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

Flat knot 6.1747

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,1,1,1,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1747', '7.39601']
Arrow polynomial of the knot is: -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.425', '6.655', '6.755', '6.769', '6.792', '6.1240', '6.1494', '6.1522', '6.1534', '6.1587', '6.1707', '6.1746', '6.1747', '6.1786', '6.1814', '6.1828', '6.1835', '6.1854', '6.1870']
Outer characteristic polynomial of the knot is: t^7+24t^5+28t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1116', '6.1747']
2-strand cable arrow polynomial of the knot is: -448*K1**6 - 512*K1**4*K2**2 + 2816*K1**4*K2 - 6304*K1**4 + 1248*K1**3*K2*K3 - 1632*K1**3*K3 + 1024*K1**2*K2**3 - 7824*K1**2*K2**2 - 928*K1**2*K2*K4 + 11432*K1**2*K2 - 1152*K1**2*K3**2 - 128*K1**2*K4**2 - 4316*K1**2 + 352*K1*K2**3*K3 - 640*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7816*K1*K2*K3 + 1288*K1*K3*K4 + 136*K1*K4*K5 - 712*K2**4 - 272*K2**2*K3**2 - 48*K2**2*K4**2 + 840*K2**2*K4 - 4092*K2**2 + 192*K2*K3*K5 + 32*K2*K4*K6 - 1872*K3**2 - 442*K4**2 - 52*K5**2 - 4*K6**2 + 4424
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1747']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4212', 'vk6.4293', 'vk6.5471', 'vk6.5584', 'vk6.7579', 'vk6.7673', 'vk6.9081', 'vk6.9162', 'vk6.11164', 'vk6.12246', 'vk6.12355', 'vk6.19370', 'vk6.19665', 'vk6.19777', 'vk6.26152', 'vk6.26210', 'vk6.26570', 'vk6.26655', 'vk6.30762', 'vk6.31963', 'vk6.38156', 'vk6.38198', 'vk6.44817', 'vk6.44939', 'vk6.48534', 'vk6.49229', 'vk6.49342', 'vk6.50321', 'vk6.52748', 'vk6.63594', 'vk6.66312', 'vk6.66354']
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 O1O2O3U4U2O4O5U6U1O6U3U5
R3 orbit {'O1O2O3U4U2O4O5U6U1O6U3U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U1O5U3U5O4O6U2U6
Gauss code of K* O1O2U1O3O4U2U5U3O6O5U6U4
Gauss code of -K* O1O2U3O4O3U1U5O6O5U2U6U4
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 0 1 -1 2 -1],[ 1 0 1 1 0 1 1],[ 0 -1 0 0 0 1 0],[-1 -1 0 0 -1 1 -1],[ 1 0 0 1 0 2 0],[-2 -1 -1 -1 -2 0 -2],[ 1 -1 0 1 0 2 0]]
Primitive based matrix [[ 0 2 1 0 -1 -1 -1],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 -1 -1 -1],[ 0 1 0 0 -1 0 0],[ 1 1 1 1 0 1 0],[ 1 2 1 0 -1 0 0],[ 1 2 1 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,0,1,1,1,1,1,1,2,2,0,1,1,1,1,0,0,-1,0,0]
Phi over symmetry [-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,1,1,1,0,0,-1,0]
Phi of -K [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,1,1,1,1,1,1,0]
Phi of K* [-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,1,1,1,0,0,-1,0]
Phi of -K* [-1,-1,-1,0,1,2,-1,0,0,1,2,0,1,1,1,0,1,2,0,1,1]
Symmetry type of based matrix c
u-polynomial -t^2+2t
Normalized Jones-Krushkal polynomial 3z^2+24z+37
Enhanced Jones-Krushkal polynomial 3w^3z^2+24w^2z+37w
Inner characteristic polynomial t^6+16t^4+15t^2+1
Outer characteristic polynomial t^7+24t^5+28t^3+4t
Flat arrow polynomial -10*K1**2 - 4*K1*K2 + 2*K1 + 5*K2 + 2*K3 + 6
2-strand cable arrow polynomial -448*K1**6 - 512*K1**4*K2**2 + 2816*K1**4*K2 - 6304*K1**4 + 1248*K1**3*K2*K3 - 1632*K1**3*K3 + 1024*K1**2*K2**3 - 7824*K1**2*K2**2 - 928*K1**2*K2*K4 + 11432*K1**2*K2 - 1152*K1**2*K3**2 - 128*K1**2*K4**2 - 4316*K1**2 + 352*K1*K2**3*K3 - 640*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 7816*K1*K2*K3 + 1288*K1*K3*K4 + 136*K1*K4*K5 - 712*K2**4 - 272*K2**2*K3**2 - 48*K2**2*K4**2 + 840*K2**2*K4 - 4092*K2**2 + 192*K2*K3*K5 + 32*K2*K4*K6 - 1872*K3**2 - 442*K4**2 - 52*K5**2 - 4*K6**2 + 4424
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}]]
If K is slice False
Contact