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

Flat knot 6.1076

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,3,1,0,1,1,0,1,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1076']
Arrow polynomial of the knot is: 12*K1**3 - 4*K1**2 - 8*K1*K2 - 5*K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.330', '6.531', '6.1076', '6.1079', '6.1567']
Outer characteristic polynomial of the knot is: t^7+34t^5+63t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1076']
2-strand cable arrow polynomial of the knot is: -16*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 192*K1**2*K2**4 + 544*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6512*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 7608*K1**2*K2 - 80*K1**2*K3**2 - 112*K1**2*K4**2 - 6264*K1**2 + 928*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 384*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 8008*K1*K2*K3 + 752*K1*K3*K4 + 200*K1*K4*K5 - 96*K2**6 + 192*K2**4*K4 - 2320*K2**4 - 32*K2**3*K6 - 528*K2**2*K3**2 - 96*K2**2*K4**2 + 2808*K2**2*K4 - 4422*K2**2 - 32*K2*K3**2*K4 + 344*K2*K3*K5 + 72*K2*K4*K6 + 8*K3**2*K6 - 2284*K3**2 - 844*K4**2 - 100*K5**2 - 18*K6**2 + 4754
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1076']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11075', 'vk6.11153', 'vk6.12237', 'vk6.12344', 'vk6.18331', 'vk6.18670', 'vk6.24765', 'vk6.25224', 'vk6.30650', 'vk6.30743', 'vk6.31878', 'vk6.31947', 'vk6.36947', 'vk6.37411', 'vk6.44138', 'vk6.44461', 'vk6.51874', 'vk6.51921', 'vk6.52739', 'vk6.52850', 'vk6.56109', 'vk6.56330', 'vk6.60624', 'vk6.60957', 'vk6.63532', 'vk6.63576', 'vk6.64010', 'vk6.64054', 'vk6.65751', 'vk6.66017', 'vk6.68759', 'vk6.68969']
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 O1O2O3O4U5U4O6U2O5U1U6U3
R3 orbit {'O1O2O3O4U5U4O6U2O5U1U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U4O6U3O5U1U6
Gauss code of K* O1O2O3U1U4U3U5O6O5U2O4U6
Gauss code of -K* O1O2O3U4O5U2O6O4U6U1U5U3
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 2 1 -1 1],[ 2 0 1 3 1 0 1],[ 1 -1 0 1 0 0 0],[-2 -3 -1 0 1 -2 -1],[-1 -1 0 -1 0 -1 -1],[ 1 0 0 2 1 0 1],[-1 -1 0 1 1 -1 0]]
Primitive based matrix [[ 0 2 1 1 -1 -1 -2],[-2 0 1 -1 -1 -2 -3],[-1 -1 0 -1 0 -1 -1],[-1 1 1 0 0 -1 -1],[ 1 1 0 0 0 0 -1],[ 1 2 1 1 0 0 0],[ 2 3 1 1 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,1,1,2,-1,1,1,2,3,1,0,1,1,0,1,1,0,1,0]
Phi over symmetry [-2,-1,-1,1,1,2,-1,1,1,2,3,1,0,1,1,0,1,1,0,1,0]
Phi of -K [-2,-1,-1,1,1,2,0,1,2,2,1,0,2,2,2,1,1,1,-1,0,2]
Phi of K* [-2,-1,-1,1,1,2,0,2,1,2,1,1,1,2,2,1,2,2,0,1,0]
Phi of -K* [-2,-1,-1,1,1,2,0,1,1,1,3,0,1,1,2,0,0,1,-1,-1,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+28z+25
Enhanced Jones-Krushkal polynomial 8w^3z^2+28w^2z+25w
Inner characteristic polynomial t^6+22t^4+17t^2
Outer characteristic polynomial t^7+34t^5+63t^3+6t
Flat arrow polynomial 12*K1**3 - 4*K1**2 - 8*K1*K2 - 5*K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -16*K1**4 + 32*K1**3*K2*K3 - 64*K1**3*K3 - 192*K1**2*K2**4 + 544*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 6512*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 320*K1**2*K2*K4 + 7608*K1**2*K2 - 80*K1**2*K3**2 - 112*K1**2*K4**2 - 6264*K1**2 + 928*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 384*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 8008*K1*K2*K3 + 752*K1*K3*K4 + 200*K1*K4*K5 - 96*K2**6 + 192*K2**4*K4 - 2320*K2**4 - 32*K2**3*K6 - 528*K2**2*K3**2 - 96*K2**2*K4**2 + 2808*K2**2*K4 - 4422*K2**2 - 32*K2*K3**2*K4 + 344*K2*K3*K5 + 72*K2*K4*K6 + 8*K3**2*K6 - 2284*K3**2 - 844*K4**2 - 100*K5**2 - 18*K6**2 + 4754
Genus of based matrix 0
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}]]
If K is slice True
Contact