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

Flat knot 6.826

Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,0,1,3,2,3,2,2,2,3,1,1,0,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.826']
Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.209', '6.231', '6.391', '6.419', '6.600', '6.661', '6.744', '6.812', '6.826', '6.1114', '6.1125', '6.1202', '6.1275', '6.1292', '6.1305', '6.1322', '6.1365', '6.1481', '6.1483', '6.1497', '6.1543', '6.1549', '6.1572', '6.1577', '6.1580', '6.1594', '6.1641', '6.1658', '6.1683', '6.1753', '6.1830', '6.1907', '6.1928']
Outer characteristic polynomial of the knot is: t^7+76t^5+159t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.826']
2-strand cable arrow polynomial of the knot is: -800*K1**4 - 256*K1**2*K2**4 + 1024*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 4160*K1**2*K2**2 - 576*K1**2*K2*K4 + 6512*K1**2*K2 - 32*K1**2*K3**2 - 128*K1**2*K4**2 - 5024*K1**2 + 256*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 192*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 4912*K1*K2*K3 - 64*K1*K2*K4*K5 + 1280*K1*K3*K4 + 336*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 1568*K2**4 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 2520*K2**2*K4 - 3878*K2**2 + 320*K2*K3*K5 + 72*K2*K4*K6 - 1680*K3**2 - 1172*K4**2 - 224*K5**2 - 26*K6**2 + 4098
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.826']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71588', 'vk6.71712', 'vk6.72131', 'vk6.72325', 'vk6.74060', 'vk6.74622', 'vk6.76812', 'vk6.77210', 'vk6.77520', 'vk6.77666', 'vk6.79060', 'vk6.79627', 'vk6.80583', 'vk6.81034', 'vk6.81358', 'vk6.81401', 'vk6.85413', 'vk6.85492', 'vk6.87994', 'vk6.89326']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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 O1O2O3O4U2O5U3U6U1U4O6U5
R3 orbit {'O1O2O3O4U2O5U3U6U1U4O6U5'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5O6U1U4U6U2O5U3
Gauss code of K* Same
Gauss code of -K* O1O2O3O4U5O6U1U4U6U2O5U3
Diagrammatic symmetry type +
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 -2 -1 3 3 -2],[ 1 0 -1 1 3 2 0],[ 2 1 0 1 2 2 1],[ 1 -1 -1 0 1 2 0],[-3 -3 -2 -1 0 0 -3],[-3 -2 -2 -2 0 0 -3],[ 2 0 -1 0 3 3 0]]
Primitive based matrix [[ 0 3 3 -1 -1 -2 -2],[-3 0 0 -1 -3 -2 -3],[-3 0 0 -2 -2 -2 -3],[ 1 1 2 0 -1 -1 0],[ 1 3 2 1 0 -1 0],[ 2 2 2 1 1 0 1],[ 2 3 3 0 0 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,1,1,2,2,0,1,3,2,3,2,2,2,3,1,1,0,1,0,-1]
Phi over symmetry [-3,-3,1,1,2,2,0,1,3,2,3,2,2,2,3,1,1,0,1,0,-1]
Phi of -K [-2,-2,-1,-1,3,3,-1,0,0,3,3,1,1,2,2,-1,1,2,3,2,0]
Phi of K* [-3,-3,1,1,2,2,0,1,3,2,3,2,2,2,3,1,1,0,1,0,-1]
Phi of -K* [-2,-2,-1,-1,3,3,-1,0,0,3,3,1,1,2,2,-1,1,2,3,2,0]
Symmetry type of based matrix +
u-polynomial -2t^3+2t^2+2t
Normalized Jones-Krushkal polynomial 4z^2+21z+27
Enhanced Jones-Krushkal polynomial 4w^3z^2-4w^3z+25w^2z+27w
Inner characteristic polynomial t^6+48t^4+71t^2+4
Outer characteristic polynomial t^7+76t^5+159t^3+16t
Flat arrow polynomial 4*K1**3 - 4*K1**2 - 4*K1*K2 - K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial -800*K1**4 - 256*K1**2*K2**4 + 1024*K1**2*K2**3 + 320*K1**2*K2**2*K4 - 4160*K1**2*K2**2 - 576*K1**2*K2*K4 + 6512*K1**2*K2 - 32*K1**2*K3**2 - 128*K1**2*K4**2 - 5024*K1**2 + 256*K1*K2**3*K3 - 1536*K1*K2**2*K3 - 192*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 4912*K1*K2*K3 - 64*K1*K2*K4*K5 + 1280*K1*K3*K4 + 336*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 32*K2**4*K4 - 1568*K2**4 - 64*K2**2*K3**2 - 48*K2**2*K4**2 + 2520*K2**2*K4 - 3878*K2**2 + 320*K2*K3*K5 + 72*K2*K4*K6 - 1680*K3**2 - 1172*K4**2 - 224*K5**2 - 26*K6**2 + 4098
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}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
Contact