Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo

Flat knot 6.1682

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,-1,-1,-1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1682']
Arrow polynomial of the knot is: -4K1K2 + 2K1 + 2K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922']
Outer characteristic polynomial of the knot is: t^7+32t^5+45t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1682']
2-strand cable arrow polynomial of the knot is: 896K14K2 - 1904K14 - 768K1*3K2*2K3 + 1280K13K2K3 + 32K1*3K3K4 - 2400K1*3K3 + 1152K12K2*3 - 3136K1*2K2*2 + 768K1*2K2K32 + 96K1*2K2K42 - 1248K1*2K2K4 + 6072K1*2K2 - 1968K12K3*2 - 192K1*2K3K5 - 272K1*2K4*2 - 96K1*2K4K6 - 4624K1*2 + 576K1K23K3 - 960K1K2*2K3 - 64K1K2*2K5 - 320K1K2K3K4 + 6624K1K2K3 + 2504K1K3K4 + 336K1K4K5 + 24K1K5K6 - 288K24 - 608K2*2K3*2 - 112K2*2K4*2 + 496K2*2K4 - 2668K22 + 304K2K3K5 + 112K2K4K6 - 2036K3*2 - 660K4*2 - 68K5*2 - 28K6**2 + 3146
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1682']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19917', 'vk6.19922', 'vk6.21148', 'vk6.21151', 'vk6.26848', 'vk6.26857', 'vk6.28622', 'vk6.28628', 'vk6.38284', 'vk6.38293', 'vk6.40416', 'vk6.40421', 'vk6.45155', 'vk6.45160', 'vk6.47002', 'vk6.47004', 'vk6.56700', 'vk6.56712', 'vk6.57790', 'vk6.57799', 'vk6.61107', 'vk6.61124', 'vk6.62360', 'vk6.62376', 'vk6.66390', 'vk6.66407', 'vk6.67154', 'vk6.67169', 'vk6.69049', 'vk6.69058', 'vk6.69839', 'vk6.69846']
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

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,-1,-1,-1,0,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1682']

Arrow polynomial of the knot is: -4*K1*K2 + 2*K1 + 2*K3 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.540', '6.925', '6.1021', '6.1117', '6.1120', '6.1135', '6.1227', '6.1230', '6.1260', '6.1682', '6.1685', '6.1922']

Outer characteristic polynomial of the knot is: t^7+32t^5+45t^3+13t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1682']

2-strand cable arrow polynomial of the knot is: 896*K1**4*K2 - 1904*K1**4 - 768*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 + 32*K1**3*K3*K4 - 2400*K1**3*K3 + 1152*K1**2*K2**3 - 3136*K1**2*K2**2 + 768*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 1248*K1**2*K2*K4 + 6072*K1**2*K2 - 1968*K1**2*K3**2 - 192*K1**2*K3*K5 - 272*K1**2*K4**2 - 96*K1**2*K4*K6 - 4624*K1**2 + 576*K1*K2**3*K3 - 960*K1*K2**2*K3 - 64*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 6624*K1*K2*K3 + 2504*K1*K3*K4 + 336*K1*K4*K5 + 24*K1*K5*K6 - 288*K2**4 - 608*K2**2*K3**2 - 112*K2**2*K4**2 + 496*K2**2*K4 - 2668*K2**2 + 304*K2*K3*K5 + 112*K2*K4*K6 - 2036*K3**2 - 660*K4**2 - 68*K5**2 - 28*K6**2 + 3146

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1682']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19917', 'vk6.19922', 'vk6.21148', 'vk6.21151', 'vk6.26848', 'vk6.26857', 'vk6.28622', 'vk6.28628', 'vk6.38284', 'vk6.38293', 'vk6.40416', 'vk6.40421', 'vk6.45155', 'vk6.45160', 'vk6.47002', 'vk6.47004', 'vk6.56700', 'vk6.56712', 'vk6.57790', 'vk6.57799', 'vk6.61107', 'vk6.61124', 'vk6.62360', 'vk6.62376', 'vk6.66390', 'vk6.66407', 'vk6.67154', 'vk6.67169', 'vk6.69049', 'vk6.69058', 'vk6.69839', 'vk6.69846']

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	O1O2O3U4O5U1U3U2O4O6U5U6
R3 orbit	{'O1O2O3U4O5U1U3U2O4O6U5U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U5O4O6U2U1U3O5U6
Gauss code of K*	O1O2O3U4U5O6O5U1U3U2O4U6
Gauss code of -K*	O1O2O3U4O5U2U1U3O6O4U6U5
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 -2 1 1 -2 1 1],[ 2 0 2 1 0 2 1],[-1 -2 0 0 -2 1 1],[-1 -1 0 0 -1 0 1],[ 2 0 2 1 0 1 0],[-1 -2 -1 0 -1 0 1],[-1 -1 -1 -1 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 1 -2 -2],[-1 0 1 1 0 -2 -2],[-1 -1 0 1 0 -1 -2],[-1 -1 -1 0 -1 0 -1],[-1 0 0 1 0 -1 -1],[ 2 2 1 0 1 0 0],[ 2 2 2 1 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,2,2,-1,-1,0,2,2,-1,0,1,2,1,0,1,1,1,0]
Phi over symmetry	[-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,-1,-1,-1,0,0,-1]
Phi of -K	[-2,-2,1,1,1,1,0,1,1,2,2,1,2,2,3,-1,0,-1,0,-1,-1]
Phi of K*	[-1,-1,-1,-1,2,2,-1,-1,-1,2,3,-1,0,1,2,0,1,1,2,2,0]
Phi of -K*	[-2,-2,1,1,1,1,0,0,1,1,2,1,1,2,2,-1,-1,-1,0,0,-1]
Symmetry type of based matrix	c
u-polynomial	2t^2-4t
Normalized Jones-Krushkal polynomial	7z^2+26z+25
Enhanced Jones-Krushkal polynomial	-2w^4z^2+9w^3z^2+26w^2z+25w
Inner characteristic polynomial	t^6+20t^4+19t^2+4
Outer characteristic polynomial	t^7+32t^5+45t^3+13t
Flat arrow polynomial	-4K1K2 + 2K1 + 2K3 + 1
2-strand cable arrow polynomial	896K14K2 - 1904K14 - 768K1*3K2*2K3 + 1280K13K2K3 + 32K1*3K3K4 - 2400K1*3K3 + 1152K12K2*3 - 3136K1*2K2*2 + 768K1*2K2K32 + 96K1*2K2K42 - 1248K1*2K2K4 + 6072K1*2K2 - 1968K12K3*2 - 192K1*2K3K5 - 272K1*2K4*2 - 96K1*2K4K6 - 4624K1*2 + 576K1K23K3 - 960K1K2*2K3 - 64K1K2*2K5 - 320K1K2K3K4 + 6624K1K2K3 + 2504K1K3K4 + 336K1K4K5 + 24K1K5K6 - 288K24 - 608K2*2K3*2 - 112K2*2K4*2 + 496K2*2K4 - 2668K22 + 304K2K3K5 + 112K2K4K6 - 2036K3*2 - 660K4*2 - 68K5*2 - 28K6**2 + 3146
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice	False

Contact