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

Flat knot 6.28

Min(phi) over symmetries of the knot is: [-5,-2,0,2,2,3,1,4,2,3,5,2,1,2,3,1,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.28']
Arrow polynomial of the knot is: -2K12 - 2K1*K4 + K2 + K3 + K5 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.7', '6.28']
Outer characteristic polynomial of the knot is: t^7+125t^5+94t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.28', '6.85']
2-strand cable arrow polynomial of the knot is: -224K12K2*2 + 1824K1*2K2 - 256K12K3*2 - 192K1*2K3K5 - 2792K1*2 - 576K1K22K3 - 192K1K2*2K5 + 32K1K2K33 - 320K1K2K3K4 + 2552K1K2K3 - 64K1K3*2K5 + 1216K1K3K4 + 568K1K4K5 + 8K1K5K6 + 24K1K7K8 - 2K102 + 8K10K2K8 - 440K24 + 96K2*2K3*2K4 - 544K22K3*2 - 64K2*2K3K7 - 32K2*2K4*2 - 32K2*2K4K8 + 1360K2*2K4 - 64K22K5*2 - 8K2*2K8*2 - 2708K2*2 - 96K2K32K4 - 64K2K3K4K5 + 1248K2K3K5 + 96K2K4K6 + 152K2K5K7 + 48K2K6K8 - 192K34 - 64K3*2K4*2 + 216K3*2K6 - 1632K32 + 72K3K4K7 + 16K3K5K8 + 16K4*2K8 - 1038K42 - 616K5*2 - 74K6*2 - 72K7*2 - 44K8**2 + 2848
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.28']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71580', 'vk6.71699', 'vk6.72117', 'vk6.72317', 'vk6.73472', 'vk6.74126', 'vk6.74132', 'vk6.74695', 'vk6.74702', 'vk6.75228', 'vk6.75478', 'vk6.76163', 'vk6.76175', 'vk6.77198', 'vk6.77301', 'vk6.77504', 'vk6.77656', 'vk6.78442', 'vk6.79123', 'vk6.79133', 'vk6.80028', 'vk6.80176', 'vk6.80627', 'vk6.80634', 'vk6.83728', 'vk6.83852', 'vk6.85062', 'vk6.85323', 'vk6.86664', 'vk6.86966', 'vk6.87414', 'vk6.89530']
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: [-5,-2,0,2,2,3,1,4,2,3,5,2,1,2,3,1,1,2,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.28']

Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K4 + K2 + K3 + K5 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.7', '6.28']

Outer characteristic polynomial of the knot is: t^7+125t^5+94t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.28', '6.85']

2-strand cable arrow polynomial of the knot is: -224*K1**2*K2**2 + 1824*K1**2*K2 - 256*K1**2*K3**2 - 192*K1**2*K3*K5 - 2792*K1**2 - 576*K1*K2**2*K3 - 192*K1*K2**2*K5 + 32*K1*K2*K3**3 - 320*K1*K2*K3*K4 + 2552*K1*K2*K3 - 64*K1*K3**2*K5 + 1216*K1*K3*K4 + 568*K1*K4*K5 + 8*K1*K5*K6 + 24*K1*K7*K8 - 2*K10**2 + 8*K10*K2*K8 - 440*K2**4 + 96*K2**2*K3**2*K4 - 544*K2**2*K3**2 - 64*K2**2*K3*K7 - 32*K2**2*K4**2 - 32*K2**2*K4*K8 + 1360*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K8**2 - 2708*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1248*K2*K3*K5 + 96*K2*K4*K6 + 152*K2*K5*K7 + 48*K2*K6*K8 - 192*K3**4 - 64*K3**2*K4**2 + 216*K3**2*K6 - 1632*K3**2 + 72*K3*K4*K7 + 16*K3*K5*K8 + 16*K4**2*K8 - 1038*K4**2 - 616*K5**2 - 74*K6**2 - 72*K7**2 - 44*K8**2 + 2848

