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

Flat knot 6.559

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,2,1,3,3,1,1,1,2,1,3,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.559']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.522', '6.559']
Outer characteristic polynomial of the knot is: t^7+52t^5+67t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.559']
2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 608*K1**4*K2 - 720*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 288*K1**3*K2*K3 - 288*K1**3*K3 - 1600*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3872*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 9536*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 7088*K1**2*K2 - 80*K1**2*K3**2 - 4728*K1**2 - 256*K1*K2**4*K3 - 128*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 3712*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 7808*K1*K2*K3 + 728*K1*K3*K4 + 112*K1*K4*K5 - 192*K2**6 - 128*K2**4*K3**2 - 192*K2**4*K4**2 + 1120*K2**4*K4 - 4112*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 - 64*K2**3*K6 - 2144*K2**2*K3**2 - 728*K2**2*K4**2 + 2560*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 1738*K2**2 + 888*K2*K3*K5 + 152*K2*K4*K6 + 8*K2*K5*K7 - 1908*K3**2 - 644*K4**2 - 140*K5**2 - 14*K6**2 + 3738
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.559']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4716', 'vk6.5031', 'vk6.6237', 'vk6.6691', 'vk6.8217', 'vk6.8655', 'vk6.9595', 'vk6.9924', 'vk6.20306', 'vk6.21641', 'vk6.27598', 'vk6.29152', 'vk6.39024', 'vk6.41274', 'vk6.45788', 'vk6.47467', 'vk6.48748', 'vk6.48945', 'vk6.49541', 'vk6.49761', 'vk6.50762', 'vk6.50964', 'vk6.51235', 'vk6.51446', 'vk6.57161', 'vk6.58351', 'vk6.61783', 'vk6.62904', 'vk6.66778', 'vk6.67656', 'vk6.69422', 'vk6.70146']
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 O1O2O3O4O5U4U1U5O6U2U3U6
R3 orbit {'O1O2O3O4O5U4U1U5O6U2U3U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U3U4O6U1U5U2
Gauss code of K* O1O2O3U4O5O6O4U2U5U6U1U3
Gauss code of -K* O1O2O3U1O4O5O6U4U6U2U3U5
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 -3 -1 1 -1 2 2],[ 3 0 2 3 0 2 2],[ 1 -2 0 1 -1 1 2],[-1 -3 -1 0 -1 1 1],[ 1 0 1 1 0 1 0],[-2 -2 -1 -1 -1 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 -1 0 -2 -2],[-2 0 0 -1 -1 -1 -2],[-1 1 1 0 -1 -1 -3],[ 1 0 1 1 0 1 0],[ 1 2 1 1 -1 0 -2],[ 3 2 2 3 0 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,1,0,2,2,1,1,1,2,1,1,3,-1,0,2]
Phi over symmetry [-3,-1,-1,1,2,2,0,2,1,3,3,1,1,1,2,1,3,2,0,0,0]
Phi of -K [-3,-1,-1,1,2,2,0,2,1,3,3,1,1,1,2,1,3,2,0,0,0]
Phi of K* [-2,-2,-1,1,1,3,0,0,1,3,3,0,2,2,3,1,1,1,-1,0,2]
Phi of -K* [-3,-1,-1,1,2,2,0,2,3,2,2,1,1,0,1,1,2,1,1,1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 6z^2+23z+23
Enhanced Jones-Krushkal polynomial 6w^3z^2-4w^3z+27w^2z+23w
Inner characteristic polynomial t^6+32t^4+31t^2
Outer characteristic polynomial t^7+52t^5+67t^3+8t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -384*K1**4*K2**2 + 608*K1**4*K2 - 720*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 288*K1**3*K2*K3 - 288*K1**3*K3 - 1600*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 3872*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 - 9536*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 768*K1**2*K2*K4 + 7088*K1**2*K2 - 80*K1**2*K3**2 - 4728*K1**2 - 256*K1*K2**4*K3 - 128*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 3712*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1824*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 384*K1*K2*K3*K4 + 7808*K1*K2*K3 + 728*K1*K3*K4 + 112*K1*K4*K5 - 192*K2**6 - 128*K2**4*K3**2 - 192*K2**4*K4**2 + 1120*K2**4*K4 - 4112*K2**4 + 320*K2**3*K3*K5 + 128*K2**3*K4*K6 - 64*K2**3*K6 - 2144*K2**2*K3**2 - 728*K2**2*K4**2 + 2560*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 1738*K2**2 + 888*K2*K3*K5 + 152*K2*K4*K6 + 8*K2*K5*K7 - 1908*K3**2 - 644*K4**2 - 140*K5**2 - 14*K6**2 + 3738
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
Contact