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

Flat knot 6.10

Min(phi) over symmetries of the knot is: [-5,-3,1,2,2,3,1,2,4,5,3,1,3,4,2,1,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.10']
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+142t^5+215t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.10']
2-strand cable arrow polynomial of the knot is: -128*K1**2*K2**3*K4 + 416*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 3072*K1**2*K2**2 - 384*K1**2*K2*K4 + 3096*K1**2*K2 - 64*K1**2*K4**2 - 2504*K1**2 - 128*K1*K2**3*K3*K4 + 768*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 992*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 3544*K1*K2*K3 + 544*K1*K3*K4 + 144*K1*K4*K5 - 128*K2**6*K4**2 + 512*K2**6*K4 - 1152*K2**6 + 128*K2**4*K4**3 - 1216*K2**4*K4**2 + 3904*K2**4*K4 - 5248*K2**4 + 96*K2**3*K3*K5 + 384*K2**3*K4*K6 - 608*K2**3*K6 - 672*K2**2*K3**2 - 32*K2**2*K4**4 + 128*K2**2*K4**3 - 1864*K2**2*K4**2 + 3696*K2**2*K4 - 64*K2**2*K5**2 - 422*K2**2 + 392*K2*K3*K5 + 400*K2*K4*K6 + 8*K2*K5*K7 - 1056*K3**2 - 16*K4**4 - 844*K4**2 - 104*K5**2 - 2*K6**2 + 2378
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.10']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73993', 'vk6.73997', 'vk6.74517', 'vk6.74523', 'vk6.75983', 'vk6.75991', 'vk6.76731', 'vk6.76734', 'vk6.78969', 'vk6.78971', 'vk6.79517', 'vk6.79521', 'vk6.80493', 'vk6.80497', 'vk6.80965', 'vk6.80967', 'vk6.83023', 'vk6.83477', 'vk6.83716', 'vk6.83718', 'vk6.83956', 'vk6.84002', 'vk6.84017', 'vk6.85084', 'vk6.86281', 'vk6.86650', 'vk6.86652', 'vk6.87470', 'vk6.87476', 'vk6.87917', 'vk6.88865', 'vk6.88872']
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 O1O2O3O4O5O6U1U2U5U6U4U3
R3 orbit {'O1O2O3O4O5O6U1U2U5U6U4U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U4U3U1U2U5U6
Gauss code of K* O1O2O3O4O5O6U1U2U6U5U3U4
Gauss code of -K* O1O2O3O4O5O6U3U4U2U1U5U6
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 -5 -3 2 2 1 3],[ 5 0 1 5 4 2 3],[ 3 -1 0 4 3 1 2],[-2 -5 -4 0 0 -1 1],[-2 -4 -3 0 0 -1 1],[-1 -2 -1 1 1 0 1],[-3 -3 -2 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 2 2 1 -3 -5],[-3 0 -1 -1 -1 -2 -3],[-2 1 0 0 -1 -3 -4],[-2 1 0 0 -1 -4 -5],[-1 1 1 1 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,-2,-2,-1,3,5,1,1,1,2,3,0,1,3,4,1,4,5,1,2,1]
Phi over symmetry [-5,-3,1,2,2,3,1,2,4,5,3,1,3,4,2,1,1,1,0,1,1]
Phi of -K [-5,-3,1,2,2,3,1,4,2,3,5,3,1,2,4,0,0,1,0,0,0]
Phi of K* [-3,-2,-2,-1,3,5,0,0,1,4,5,0,0,1,2,0,2,3,3,4,1]
Phi of -K* [-5,-3,1,2,2,3,1,2,4,5,3,1,3,4,2,1,1,1,0,1,1]
Symmetry type of based matrix c
u-polynomial t^5-2t^2-t
Normalized Jones-Krushkal polynomial 2z^2+7z+7
Enhanced Jones-Krushkal polynomial -6w^4z^2+8w^3z^2-16w^3z+23w^2z+7w
Inner characteristic polynomial t^6+90t^4+25t^2+1
Outer characteristic polynomial t^7+142t^5+215t^3+19t
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 -128*K1**2*K2**3*K4 + 416*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 3072*K1**2*K2**2 - 384*K1**2*K2*K4 + 3096*K1**2*K2 - 64*K1**2*K4**2 - 2504*K1**2 - 128*K1*K2**3*K3*K4 + 768*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 992*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 3544*K1*K2*K3 + 544*K1*K3*K4 + 144*K1*K4*K5 - 128*K2**6*K4**2 + 512*K2**6*K4 - 1152*K2**6 + 128*K2**4*K4**3 - 1216*K2**4*K4**2 + 3904*K2**4*K4 - 5248*K2**4 + 96*K2**3*K3*K5 + 384*K2**3*K4*K6 - 608*K2**3*K6 - 672*K2**2*K3**2 - 32*K2**2*K4**4 + 128*K2**2*K4**3 - 1864*K2**2*K4**2 + 3696*K2**2*K4 - 64*K2**2*K5**2 - 422*K2**2 + 392*K2*K3*K5 + 400*K2*K4*K6 + 8*K2*K5*K7 - 1056*K3**2 - 16*K4**4 - 844*K4**2 - 104*K5**2 - 2*K6**2 + 2378
Genus of based matrix 1
Fillings of based matrix [[{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice False
Contact