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

Flat knot 6.1383

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,1,2,1,2,3,1,1,0,1,0,1,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1383']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']
Outer characteristic polynomial of the knot is: t^7+36t^5+80t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1383']
2-strand cable arrow polynomial of the knot is: 192*K1**4*K2 - 1344*K1**4 - 128*K1**3*K3 - 256*K1**2*K2**4 + 1024*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 5824*K1**2*K2**2 - 576*K1**2*K2*K4 + 8720*K1**2*K2 - 64*K1**2*K4**2 - 5752*K1**2 + 256*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 128*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 5616*K1*K2*K3 + 624*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1264*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 1584*K2**2*K4 - 3848*K2**2 + 64*K2*K3*K5 - 1320*K3**2 - 468*K4**2 - 16*K5**2 + 4002
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1383']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71595', 'vk6.71719', 'vk6.72136', 'vk6.72331', 'vk6.74047', 'vk6.74614', 'vk6.76805', 'vk6.77218', 'vk6.77526', 'vk6.77670', 'vk6.79046', 'vk6.79611', 'vk6.80570', 'vk6.81020', 'vk6.81356', 'vk6.81399', 'vk6.85409', 'vk6.85488', 'vk6.87993', 'vk6.89323']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3U4O5O6U3U5U1O4U2U6
R3 orbit {'O1O2O3U4O5O6U3U5U1O4U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U2O5U3U6U1O4O6U5
Gauss code of K* Same
Gauss code of -K* O1O2O3U4U2O5U3U6U1O4O6U5
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 -1 0 -1 -1 0 3],[ 1 0 0 -1 0 1 3],[ 0 0 0 0 -1 1 2],[ 1 1 0 0 0 1 1],[ 1 0 1 0 0 0 2],[ 0 -1 -1 -1 0 0 1],[-3 -3 -2 -1 -2 -1 0]]
Primitive based matrix [[ 0 3 0 0 -1 -1 -1],[-3 0 -1 -2 -1 -2 -3],[ 0 1 0 -1 -1 0 -1],[ 0 2 1 0 0 -1 0],[ 1 1 1 0 0 0 1],[ 1 2 0 1 0 0 0],[ 1 3 1 0 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,0,0,1,1,1,1,2,1,2,3,1,1,0,1,0,1,0,0,-1,0]
Phi over symmetry [-3,0,0,1,1,1,1,2,1,2,3,1,1,0,1,0,1,0,0,-1,0]
Phi of -K [-1,-1,-1,0,0,3,-1,0,0,1,3,0,0,1,1,1,0,2,1,2,1]
Phi of K* [-3,0,0,1,1,1,1,2,1,2,3,1,1,0,1,0,1,0,0,-1,0]
Phi of -K* [-1,-1,-1,0,0,3,-1,0,0,1,3,0,0,1,1,1,0,2,1,2,1]
Symmetry type of based matrix +
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 6z^2+26z+29
Enhanced Jones-Krushkal polynomial 6w^3z^2+26w^2z+29w
Inner characteristic polynomial t^6+24t^4+42t^2+1
Outer characteristic polynomial t^7+36t^5+80t^3+7t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial 192*K1**4*K2 - 1344*K1**4 - 128*K1**3*K3 - 256*K1**2*K2**4 + 1024*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 5824*K1**2*K2**2 - 576*K1**2*K2*K4 + 8720*K1**2*K2 - 64*K1**2*K4**2 - 5752*K1**2 + 256*K1*K2**3*K3 - 1216*K1*K2**2*K3 - 128*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 5616*K1*K2*K3 + 624*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1264*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 1584*K2**2*K4 - 3848*K2**2 + 64*K2*K3*K5 - 1320*K3**2 - 468*K4**2 - 16*K5**2 + 4002
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}]]
If K is slice False
Contact