,3,1,1,1,1,1,-1]
Symmetry type of based matrix c
u-polynomial t^4-2t^2
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+63t^4+40t^2
Outer characteristic polynomial t^7+89t^5+76t^3+7t
Flat arrow polynomial 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 8*K1*K2 - 2*K1*K3 - 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -256*K1**6 - 832*K1**4*K2**2 + 2560*K1**4*K2 - 4656*K1**4 - 384*K1**3*K2**2*K3 + 1472*K1**3*K2*K3 - 960*K1**3*K3 - 448*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2784*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 512*K1**2*K2**2*K4 - 11024*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 1632*K1**2*K2*K4 + 11472*K1**2*K2 - 848*K1**2*K3**2 - 32*K1**2*K3*K5 - 144*K1**2*K4**2 - 5580*K1**2 - 128*K1*K2**3*K3*K4 + 2368*K1*K2**3*K3 + 800*K1*K2**2*K3*K4 - 1856*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 672*K1*K2**2*K5 - 480*K1*K2*K3*K4 + 9552*K1*K2*K3 - 96*K1*K2*K4*K5 + 1784*K1*K3*K4 + 280*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 512*K2**4*K4 - 2728*K2**4 + 288*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1616*K2**2*K3**2 - 704*K2**2*K4**2 + 2496*K2**2*K4 - 176*K2**2*K5**2 - 8*K2**2*K6**2 - 3984*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 968*K2*K3*K5 + 208*K2*K4*K6 + 24*K2*K5*K7 + 8*K3**2*K6 - 2456*K3**2 - 964*K4**2 - 252*K5**2 - 24*K6**2 + 5322
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}]]
If K is slice False
Contact