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

Flat knot 6.521

Min(phi) over symmetries of the knot is: [-3,-2,-1,1,2,3,-1,1,4,3,3,1,2,2,1,2,2,2,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.521']
Arrow polynomial of the knot is: 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.521', '6.920', '6.1255', '6.1917']
Outer characteristic polynomial of the knot is: t^7+72t^5+56t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.521']
2-strand cable arrow polynomial of the knot is: 512*K1**4*K2 - 1168*K1**4 - 512*K1**3*K2**2*K3 + 1632*K1**3*K2*K3 - 992*K1**3*K3 - 128*K1**2*K2**4 + 2080*K1**2*K2**3 - 704*K1**2*K2**2*K3**2 + 288*K1**2*K2**2*K4 - 8304*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1312*K1**2*K2*K4 + 7928*K1**2*K2 - 848*K1**2*K3**2 - 32*K1**2*K4**2 - 5408*K1**2 - 256*K1*K2**3*K3*K4 + 2592*K1*K2**3*K3 + 832*K1*K2**2*K3*K4 - 2208*K1*K2**2*K3 + 288*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 + 32*K1*K2*K3**3 - 832*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8592*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1360*K1*K3*K4 + 344*K1*K4*K5 - 32*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 352*K2**4*K4 - 2384*K2**4 + 352*K2**3*K3*K5 + 128*K2**3*K4*K6 - 96*K2**3*K6 - 1840*K2**2*K3**2 - 688*K2**2*K4**2 + 2448*K2**2*K4 - 224*K2**2*K5**2 - 48*K2**2*K6**2 - 3258*K2**2 + 1096*K2*K3*K5 + 272*K2*K4*K6 + 16*K2*K5*K7 - 2316*K3**2 - 812*K4**2 - 252*K5**2 - 38*K6**2 + 4218
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.521']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4195', 'vk6.4276', 'vk6.5449', 'vk6.5563', 'vk6.7560', 'vk6.7647', 'vk6.9064', 'vk6.9145', 'vk6.18236', 'vk6.18571', 'vk6.24708', 'vk6.25121', 'vk6.36831', 'vk6.37294', 'vk6.44071', 'vk6.44410', 'vk6.48507', 'vk6.48588', 'vk6.49199', 'vk6.49307', 'vk6.50296', 'vk6.50372', 'vk6.51061', 'vk6.51094', 'vk6.56039', 'vk6.56313', 'vk6.60592', 'vk6.60931', 'vk6.65705', 'vk6.65999', 'vk6.68750', 'vk6.68958']
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 O1O2O3O4O5U2U3U5O6U1U6U4
R3 orbit {'O1O2O3O4O5U2U3U5O6U1U6U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U6U5O6U1U3U4
Gauss code of K* O1O2O3U4O5O4O6U5U1U2U6U3
Gauss code of -K* O1O2O3U2O4O5O6U4U1U5U6U3
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 -3 -1 3 2 1],[ 2 0 -2 0 4 2 1],[ 3 2 0 1 3 2 0],[ 1 0 -1 0 2 1 0],[-3 -4 -3 -2 0 0 0],[-2 -2 -2 -1 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix [[ 0 3 2 1 -1 -2 -3],[-3 0 0 0 -2 -4 -3],[-2 0 0 0 -1 -2 -2],[-1 0 0 0 0 -1 0],[ 1 2 1 0 0 0 -1],[ 2 4 2 1 0 0 -2],[ 3 3 2 0 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,1,2,3,0,0,2,4,3,0,1,2,2,0,1,0,0,1,2]
Phi over symmetry [-3,-2,-1,1,2,3,-1,1,4,3,3,1,2,2,1,2,2,2,1,2,1]
Phi of -K [-3,-2,-1,1,2,3,-1,1,4,3,3,1,2,2,1,2,2,2,1,2,1]
Phi of K* [-3,-2,-1,1,2,3,1,2,2,1,3,1,2,2,3,2,2,4,1,1,-1]
Phi of -K* [-3,-2,-1,1,2,3,2,1,0,2,3,0,1,2,4,0,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial 0
Normalized Jones-Krushkal polynomial 8z^2+29z+27
Enhanced Jones-Krushkal polynomial 8w^3z^2+29w^2z+27w
Inner characteristic polynomial t^6+44t^4+24t^2+1
Outer characteristic polynomial t^7+72t^5+56t^3+8t
Flat arrow polynomial 4*K1**3 + 8*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial 512*K1**4*K2 - 1168*K1**4 - 512*K1**3*K2**2*K3 + 1632*K1**3*K2*K3 - 992*K1**3*K3 - 128*K1**2*K2**4 + 2080*K1**2*K2**3 - 704*K1**2*K2**2*K3**2 + 288*K1**2*K2**2*K4 - 8304*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 1312*K1**2*K2*K4 + 7928*K1**2*K2 - 848*K1**2*K3**2 - 32*K1**2*K4**2 - 5408*K1**2 - 256*K1*K2**3*K3*K4 + 2592*K1*K2**3*K3 + 832*K1*K2**2*K3*K4 - 2208*K1*K2**2*K3 + 288*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 + 32*K1*K2*K3**3 - 832*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8592*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1360*K1*K3*K4 + 344*K1*K4*K5 - 32*K2**6 - 128*K2**4*K3**2 - 64*K2**4*K4**2 + 352*K2**4*K4 - 2384*K2**4 + 352*K2**3*K3*K5 + 128*K2**3*K4*K6 - 96*K2**3*K6 - 1840*K2**2*K3**2 - 688*K2**2*K4**2 + 2448*K2**2*K4 - 224*K2**2*K5**2 - 48*K2**2*K6**2 - 3258*K2**2 + 1096*K2*K3*K5 + 272*K2*K4*K6 + 16*K2*K5*K7 - 2316*K3**2 - 812*K4**2 - 252*K5**2 - 38*K6**2 + 4218
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice False
Contact