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

Flat knot 6.329

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,0,3,2,4,0,2,1,2,2,2,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.329']
Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 8*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + 2*K2 + 3*K3 + 2*K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.329']
Outer characteristic polynomial of the knot is: t^7+76t^5+92t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.329']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 336*K1**4 + 128*K1**3*K2**3*K3 + 672*K1**3*K2*K3 - 320*K1**2*K2**4 + 192*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 2528*K1**2*K2**2 + 1496*K1**2*K2 - 848*K1**2*K3**2 - 48*K1**2*K4**2 - 1804*K1**2 + 704*K1*K2**3*K3 + 192*K1*K2*K3**3 + 3552*K1*K2*K3 + 32*K1*K3**3*K4 + 1272*K1*K3*K4 + 256*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 128*K2**4*K4 - 840*K2**4 + 32*K2**3*K3*K5 - 720*K2**2*K3**2 - 160*K2**2*K4**2 + 936*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 1590*K2**2 + 592*K2*K3*K5 + 96*K2*K4*K6 + 72*K2*K5*K7 + 16*K2*K6*K8 - 128*K3**4 - 96*K3**2*K4**2 + 88*K3**2*K6 - 1692*K3**2 + 72*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 956*K4**2 - 324*K5**2 - 66*K6**2 - 36*K7**2 - 12*K8**2 + 2478
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.329']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11436', 'vk6.11733', 'vk6.12746', 'vk6.13091', 'vk6.20333', 'vk6.21675', 'vk6.27634', 'vk6.29179', 'vk6.31183', 'vk6.31526', 'vk6.32347', 'vk6.32766', 'vk6.39059', 'vk6.41317', 'vk6.45811', 'vk6.47484', 'vk6.52189', 'vk6.52448', 'vk6.53016', 'vk6.53334', 'vk6.57204', 'vk6.58421', 'vk6.61815', 'vk6.62941', 'vk6.63755', 'vk6.63867', 'vk6.64179', 'vk6.64367', 'vk6.66811', 'vk6.67680', 'vk6.69448', 'vk6.70171']
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 O1O2O3O4O5U2U5O6U3U1U4U6
R3 orbit {'O1O2O3O4O5U2U5O6U3U1U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U2U5U3O6U1U4
Gauss code of K* O1O2O3O4U2U5U1U3U6O5O6U4
Gauss code of -K* O1O2O3O4U1O5O6U5U2U4U6U3
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 -2 -3 -1 2 1 3],[ 2 0 -2 1 3 1 3],[ 3 2 0 2 3 1 2],[ 1 -1 -2 0 1 0 2],[-2 -3 -3 -1 0 0 1],[-1 -1 -1 0 0 0 0],[-3 -3 -2 -2 -1 0 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 -1 0 -2 -3 -2],[-2 1 0 0 -1 -3 -3],[-1 0 0 0 0 -1 -1],[ 1 2 1 0 0 -1 -2],[ 2 3 3 1 1 0 -2],[ 3 2 3 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,1,0,2,3,2,0,1,3,3,0,1,1,1,2,2]
Phi over symmetry [-3,-2,-1,1,2,3,-1,0,3,2,4,0,2,1,2,2,2,2,1,2,0]
Phi of -K [-3,-2,-1,1,2,3,-1,0,3,2,4,0,2,1,2,2,2,2,1,2,0]
Phi of K* [-3,-2,-1,1,2,3,0,2,2,2,4,1,2,1,2,2,2,3,0,0,-1]
Phi of -K* [-3,-2,-1,1,2,3,2,2,1,3,2,1,1,3,3,0,1,2,0,0,1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z+17
Enhanced Jones-Krushkal polynomial -8w^3z+16w^2z+17w
Inner characteristic polynomial t^6+48t^4+24t^2
Outer characteristic polynomial t^7+76t^5+92t^3
Flat arrow polynomial 4*K1**3 - 2*K1**2 - 8*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + 2*K2 + 3*K3 + 2*K4 + 3
2-strand cable arrow polynomial -256*K1**4*K2**2 + 96*K1**4*K2 - 336*K1**4 + 128*K1**3*K2**3*K3 + 672*K1**3*K2*K3 - 320*K1**2*K2**4 + 192*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 2528*K1**2*K2**2 + 1496*K1**2*K2 - 848*K1**2*K3**2 - 48*K1**2*K4**2 - 1804*K1**2 + 704*K1*K2**3*K3 + 192*K1*K2*K3**3 + 3552*K1*K2*K3 + 32*K1*K3**3*K4 + 1272*K1*K3*K4 + 256*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 128*K2**4*K4 - 840*K2**4 + 32*K2**3*K3*K5 - 720*K2**2*K3**2 - 160*K2**2*K4**2 + 936*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 1590*K2**2 + 592*K2*K3*K5 + 96*K2*K4*K6 + 72*K2*K5*K7 + 16*K2*K6*K8 - 128*K3**4 - 96*K3**2*K4**2 + 88*K3**2*K6 - 1692*K3**2 + 72*K3*K4*K7 - 8*K4**4 + 16*K4**2*K8 - 956*K4**2 - 324*K5**2 - 66*K6**2 - 36*K7**2 - 12*K8**2 + 2478
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
Contact