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

Flat knot 6.538

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,3,3,2,2,2,2,1,1,1,1,2,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.538']
Arrow polynomial of the knot is: 4*K1**2*K2 - 12*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + 5*K2 + K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.538']
Outer characteristic polynomial of the knot is: t^7+65t^5+70t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.538']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 1184*K1**4 - 64*K1**3*K3 + 32*K1**2*K2**3 - 2352*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 544*K1**2*K2*K4 + 6080*K1**2*K2 - 320*K1**2*K3**2 - 32*K1**2*K3*K5 - 192*K1**2*K4**2 - 5700*K1**2 + 128*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 256*K1*K2*K3*K4 + 5352*K1*K2*K3 - 192*K1*K2*K4*K5 + 2160*K1*K3*K4 + 392*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**4*K4**2 + 96*K2**4*K4 - 448*K2**4 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 224*K2**2*K3**2 + 32*K2**2*K4**3 - 296*K2**2*K4**2 - 32*K2**2*K4*K8 + 1600*K2**2*K4 - 8*K2**2*K6**2 - 4538*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 464*K2*K3*K5 + 304*K2*K4*K6 + 8*K2*K6*K8 - 32*K3**4 - 64*K3**2*K4**2 + 56*K3**2*K6 - 2452*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1294*K4**2 - 240*K5**2 - 62*K6**2 - 2*K8**2 + 4702
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.538']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72421', 'vk6.72422', 'vk6.72470', 'vk6.72473', 'vk6.72480', 'vk6.72491', 'vk6.72832', 'vk6.72835', 'vk6.72841', 'vk6.72854', 'vk6.72895', 'vk6.72896', 'vk6.74449', 'vk6.74464', 'vk6.74476', 'vk6.74479', 'vk6.75064', 'vk6.75076', 'vk6.76963', 'vk6.77774', 'vk6.77783', 'vk6.77966', 'vk6.79457', 'vk6.79463', 'vk6.79907', 'vk6.79912', 'vk6.79929', 'vk6.79932', 'vk6.80930', 'vk6.80942', 'vk6.87232', 'vk6.89365']
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 O1O2O3O4O5U3U1U5O6U4U2U6
R3 orbit {'O1O2O3O4O5U3U1U5O6U4U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U6U4U2O6U1U5U3
Gauss code of K* O1O2O3U4O5O6O4U2U6U1U5U3
Gauss code of -K* O1O2O3U1O4O5O6U4U3U6U2U5
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 -3 0 -2 1 2 2],[ 3 0 3 0 3 2 2],[ 0 -3 0 -2 1 1 2],[ 2 0 2 0 2 1 1],[-1 -3 -1 -2 0 0 1],[-2 -2 -1 -1 0 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix [[ 0 2 2 1 0 -2 -3],[-2 0 0 0 -1 -1 -2],[-2 0 0 -1 -2 -1 -2],[-1 0 1 0 -1 -2 -3],[ 0 1 2 1 0 -2 -3],[ 2 1 1 2 2 0 0],[ 3 2 2 3 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,0,2,3,0,0,1,1,2,1,2,1,2,1,2,3,2,3,0]
Phi over symmetry [-3,-2,0,1,2,2,0,3,3,2,2,2,2,1,1,1,1,2,0,1,0]
Phi of -K [-3,-2,0,1,2,2,1,0,1,3,3,0,1,3,3,0,0,1,0,1,0]
Phi of K* [-2,-2,-1,0,2,3,0,0,0,3,3,1,1,3,3,0,1,1,0,0,1]
Phi of -K* [-3,-2,0,1,2,2,0,3,3,2,2,2,2,1,1,1,1,2,0,1,0]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial z^2+18z+33
Enhanced Jones-Krushkal polynomial w^3z^2+18w^2z+33w
Inner characteristic polynomial t^6+43t^4+19t^2
Outer characteristic polynomial t^7+65t^5+70t^3+4t
Flat arrow polynomial 4*K1**2*K2 - 12*K1**2 - 2*K1*K2 - 2*K1*K3 + K1 - 2*K2**2 + 5*K2 + K3 + K4 + 7
2-strand cable arrow polynomial 96*K1**4*K2 - 1184*K1**4 - 64*K1**3*K3 + 32*K1**2*K2**3 - 2352*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 544*K1**2*K2*K4 + 6080*K1**2*K2 - 320*K1**2*K3**2 - 32*K1**2*K3*K5 - 192*K1**2*K4**2 - 5700*K1**2 + 128*K1*K2**2*K3*K4 - 864*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 256*K1*K2*K3*K4 + 5352*K1*K2*K3 - 192*K1*K2*K4*K5 + 2160*K1*K3*K4 + 392*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**4*K4**2 + 96*K2**4*K4 - 448*K2**4 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 224*K2**2*K3**2 + 32*K2**2*K4**3 - 296*K2**2*K4**2 - 32*K2**2*K4*K8 + 1600*K2**2*K4 - 8*K2**2*K6**2 - 4538*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 464*K2*K3*K5 + 304*K2*K4*K6 + 8*K2*K6*K8 - 32*K3**4 - 64*K3**2*K4**2 + 56*K3**2*K6 - 2452*K3**2 + 32*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1294*K4**2 - 240*K5**2 - 62*K6**2 - 2*K8**2 + 4702
Genus of based matrix 1
Fillings of based matrix [[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
Contact