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

Flat knot 6.3

Min(phi) over symmetries of the knot is: [-5,-3,-1,3,3,3,1,2,3,4,5,1,2,3,4,1,2,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.3']
Arrow polynomial of the knot is: -8*K1**3*K2 + 8*K1**3 + 4*K1*K2**2 - 2*K1*K2 - 3*K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.3', '6.10']
Outer characteristic polynomial of the knot is: t^7+161t^5+246t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.3']
2-strand cable arrow polynomial of the knot is: -256*K1**2*K2**2 + 288*K1**2*K2 - 64*K1**2*K4**2 - 520*K1**2 + 480*K1*K2*K3 + 288*K1*K3*K4 + 144*K1*K4*K5 - 128*K2**6*K4**2 + 768*K2**6*K4 - 1408*K2**6 + 128*K2**4*K4**3 - 832*K2**4*K4**2 + 1152*K2**4*K4 - 256*K2**4 + 192*K2**3*K4*K6 - 32*K2**2*K4**4 + 64*K2**2*K4**3 - 552*K2**2*K4**2 + 744*K2**2*K4 - 278*K2**2 + 96*K2*K3*K5 + 136*K2*K4*K6 - 352*K3**2 - 16*K4**4 - 396*K4**2 - 104*K5**2 - 2*K6**2 + 714
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.3']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81821', 'vk6.81823', 'vk6.82041', 'vk6.82043', 'vk6.82772', 'vk6.82773', 'vk6.82972', 'vk6.82974', 'vk6.83448', 'vk6.83450', 'vk6.83902', 'vk6.83906', 'vk6.84519', 'vk6.84521', 'vk6.84762', 'vk6.84764', 'vk6.86240', 'vk6.86246', 'vk6.86843', 'vk6.86844', 'vk6.88507', 'vk6.88514', 'vk6.89872', 'vk6.89874']
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 O1O2O3O4O5O6U1U2U3U6U5U4
R3 orbit {'O1O2O3O4O5O6U1U2U3U6U5U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U3U2U1U4U5U6
Gauss code of K* Same
Gauss code of -K* O1O2O3O4O5O6U3U2U1U4U5U6
Diagrammatic symmetry type +
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -5 -3 -1 3 3 3],[ 5 0 1 2 5 4 3],[ 3 -1 0 1 4 3 2],[ 1 -2 -1 0 3 2 1],[-3 -5 -4 -3 0 0 0],[-3 -4 -3 -2 0 0 0],[-3 -3 -2 -1 0 0 0]]
Primitive based matrix [[ 0 3 3 3 -1 -3 -5],[-3 0 0 0 -1 -2 -3],[-3 0 0 0 -2 -3 -4],[-3 0 0 0 -3 -4 -5],[ 1 1 2 3 0 -1 -2],[ 3 2 3 4 1 0 -1],[ 5 3 4 5 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-3,-3,1,3,5,0,0,1,2,3,0,2,3,4,3,4,5,1,2,1]
Phi over symmetry [-5,-3,-1,3,3,3,1,2,3,4,5,1,2,3,4,1,2,3,0,0,0]
Phi of -K [-5,-3,-1,3,3,3,1,2,3,4,5,1,2,3,4,1,2,3,0,0,0]
Phi of K* [-3,-3,-3,1,3,5,0,0,1,2,3,0,2,3,4,3,4,5,1,2,1]
Phi of -K* [-5,-3,-1,3,3,3,1,2,3,4,5,1,2,3,4,1,2,3,0,0,0]
Symmetry type of based matrix +
u-polynomial t^5-2t^3+t
Normalized Jones-Krushkal polynomial z+3
Enhanced Jones-Krushkal polynomial -8w^5z+12w^4z-8w^3z+4w^3+5w^2z-w
Inner characteristic polynomial t^6+99t^4+36t^2
Outer characteristic polynomial t^7+161t^5+246t^3
Flat arrow polynomial -8*K1**3*K2 + 8*K1**3 + 4*K1*K2**2 - 2*K1*K2 - 3*K1 + K3 + 1
2-strand cable arrow polynomial -256*K1**2*K2**2 + 288*K1**2*K2 - 64*K1**2*K4**2 - 520*K1**2 + 480*K1*K2*K3 + 288*K1*K3*K4 + 144*K1*K4*K5 - 128*K2**6*K4**2 + 768*K2**6*K4 - 1408*K2**6 + 128*K2**4*K4**3 - 832*K2**4*K4**2 + 1152*K2**4*K4 - 256*K2**4 + 192*K2**3*K4*K6 - 32*K2**2*K4**4 + 64*K2**2*K4**3 - 552*K2**2*K4**2 + 744*K2**2*K4 - 278*K2**2 + 96*K2*K3*K5 + 136*K2*K4*K6 - 352*K3**2 - 16*K4**4 - 396*K4**2 - 104*K5**2 - 2*K6**2 + 714
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{4, 6}, {5}, {3}, {2}, {1}]]
If K is slice False
Contact