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

Flat knot 6.1923

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,0,1,0,2,1,1,1,1,0,1,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1923', '7.31310', '7.41361']
Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923']
Outer characteristic polynomial of the knot is: t^7+32t^5+44t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1923', '7.15570', '7.31310']
2-strand cable arrow polynomial of the knot is: -512*K1**6 - 640*K1**4*K2**2 + 4224*K1**4*K2 - 8608*K1**4 + 2304*K1**3*K2*K3 + 256*K1**3*K3*K4 - 2176*K1**3*K3 - 384*K1**2*K2**4 + 3456*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 13472*K1**2*K2**2 - 1856*K1**2*K2*K4 + 12560*K1**2*K2 - 1888*K1**2*K3**2 - 64*K1**2*K3*K5 - 288*K1**2*K4**2 - 1416*K1**2 + 1792*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 - 512*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9216*K1*K2*K3 - 64*K1*K2*K4*K5 + 1408*K1*K3*K4 + 160*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 3040*K2**4 - 64*K2**3*K6 - 1088*K2**2*K3**2 - 160*K2**2*K4**2 + 2096*K2**2*K4 - 1780*K2**2 + 512*K2*K3*K5 + 96*K2*K4*K6 - 1240*K3**2 - 312*K4**2 - 16*K5**2 - 4*K6**2 + 3102
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1923']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.21', 'vk6.32', 'vk6.155', 'vk6.170', 'vk6.1208', 'vk6.1301', 'vk6.1316', 'vk6.2353', 'vk6.2389', 'vk6.2960', 'vk6.3549', 'vk6.6933', 'vk6.6964', 'vk6.15387', 'vk6.15504', 'vk6.33435', 'vk6.33490', 'vk6.33604', 'vk6.49941', 'vk6.53746']
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 O1O2O3U1U4U3O4O5O6U5U6U2
R3 orbit {'O1O2O3U1U4U3O4O5O6U5U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U5O4O5O6U1U6U3
Gauss code of K* O1O2O3U2U4U5O4O5O6U1U6U3
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 -2 1 2 -1 -1 1],[ 2 0 2 1 1 0 0],[-1 -2 0 1 -2 -1 1],[-2 -1 -1 0 -2 0 0],[ 1 -1 2 2 0 -1 1],[ 1 0 1 0 1 0 1],[-1 0 -1 0 -1 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 0 -1 0 -2 -1],[-1 0 0 -1 -1 -1 0],[-1 1 1 0 -1 -2 -2],[ 1 0 1 1 0 1 0],[ 1 2 1 2 -1 0 -1],[ 2 1 0 2 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,0,1,0,2,1,1,1,1,0,1,2,2,-1,0,1]
Phi over symmetry [-2,-1,-1,1,1,2,0,1,0,2,1,1,1,1,0,1,2,2,-1,0,1]
Phi of -K [-2,-1,-1,1,1,2,0,1,1,3,3,1,0,1,1,1,1,3,-1,0,1]
Phi of K* [-2,-1,-1,1,1,2,0,1,1,3,3,1,0,1,1,1,1,3,-1,0,1]
Phi of -K* [-2,-1,-1,1,1,2,0,1,0,2,1,1,1,1,0,1,2,2,-1,0,1]
Symmetry type of based matrix -
u-polynomial 0
Normalized Jones-Krushkal polynomial 4z^2+24z+33
Enhanced Jones-Krushkal polynomial 4w^3z^2+24w^2z+33w
Inner characteristic polynomial t^6+20t^4+24t^2+1
Outer characteristic polynomial t^7+32t^5+44t^3+7t
Flat arrow polynomial 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -512*K1**6 - 640*K1**4*K2**2 + 4224*K1**4*K2 - 8608*K1**4 + 2304*K1**3*K2*K3 + 256*K1**3*K3*K4 - 2176*K1**3*K3 - 384*K1**2*K2**4 + 3456*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 13472*K1**2*K2**2 - 1856*K1**2*K2*K4 + 12560*K1**2*K2 - 1888*K1**2*K3**2 - 64*K1**2*K3*K5 - 288*K1**2*K4**2 - 1416*K1**2 + 1792*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 - 512*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 9216*K1*K2*K3 - 64*K1*K2*K4*K5 + 1408*K1*K3*K4 + 160*K1*K4*K5 - 64*K2**6 + 128*K2**4*K4 - 3040*K2**4 - 64*K2**3*K6 - 1088*K2**2*K3**2 - 160*K2**2*K4**2 + 2096*K2**2*K4 - 1780*K2**2 + 512*K2*K3*K5 + 96*K2*K4*K6 - 1240*K3**2 - 312*K4**2 - 16*K5**2 - 4*K6**2 + 3102
Genus of based matrix 0
Fillings of based matrix [[{5, 6}, {2, 4}, {1, 3}]]
If K is slice True
Contact