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

Flat knot 6.1876

Min(phi) over symmetries of the knot is: [-1,-1,0,0,1,1,-1,0,0,0,1,0,1,1,1,1,1,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1876']
Arrow polynomial of the knot is: 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.927', '6.1364', '6.1367', '6.1540', '6.1675', '6.1779', '6.1811', '6.1876', '6.2075']
Outer characteristic polynomial of the knot is: t^7+14t^5+30t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1876', '6.1998', '6.2007']
2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 3520*K1**4*K2 - 6368*K1**4 + 672*K1**3*K2*K3 - 1632*K1**3*K3 - 128*K1**2*K2**4 + 1536*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8000*K1**2*K2**2 - 832*K1**2*K2*K4 + 10512*K1**2*K2 - 960*K1**2*K3**2 - 32*K1**2*K4**2 - 3984*K1**2 + 448*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 320*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7896*K1*K2*K3 + 1560*K1*K3*K4 + 184*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1232*K2**4 - 64*K2**3*K6 - 288*K2**2*K3**2 - 128*K2**2*K4**2 + 1752*K2**2*K4 - 4172*K2**2 + 440*K2*K3*K5 + 80*K2*K4*K6 - 2040*K3**2 - 748*K4**2 - 136*K5**2 - 12*K6**2 + 4394
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1876']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13922', 'vk6.14017', 'vk6.14176', 'vk6.14417', 'vk6.14989', 'vk6.15110', 'vk6.15648', 'vk6.16104', 'vk6.16715', 'vk6.16744', 'vk6.16846', 'vk6.18799', 'vk6.19277', 'vk6.19571', 'vk6.23157', 'vk6.23231', 'vk6.25397', 'vk6.26468', 'vk6.33741', 'vk6.33816', 'vk6.34289', 'vk6.35143', 'vk6.37518', 'vk6.42734', 'vk6.44686', 'vk6.54115', 'vk6.54932', 'vk6.54962', 'vk6.56393', 'vk6.56602', 'vk6.59360', 'vk6.64585']
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 O1O2O3U4U3O5U1O6O4U6U5U2
R3 orbit {'O1O2O3U4U3O5U1O6O4U6U5U2'}
R3 orbit length 1
Gauss code of -K O1O2O3U2U4U5O6O5U3O4U1U6
Gauss code of K* O1O2O3U4U3U5O6O5U2O4U1U6
Gauss code of -K* O1O2O3U4U3O5U2O6O4U6U1U5
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 -1 1 1 0 0 -1],[ 1 0 1 0 0 0 -1],[-1 -1 0 1 0 -1 -1],[-1 0 -1 0 -1 -1 -1],[ 0 0 0 1 0 -1 -1],[ 0 0 1 1 1 0 0],[ 1 1 1 1 1 0 0]]
Primitive based matrix [[ 0 1 1 0 0 -1 -1],[-1 0 1 0 -1 -1 -1],[-1 -1 0 -1 -1 0 -1],[ 0 0 1 0 -1 0 -1],[ 0 1 1 1 0 0 0],[ 1 1 0 0 0 0 -1],[ 1 1 1 1 0 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,1,1,-1,0,1,1,1,1,1,0,1,1,0,1,0,0,1]
Phi over symmetry [-1,-1,0,0,1,1,-1,0,0,0,1,0,1,1,1,1,1,1,1,0,-1]
Phi of -K [-1,-1,0,0,1,1,-1,0,1,1,1,1,1,1,2,1,1,0,0,0,-1]
Phi of K* [-1,-1,0,0,1,1,-1,0,0,1,2,0,1,1,1,1,1,1,0,1,1]
Phi of -K* [-1,-1,0,0,1,1,-1,0,0,0,1,0,1,1,1,1,1,1,1,0,-1]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 7z^2+28z+29
Enhanced Jones-Krushkal polynomial 7w^3z^2+28w^2z+29w
Inner characteristic polynomial t^6+10t^4+12t^2
Outer characteristic polynomial t^7+14t^5+30t^3+4t
Flat arrow polynomial 8*K1**3 - 4*K1**2 - 8*K1*K2 - 2*K1 + 2*K2 + 2*K3 + 3
2-strand cable arrow polynomial -256*K1**4*K2**2 + 3520*K1**4*K2 - 6368*K1**4 + 672*K1**3*K2*K3 - 1632*K1**3*K3 - 128*K1**2*K2**4 + 1536*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 8000*K1**2*K2**2 - 832*K1**2*K2*K4 + 10512*K1**2*K2 - 960*K1**2*K3**2 - 32*K1**2*K4**2 - 3984*K1**2 + 448*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 320*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 7896*K1*K2*K3 + 1560*K1*K3*K4 + 184*K1*K4*K5 - 64*K2**6 + 192*K2**4*K4 - 1232*K2**4 - 64*K2**3*K6 - 288*K2**2*K3**2 - 128*K2**2*K4**2 + 1752*K2**2*K4 - 4172*K2**2 + 440*K2*K3*K5 + 80*K2*K4*K6 - 2040*K3**2 - 748*K4**2 - 136*K5**2 - 12*K6**2 + 4394
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
Contact