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

Flat knot 6.1185

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,2,2,1,3,1,1,1,1,0,1,1,2,2,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1185']
Arrow polynomial of the knot is: 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.122', '6.327', '6.371', '6.1185']
Outer characteristic polynomial of the knot is: t^7+51t^5+44t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1185']
2-strand cable arrow polynomial of the knot is: -576*K1**4*K2**2 + 1024*K1**4*K2 - 3648*K1**4 + 1088*K1**3*K2*K3 - 544*K1**3*K3 - 384*K1**2*K2**4 + 1728*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 - 512*K1**2*K2**2*K4**2 + 864*K1**2*K2**2*K4 - 11552*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 2240*K1**2*K2*K4 + 13248*K1**2*K2 - 352*K1**2*K3**2 - 464*K1**2*K4**2 - 7520*K1**2 + 2464*K1*K2**3*K3 + 1152*K1*K2**2*K3*K4 - 2368*K1*K2**2*K3 + 416*K1*K2**2*K4*K5 - 960*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 704*K1*K2*K3*K4 + 11488*K1*K2*K3 - 160*K1*K2*K4*K5 + 2152*K1*K3*K4 + 512*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 800*K2**4*K4 - 3264*K2**4 + 160*K2**3*K3*K5 + 128*K2**3*K4*K6 - 160*K2**3*K6 - 1616*K2**2*K3**2 - 1096*K2**2*K4**2 + 3704*K2**2*K4 - 112*K2**2*K5**2 - 16*K2**2*K6**2 - 5198*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 792*K2*K3*K5 + 320*K2*K4*K6 + 8*K2*K5*K7 - 2888*K3**2 - 1312*K4**2 - 208*K5**2 - 34*K6**2 + 6382
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1185']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3616', 'vk6.3689', 'vk6.3880', 'vk6.3999', 'vk6.7042', 'vk6.7085', 'vk6.7260', 'vk6.7373', 'vk6.17696', 'vk6.17745', 'vk6.24247', 'vk6.24308', 'vk6.36538', 'vk6.36615', 'vk6.43648', 'vk6.43755', 'vk6.48252', 'vk6.48329', 'vk6.48412', 'vk6.48431', 'vk6.50012', 'vk6.50055', 'vk6.50138', 'vk6.50155', 'vk6.55720', 'vk6.55777', 'vk6.60296', 'vk6.60379', 'vk6.65428', 'vk6.65457', 'vk6.68560', 'vk6.68589']
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 O1O2O3O4U1U5U4O6O5U6U2U3
R3 orbit {'O1O2O3O4U1U5U4O6O5U6U2U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U3U5O6O5U1U6U4
Gauss code of K* O1O2O3U4U2O4O5O6U1U5U6U3
Gauss code of -K* O1O2O3U4U3O5O4O6U5U1U2U6
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 2 0 -1],[ 3 0 2 3 1 2 0],[ 0 -2 0 1 1 0 -1],[-2 -3 -1 0 1 -2 -1],[-2 -1 -1 -1 0 -2 -1],[ 0 -2 0 2 2 0 -1],[ 1 0 1 1 1 1 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 1 -1 -2 -1 -3],[-2 -1 0 -1 -2 -1 -1],[ 0 1 1 0 0 -1 -2],[ 0 2 2 0 0 -1 -2],[ 1 1 1 1 1 0 0],[ 3 3 1 2 2 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,-1,1,2,1,3,1,2,1,1,0,1,2,1,2,0]
Phi over symmetry [-3,-1,0,0,2,2,0,2,2,1,3,1,1,1,1,0,1,1,2,2,-1]
Phi of -K [-3,-1,0,0,2,2,2,1,1,2,4,0,0,2,2,0,0,0,1,1,-1]
Phi of K* [-2,-2,0,0,1,3,-1,0,1,2,4,0,1,2,2,0,0,1,0,1,2]
Phi of -K* [-3,-1,0,0,2,2,0,2,2,1,3,1,1,1,1,0,1,1,2,2,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+33t^4+24t^2+4
Outer characteristic polynomial t^7+51t^5+44t^3+8t
Flat arrow polynomial 8*K1**3 + 8*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial -576*K1**4*K2**2 + 1024*K1**4*K2 - 3648*K1**4 + 1088*K1**3*K2*K3 - 544*K1**3*K3 - 384*K1**2*K2**4 + 1728*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 - 512*K1**2*K2**2*K4**2 + 864*K1**2*K2**2*K4 - 11552*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 192*K1**2*K2*K4**2 - 2240*K1**2*K2*K4 + 13248*K1**2*K2 - 352*K1**2*K3**2 - 464*K1**2*K4**2 - 7520*K1**2 + 2464*K1*K2**3*K3 + 1152*K1*K2**2*K3*K4 - 2368*K1*K2**2*K3 + 416*K1*K2**2*K4*K5 - 960*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 704*K1*K2*K3*K4 + 11488*K1*K2*K3 - 160*K1*K2*K4*K5 + 2152*K1*K3*K4 + 512*K1*K4*K5 + 32*K1*K5*K6 - 64*K2**6 - 192*K2**4*K3**2 - 192*K2**4*K4**2 + 800*K2**4*K4 - 3264*K2**4 + 160*K2**3*K3*K5 + 128*K2**3*K4*K6 - 160*K2**3*K6 - 1616*K2**2*K3**2 - 1096*K2**2*K4**2 + 3704*K2**2*K4 - 112*K2**2*K5**2 - 16*K2**2*K6**2 - 5198*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 792*K2*K3*K5 + 320*K2*K4*K6 + 8*K2*K5*K7 - 2888*K3**2 - 1312*K4**2 - 208*K5**2 - 34*K6**2 + 6382
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
Contact