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.28']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71580', 'vk6.71699', 'vk6.72117', 'vk6.72317', 'vk6.73472', 'vk6.74126', 'vk6.74132', 'vk6.74695', 'vk6.74702', 'vk6.75228', 'vk6.75478', 'vk6.76163', 'vk6.76175', 'vk6.77198', 'vk6.77301', 'vk6.77504', 'vk6.77656', 'vk6.78442', 'vk6.79123', 'vk6.79133', 'vk6.80028', 'vk6.80176', 'vk6.80627', 'vk6.80634', 'vk6.83728', 'vk6.83852', 'vk6.85062', 'vk6.85323', 'vk6.86664', 'vk6.86966', 'vk6.87414', 'vk6.89530']

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	O1O2O3O4O5O6U1U3U6U5U2U4
R3 orbit	{'O1O2O3O4O5O6U1U3U6U5U2U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U3U5U2U1U4U6
Gauss code of K*	O1O2O3O4O5O6U1U5U2U6U4U3
Gauss code of -K*	O1O2O3O4O5O6U4U3U1U5U2U6
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 0 -2 3 2 2],[ 5 0 4 1 5 3 2],[ 0 -4 0 -2 2 1 1],[ 2 -1 2 0 3 2 1],[-3 -5 -2 -3 0 0 0],[-2 -3 -1 -2 0 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix	[[ 0 3 2 2 0 -2 -5],[-3 0 0 0 -2 -3 -5],[-2 0 0 0 -1 -1 -2],[-2 0 0 0 -1 -2 -3],[ 0 2 1 1 0 -2 -4],[ 2 3 1 2 2 0 -1],[ 5 5 2 3 4 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-2,0,2,5,0,0,2,3,5,0,1,1,2,1,2,3,2,4,1]
Phi over symmetry	[-5,-2,0,2,2,3,1,4,2,3,5,2,1,2,3,1,1,2,0,0,0]
Phi of -K	[-5,-2,0,2,2,3,2,1,4,5,3,0,2,3,2,1,1,1,0,1,1]
Phi of K*	[-3,-2,-2,0,2,5,1,1,1,2,3,0,1,2,4,1,3,5,0,1,2]
Phi of -K*	[-5,-2,0,2,2,3,1,4,2,3,5,2,1,2,3,1,1,2,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^5-t^3-t^2
Normalized Jones-Krushkal polynomial	6z^2+23z+23
Enhanced Jones-Krushkal polynomial	6w^3z^2+23w^2z+23w
Inner characteristic polynomial	t^6+79t^4+11t^2
Outer characteristic polynomial	t^7+125t^5+94t^3+4t
Flat arrow polynomial	-2K12 - 2K1*K4 + K2 + K3 + K5 + 2
2-strand cable arrow polynomial	-224K12K2*2 + 1824K1*2K2 - 256K12K3*2 - 192K1*2K3K5 - 2792K1*2 - 576K1K22K3 - 192K1K2*2K5 + 32K1K2K33 - 320K1K2K3K4 + 2552K1K2K3 - 64K1K3*2K5 + 1216K1K3K4 + 568K1K4K5 + 8K1K5K6 + 24K1K7K8 - 2K102 + 8K10K2K8 - 440K24 + 96K2*2K3*2K4 - 544K22K3*2 - 64K2*2K3K7 - 32K2*2K4*2 - 32K2*2K4K8 + 1360K2*2K4 - 64K22K5*2 - 8K2*2K8*2 - 2708K2*2 - 96K2K32K4 - 64K2K3K4K5 + 1248K2K3K5 + 96K2K4K6 + 152K2K5K7 + 48K2K6K8 - 192K34 - 64K3*2K4*2 + 216K3*2K6 - 1632K32 + 72K3K4K7 + 16K3K5K8 + 16K4*2K8 - 1038K42 - 616K5*2 - 74K6*2 - 72K7*2 - 44K8**2 + 2848
Genus of based matrix	1
Fillings of based matrix	[[{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice	False

Contact