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

Flat knot 6.967

Min(phi) over symmetries of the knot is: [-3,0,0,1,1,1,0,2,1,2,3,0,1,1,1,1,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.967']
Arrow polynomial of the knot is: -6*K1*K2 + 3*K1 - 4*K2**2 + 3*K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.881', '6.967', '6.1179']
Outer characteristic polynomial of the knot is: t^7+38t^5+57t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.967', '6.1032', '6.1134']
2-strand cable arrow polynomial of the knot is: -528*K1**4 + 224*K1**3*K3*K4 + 216*K1**2*K2 - 768*K1**2*K3**2 - 624*K1**2*K4**2 - 1220*K1**2 + 1160*K1*K2*K3 + 2512*K1*K3*K4 + 616*K1*K4*K5 - 24*K2**2*K4**2 + 192*K2**2*K4 - 662*K2**2 + 88*K2*K3*K5 + 40*K2*K4*K6 + 8*K2*K5*K7 - 48*K3**4 - 112*K3**2*K4**2 + 32*K3**2*K6 - 1396*K3**2 + 88*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 1220*K4**2 - 228*K5**2 - 18*K6**2 - 20*K7**2 - 4*K8**2 + 1710
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.967']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3634', 'vk6.3725', 'vk6.3916', 'vk6.4017', 'vk6.7056', 'vk6.7113', 'vk6.7288', 'vk6.7387', 'vk6.11390', 'vk6.12577', 'vk6.12688', 'vk6.19106', 'vk6.19153', 'vk6.19818', 'vk6.25715', 'vk6.25776', 'vk6.26257', 'vk6.26700', 'vk6.30990', 'vk6.31117', 'vk6.32174', 'vk6.37826', 'vk6.37883', 'vk6.44978', 'vk6.48270', 'vk6.48449', 'vk6.50023', 'vk6.50167', 'vk6.52147', 'vk6.63721', 'vk6.66199', 'vk6.66228']
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 O1O2O3O4U5U3O6O5U6U2U1U4
R3 orbit {'O1O2O3O4U5U3O6O5U6U2U1U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U4U3U5O6O5U2U6
Gauss code of K* O1O2O3O4U3U2U5U4O6O5U1U6
Gauss code of -K* O1O2O3O4U5U4O6O5U1U6U3U2
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -1 -1 0 3 0 -1],[ 1 0 0 1 3 1 -1],[ 1 0 0 1 2 1 -1],[ 0 -1 -1 0 0 0 -1],[-3 -3 -2 0 0 -2 -1],[ 0 -1 -1 0 2 0 -1],[ 1 1 1 1 1 1 0]]
Primitive based matrix [[ 0 3 0 0 -1 -1 -1],[-3 0 0 -2 -1 -2 -3],[ 0 0 0 0 -1 -1 -1],[ 0 2 0 0 -1 -1 -1],[ 1 1 1 1 0 1 1],[ 1 2 1 1 -1 0 0],[ 1 3 1 1 -1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,0,0,1,1,1,0,2,1,2,3,0,1,1,1,1,1,1,-1,-1,0]
Phi over symmetry [-3,0,0,1,1,1,0,2,1,2,3,0,1,1,1,1,1,1,-1,-1,0]
Phi of -K [-1,-1,-1,0,0,3,-1,-1,0,0,3,0,0,0,1,0,0,2,0,1,3]
Phi of K* [-3,0,0,1,1,1,1,3,1,2,3,0,0,0,0,0,0,0,0,-1,-1]
Phi of -K* [-1,-1,-1,0,0,3,-1,0,1,1,2,1,1,1,1,1,1,3,0,0,2]
Symmetry type of based matrix c
u-polynomial -t^3+3t
Normalized Jones-Krushkal polynomial 9z+19
Enhanced Jones-Krushkal polynomial -8w^3z+17w^2z+19w
Inner characteristic polynomial t^6+26t^4+21t^2
Outer characteristic polynomial t^7+38t^5+57t^3
Flat arrow polynomial -6*K1*K2 + 3*K1 - 4*K2**2 + 3*K3 + 2*K4 + 3
2-strand cable arrow polynomial -528*K1**4 + 224*K1**3*K3*K4 + 216*K1**2*K2 - 768*K1**2*K3**2 - 624*K1**2*K4**2 - 1220*K1**2 + 1160*K1*K2*K3 + 2512*K1*K3*K4 + 616*K1*K4*K5 - 24*K2**2*K4**2 + 192*K2**2*K4 - 662*K2**2 + 88*K2*K3*K5 + 40*K2*K4*K6 + 8*K2*K5*K7 - 48*K3**4 - 112*K3**2*K4**2 + 32*K3**2*K6 - 1396*K3**2 + 88*K3*K4*K7 - 48*K4**4 + 32*K4**2*K8 - 1220*K4**2 - 228*K5**2 - 18*K6**2 - 20*K7**2 - 4*K8**2 + 1710
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
Contact