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

Flat knot 6.225

Min(phi) over symmetries of the knot is: [-4,-2,-1,1,3,3,0,2,1,4,5,0,0,1,2,0,2,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.225']
Arrow polynomial of the knot is: 4K12K2 - 4K12 - 2K1*K3 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.166', '6.225', '6.296', '6.498']
Outer characteristic polynomial of the knot is: t^7+104t^5+181t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.225']
2-strand cable arrow polynomial of the knot is: 256K12K2*2K4 - 2016K12K2*2 - 512K1*2K2K4 + 3216K1*2K2 - 224K12K4*2 - 2784K1*2 + 320K1K23K3 + 160K1K2*2K3K4 - 1120K1K22K3 - 160K1K2*2K5 - 224K1K2K3K4 + 3592K1K2K3 - 32K1K2K4K5 + 920K1K3K4 + 112K1K4K5 - 32K2*4K4*2 + 192K2*4K4 - 1104K24 + 32K2*3K4K6 - 96K2*3K6 + 64K22K3*2K4 - 928K22K3*2 - 32K2*2K3K7 - 272K2*2K4*2 + 1712K2*2K4 - 8K22K6*2 - 2120K2*2 + 552K2K3K5 + 144K2K4K6 - 64K3*2K4*2 - 1096K3*2 + 32K3K4K7 - 566K42 - 32K5*2 - 16K6**2 + 2132
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.225']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16371', 'vk6.16414', 'vk6.18111', 'vk6.18447', 'vk6.22705', 'vk6.22808', 'vk6.24560', 'vk6.24977', 'vk6.34674', 'vk6.34755', 'vk6.36697', 'vk6.37119', 'vk6.42327', 'vk6.42373', 'vk6.43973', 'vk6.44288', 'vk6.54630', 'vk6.54657', 'vk6.55929', 'vk6.56223', 'vk6.59109', 'vk6.59178', 'vk6.60459', 'vk6.60822', 'vk6.64652', 'vk6.64702', 'vk6.65578', 'vk6.65889', 'vk6.68002', 'vk6.68030', 'vk6.68655', 'vk6.68868']
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: [-4,-2,-1,1,3,3,0,2,1,4,5,0,0,1,2,0,2,3,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.225']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.166', '6.225', '6.296', '6.498']

Outer characteristic polynomial of the knot is: t^7+104t^5+181t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.225']

2-strand cable arrow polynomial of the knot is: 256*K1**2*K2**2*K4 - 2016*K1**2*K2**2 - 512*K1**2*K2*K4 + 3216*K1**2*K2 - 224*K1**2*K4**2 - 2784*K1**2 + 320*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1120*K1*K2**2*K3 - 160*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 3592*K1*K2*K3 - 32*K1*K2*K4*K5 + 920*K1*K3*K4 + 112*K1*K4*K5 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1104*K2**4 + 32*K2**3*K4*K6 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 928*K2**2*K3**2 - 32*K2**2*K3*K7 - 272*K2**2*K4**2 + 1712*K2**2*K4 - 8*K2**2*K6**2 - 2120*K2**2 + 552*K2*K3*K5 + 144*K2*K4*K6 - 64*K3**2*K4**2 - 1096*K3**2 + 32*K3*K4*K7 - 566*K4**2 - 32*K5**2 - 16*K6**2 + 2132

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.225']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16371', 'vk6.16414', 'vk6.18111', 'vk6.18447', 'vk6.22705', 'vk6.22808', 'vk6.24560', 'vk6.24977', 'vk6.34674', 'vk6.34755', 'vk6.36697', 'vk6.37119', 'vk6.42327', 'vk6.42373', 'vk6.43973', 'vk6.44288', 'vk6.54630', 'vk6.54657', 'vk6.55929', 'vk6.56223', 'vk6.59109', 'vk6.59178', 'vk6.60459', 'vk6.60822', 'vk6.64652', 'vk6.64702', 'vk6.65578', 'vk6.65889', 'vk6.68002', 'vk6.68030', 'vk6.68655', 'vk6.68868']

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	O1O2O3O4O5U3O6U1U6U2U5U4
R3 orbit	{'O1O2O3O4O5U3O6U1U6U2U5U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U1U4U6U5O6U3
Gauss code of K*	O1O2O3O4O5U1U3U6U5U4O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U2U1U6U3U5
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 -4 -1 -2 3 3 1],[ 4 0 2 0 5 4 1],[ 1 -2 0 0 3 2 0],[ 2 0 0 0 2 1 0],[-3 -5 -3 -2 0 0 0],[-3 -4 -2 -1 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix	[[ 0 3 3 1 -1 -2 -4],[-3 0 0 0 -2 -1 -4],[-3 0 0 0 -3 -2 -5],[-1 0 0 0 0 0 -1],[ 1 2 3 0 0 0 -2],[ 2 1 2 0 0 0 0],[ 4 4 5 1 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,-1,1,2,4,0,0,2,1,4,0,3,2,5,0,0,1,0,2,0]
Phi over symmetry	[-4,-2,-1,1,3,3,0,2,1,4,5,0,0,1,2,0,2,3,0,0,0]
Phi of -K	[-4,-2,-1,1,3,3,2,1,4,2,3,1,3,3,4,2,1,2,2,2,0]
Phi of K*	[-3,-3,-1,1,2,4,0,2,1,3,2,2,2,4,3,2,3,4,1,1,2]
Phi of -K*	[-4,-2,-1,1,3,3,0,2,1,4,5,0,0,1,2,0,2,3,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^3+t^2
Normalized Jones-Krushkal polynomial	8z^2+25z+19
Enhanced Jones-Krushkal polynomial	8w^3z^2+25w^2z+19w
Inner characteristic polynomial	t^6+64t^4+52t^2+1
Outer characteristic polynomial	t^7+104t^5+181t^3+5t
Flat arrow polynomial	4K12K2 - 4K12 - 2K1*K3 + K2 + 2
2-strand cable arrow polynomial	256K12K2*2K4 - 2016K12K2*2 - 512K1*2K2K4 + 3216K1*2K2 - 224K12K4*2 - 2784K1*2 + 320K1K23K3 + 160K1K2*2K3K4 - 1120K1K22K3 - 160K1K2*2K5 - 224K1K2K3K4 + 3592K1K2K3 - 32K1K2K4K5 + 920K1K3K4 + 112K1K4K5 - 32K2*4K4*2 + 192K2*4K4 - 1104K24 + 32K2*3K4K6 - 96K2*3K6 + 64K22K3*2K4 - 928K22K3*2 - 32K2*2K3K7 - 272K2*2K4*2 + 1712K2*2K4 - 8K22K6*2 - 2120K2*2 + 552K2K3K5 + 144K2K4K6 - 64K3*2K4*2 - 1096K3*2 + 32K3K4K7 - 566K42 - 32K5*2 - 16K6**2 + 2132
Genus of based matrix	1
Fillings of based matrix	[[{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice	False

Contact