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

Flat knot 6.888

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,1,1,2,3,1,1,0,0,1,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.888', '7.28195']
Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.115', '6.407', '6.413', '6.448', '6.844', '6.879', '6.888', '6.926', '6.934', '6.1140', '6.1143', '6.1161', '6.1177']
Outer characteristic polynomial of the knot is: t^7+44t^5+58t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.888', '7.28195']
2-strand cable arrow polynomial of the knot is: -1024*K1**4*K2**2 + 1024*K1**4*K2 - 1792*K1**4 - 256*K1**3*K2**2*K3 + 2432*K1**3*K2*K3 - 896*K1**3*K3 - 1280*K1**2*K2**4 + 2048*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 384*K1**2*K2**2*K4 - 7856*K1**2*K2**2 - 1056*K1**2*K2*K4 + 5632*K1**2*K2 - 1216*K1**2*K3**2 - 64*K1**2*K3*K5 - 2028*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 3936*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 1056*K1*K2**2*K5 - 512*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6128*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1056*K1*K3*K4 + 128*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2368*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1920*K2**2*K3**2 - 264*K2**2*K4**2 + 1616*K2**2*K4 - 144*K2**2*K5**2 - 48*K2**2*K6**2 - 742*K2**2 + 976*K2*K3*K5 + 152*K2*K4*K6 + 32*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1216*K3**2 - 288*K4**2 - 112*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1920
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.888']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.471', 'vk6.533', 'vk6.568', 'vk6.932', 'vk6.978', 'vk6.1029', 'vk6.1068', 'vk6.1714', 'vk6.1788', 'vk6.2112', 'vk6.2216', 'vk6.2248', 'vk6.2545', 'vk6.2823', 'vk6.2857', 'vk6.3161', 'vk6.20314', 'vk6.20642', 'vk6.21655', 'vk6.22073', 'vk6.27616', 'vk6.28127', 'vk6.29162', 'vk6.39037', 'vk6.39562', 'vk6.41296', 'vk6.41792', 'vk6.46175', 'vk6.57176', 'vk6.57550', 'vk6.58381', 'vk6.66788']
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 O1O2O3O4U2U1O5O6U5U6U3U4
R3 orbit {'O1O2O3O4U2U1O5O6U5U6U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U1U2U5U6O5O6U4U3
Gauss code of K* O1O2O3O4U5U6U3U4O6O5U1U2
Gauss code of -K* O1O2O3O4U3U4O5O6U1U2U6U5
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 -2 1 3 -1 1],[ 2 0 0 2 3 0 0],[ 2 0 0 1 2 0 0],[-1 -2 -1 0 1 -1 1],[-3 -3 -2 -1 0 -1 1],[ 1 0 0 1 1 0 1],[-1 0 0 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 1 1 -1 -2 -2],[-3 0 1 -1 -1 -2 -3],[-1 -1 0 -1 -1 0 0],[-1 1 1 0 -1 -1 -2],[ 1 1 1 1 0 0 0],[ 2 2 0 1 0 0 0],[ 2 3 0 2 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,1,2,2,-1,1,1,2,3,1,1,0,0,1,1,2,0,0,0]
Phi over symmetry [-3,-1,-1,1,2,2,-1,1,1,2,3,1,1,0,0,1,1,2,0,0,0]
Phi of -K [-2,-2,-1,1,1,3,0,1,1,3,2,1,2,3,3,1,1,3,-1,1,3]
Phi of K* [-3,-1,-1,1,2,2,1,3,3,2,3,1,1,1,2,1,3,3,1,1,0]
Phi of -K* [-2,-2,-1,1,1,3,0,0,0,1,2,0,0,2,3,1,1,1,-1,-1,1]
Symmetry type of based matrix c
u-polynomial -t^3+2t^2-t
Normalized Jones-Krushkal polynomial 7z^2+24z+21
Enhanced Jones-Krushkal polynomial -4w^4z^2+11w^3z^2+24w^2z+21w
Inner characteristic polynomial t^6+24t^4+24t^2+1
Outer characteristic polynomial t^7+44t^5+58t^3+13t
Flat arrow polynomial 4*K1**2*K2 - 4*K1**2 - 2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial -1024*K1**4*K2**2 + 1024*K1**4*K2 - 1792*K1**4 - 256*K1**3*K2**2*K3 + 2432*K1**3*K2*K3 - 896*K1**3*K3 - 1280*K1**2*K2**4 + 2048*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 384*K1**2*K2**2*K4 - 7856*K1**2*K2**2 - 1056*K1**2*K2*K4 + 5632*K1**2*K2 - 1216*K1**2*K3**2 - 64*K1**2*K3*K5 - 2028*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 3936*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 1056*K1*K2**2*K5 - 512*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6128*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K5*K6 + 1056*K1*K3*K4 + 128*K1*K4*K5 + 16*K1*K5*K6 + 8*K1*K6*K7 - 128*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2368*K2**4 + 320*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1920*K2**2*K3**2 - 264*K2**2*K4**2 + 1616*K2**2*K4 - 144*K2**2*K5**2 - 48*K2**2*K6**2 - 742*K2**2 + 976*K2*K3*K5 + 152*K2*K4*K6 + 32*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1216*K3**2 - 288*K4**2 - 112*K5**2 - 26*K6**2 - 4*K7**2 - 2*K8**2 + 1920
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {3}, {1, 2}]]
If K is slice False
Contact