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

Flat knot 6.1182

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,2,2,4,1,1,1,2,0,0,0,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1182']
Arrow polynomial of the knot is: -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.586', '6.590', '6.958', '6.987', '6.991', '6.993', '6.999', '6.1054', '6.1065', '6.1096', '6.1168', '6.1182']
Outer characteristic polynomial of the knot is: t^7+53t^5+48t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1182']
2-strand cable arrow polynomial of the knot is: -48*K1**4 + 128*K1**3*K2*K3 - 64*K1**2*K2**2*K3**2 - 928*K1**2*K2**2 + 496*K1**2*K2 - 368*K1**2*K3**2 - 32*K1**2*K5**2 - 1212*K1**2 + 320*K1*K2**3*K3 + 96*K1*K2*K3**3 + 2608*K1*K2*K3 + 312*K1*K3*K4 + 112*K1*K4*K5 + 80*K1*K5*K6 - 192*K2**4 - 560*K2**2*K3**2 - 24*K2**2*K4**2 + 72*K2**2*K4 - 1038*K2**2 + 440*K2*K3*K5 + 40*K2*K4*K6 - 64*K3**4 + 24*K3**2*K6 - 1136*K3**2 - 164*K4**2 - 204*K5**2 - 50*K6**2 + 1354
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1182']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71478', 'vk6.71502', 'vk6.71537', 'vk6.71559', 'vk6.72016', 'vk6.72032', 'vk6.72067', 'vk6.72082', 'vk6.72524', 'vk6.72541', 'vk6.72645', 'vk6.72647', 'vk6.72921', 'vk6.72958', 'vk6.73121', 'vk6.73124', 'vk6.73643', 'vk6.73680', 'vk6.73685', 'vk6.77099', 'vk6.77120', 'vk6.77151', 'vk6.77173', 'vk6.77448', 'vk6.77470', 'vk6.77947', 'vk6.77949', 'vk6.78578', 'vk6.81425', 'vk6.86905', 'vk6.87245', 'vk6.89348']
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 O1O2O3O4U1U5U4O6O5U2U6U3
R3 orbit {'O1O2O3O4U1U5U4O6O5U2U6U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U3O6O5U1U6U4
Gauss code of K* O1O2O3U4U2O5O4O6U1U5U6U3
Gauss code of -K* O1O2O3U4U2O5O4O6U5U1U3U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -1 2 2 0 0],[ 3 0 2 3 1 2 1],[ 1 -2 0 2 1 0 0],[-2 -3 -2 0 1 -2 -1],[-2 -1 -1 -1 0 -2 -1],[ 0 -2 0 2 2 0 0],[ 0 -1 0 1 1 0 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 1 -1 -2 -2 -3],[-2 -1 0 -1 -2 -1 -1],[ 0 1 1 0 0 0 -1],[ 0 2 2 0 0 0 -2],[ 1 2 1 0 0 0 -2],[ 3 3 1 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,-1,1,2,2,3,1,2,1,1,0,0,1,0,2,2]
Phi over symmetry [-3,-1,0,0,2,2,0,1,2,2,4,1,1,1,2,0,0,0,1,1,-1]
Phi of -K [-3,-1,0,0,2,2,0,1,2,2,4,1,1,1,2,0,0,0,1,1,-1]
Phi of K* [-2,-2,0,0,1,3,-1,0,1,2,4,0,1,1,2,0,1,1,1,2,0]
Phi of -K* [-3,-1,0,0,2,2,2,1,2,1,3,0,0,1,2,0,1,1,2,2,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial -6w^3z+13w^2z+15w
Inner characteristic polynomial t^6+35t^4+16t^2
Outer characteristic polynomial t^7+53t^5+48t^3
Flat arrow polynomial -4*K1**2 - 6*K1*K2 + 3*K1 + 2*K2 + 3*K3 + 3
2-strand cable arrow polynomial -48*K1**4 + 128*K1**3*K2*K3 - 64*K1**2*K2**2*K3**2 - 928*K1**2*K2**2 + 496*K1**2*K2 - 368*K1**2*K3**2 - 32*K1**2*K5**2 - 1212*K1**2 + 320*K1*K2**3*K3 + 96*K1*K2*K3**3 + 2608*K1*K2*K3 + 312*K1*K3*K4 + 112*K1*K4*K5 + 80*K1*K5*K6 - 192*K2**4 - 560*K2**2*K3**2 - 24*K2**2*K4**2 + 72*K2**2*K4 - 1038*K2**2 + 440*K2*K3*K5 + 40*K2*K4*K6 - 64*K3**4 + 24*K3**2*K6 - 1136*K3**2 - 164*K4**2 - 204*K5**2 - 50*K6**2 + 1354
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{4, 6}, {3, 5}, {1, 2}]]
If K is slice False
Contact