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

Flat knot 6.309

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,1,0,2,3,5,0,1,2,4,0,0,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.309']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 4*K1**2*K2 + 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.250', '6.309']
Outer characteristic polynomial of the knot is: t^7+104t^5+119t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.309']
2-strand cable arrow polynomial of the knot is: -912*K1**4 - 256*K1**3*K3 - 768*K1**2*K2**6 + 1408*K1**2*K2**5 - 3264*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 4224*K1**2*K2**3 - 7168*K1**2*K2**2 - 352*K1**2*K2*K4 + 5872*K1**2*K2 - 112*K1**2*K3**2 - 3332*K1**2 + 640*K1*K2**5*K3 - 512*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2912*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1408*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 224*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4960*K1*K2*K3 + 272*K1*K3*K4 + 24*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 960*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 768*K2**4*K4 - 2824*K2**4 + 32*K2**3*K3*K5 - 976*K2**2*K3**2 - 240*K2**2*K4**2 + 1736*K2**2*K4 - 32*K2**2*K5**2 - 720*K2**2 - 32*K2*K3**2*K4 + 360*K2*K3*K5 + 24*K2*K4*K6 + 8*K2*K5*K7 - 1032*K3**2 - 222*K4**2 - 44*K5**2 + 2468
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.309']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11416', 'vk6.11703', 'vk6.12720', 'vk6.13073', 'vk6.20265', 'vk6.21580', 'vk6.27523', 'vk6.29101', 'vk6.31153', 'vk6.31486', 'vk6.32309', 'vk6.32746', 'vk6.38928', 'vk6.41146', 'vk6.45681', 'vk6.47402', 'vk6.52167', 'vk6.52398', 'vk6.52986', 'vk6.53313', 'vk6.57094', 'vk6.58259', 'vk6.61664', 'vk6.62830', 'vk6.63741', 'vk6.63841', 'vk6.64161', 'vk6.64357', 'vk6.66733', 'vk6.67604', 'vk6.69380', 'vk6.70115']
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 O1O2O3O4O5U2U3O6U1U6U4U5
R3 orbit {'O1O2O3O4O5U2U3O6U1U6U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U1U2U6U5O6U3U4
Gauss code of K* O1O2O3O4U1U5U6U3U4O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U1U2U5U6U4
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 -3 -1 2 4 1],[ 3 0 -1 1 4 5 1],[ 3 1 0 1 2 3 0],[ 1 -1 -1 0 1 2 0],[-2 -4 -2 -1 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 4 2 1 -1 -3 -3],[-4 0 -1 0 -2 -3 -5],[-2 1 0 0 -1 -2 -4],[-1 0 0 0 0 0 -1],[ 1 2 1 0 0 -1 -1],[ 3 3 2 0 1 0 1],[ 3 5 4 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-4,-2,-1,1,3,3,1,0,2,3,5,0,1,2,4,0,0,1,1,1,-1]
Phi over symmetry [-4,-2,-1,1,3,3,1,0,2,3,5,0,1,2,4,0,0,1,1,1,-1]
Phi of -K [-3,-3,-1,1,2,4,-1,1,4,3,4,1,3,1,2,2,2,3,1,3,1]
Phi of K* [-4,-2,-1,1,3,3,1,3,3,2,4,1,2,1,3,2,3,4,1,1,-1]
Phi of -K* [-3,-3,-1,1,2,4,-1,1,1,4,5,1,0,2,3,0,1,2,0,0,1]
Symmetry type of based matrix c
u-polynomial -t^4+2t^3-t^2
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial -4w^4z^2+8w^3z^2-4w^3z+21w^2z+19w
Inner characteristic polynomial t^6+64t^4+34t^2
Outer characteristic polynomial t^7+104t^5+119t^3+7t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 4*K1**2*K2 + 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2
2-strand cable arrow polynomial -912*K1**4 - 256*K1**3*K3 - 768*K1**2*K2**6 + 1408*K1**2*K2**5 - 3264*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 4224*K1**2*K2**3 - 7168*K1**2*K2**2 - 352*K1**2*K2*K4 + 5872*K1**2*K2 - 112*K1**2*K3**2 - 3332*K1**2 + 640*K1*K2**5*K3 - 512*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2912*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1408*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 224*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 4960*K1*K2*K3 + 272*K1*K3*K4 + 24*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 960*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 768*K2**4*K4 - 2824*K2**4 + 32*K2**3*K3*K5 - 976*K2**2*K3**2 - 240*K2**2*K4**2 + 1736*K2**2*K4 - 32*K2**2*K5**2 - 720*K2**2 - 32*K2*K3**2*K4 + 360*K2*K3*K5 + 24*K2*K4*K6 + 8*K2*K5*K7 - 1032*K3**2 - 222*K4**2 - 44*K5**2 + 2468
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice False
Contact