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

Flat knot 6.1262

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,0,0,1,1,2,1,2,1,2,0,-1,1,0,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1262']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 8K12 - 4K1K2 - 4K1K3 - 4K1 + 4*K2 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1247', '6.1262']
Outer characteristic polynomial of the knot is: t^7+33t^5+97t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1262']
2-strand cable arrow polynomial of the knot is: -128K14K2*2 + 128K1*4K2 - 864K14 + 256K1*3K2K3 - 256K1*3K3 - 1536K12K2*4 + 2816K1*2K2*3 - 128K1*2K2*2K3*2 + 128K1*2K2*2K4 - 8000K12K2*2 - 192K1*2K2K4 + 6976K1*2K2 - 512K12K3*2 - 4456K1*2 + 3776K1K23K3 + 192K1K2*2K3K4 - 1792K1K22K3 - 768K1K2*2K5 + 128K1K2K33 - 256K1K2K3K4 - 192K1K2K3K6 + 7040K1K2K3 + 576K1K3K4 + 32K1K4K5 + 16K1K5K6 - 704K2*6 - 384K2*4K3*2 - 32K2*4K4*2 + 512K2*4K4 - 3792K24 + 384K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 2080K22K3*2 - 144K2*2K4*2 + 2416K2*2K4 - 128K22K5*2 - 48K2*2K6*2 - 1400K2*2 + 960K2K3K5 + 112K2K4K6 + 16K2K5K7 + 16K2K6K8 - 32K3*4 + 32K3*2K6 - 1664K32 - 296K4*2 - 88K5*2 - 24K6*2 - 2K8**2 + 3384
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1262']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70417', 'vk6.70429', 'vk6.70444', 'vk6.70455', 'vk6.70537', 'vk6.70618', 'vk6.70773', 'vk6.70856', 'vk6.70887', 'vk6.70902', 'vk6.70910', 'vk6.71016', 'vk6.71126', 'vk6.71255', 'vk6.71851', 'vk6.72295', 'vk6.76663', 'vk6.77642', 'vk6.87960', 'vk6.89216']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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,-1,0,0,1,2,0,0,1,1,2,1,2,1,2,0,-1,1,0,2,1]

Flat knots (up to 7 crossings) with same phi are :['6.1262']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 8*K1**2 - 4*K1*K2 - 4*K1*K3 - 4*K1 + 4*K2 + K4 + 4

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1247', '6.1262']

Outer characteristic polynomial of the knot is: t^7+33t^5+97t^3+16t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1262']

2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 128*K1**4*K2 - 864*K1**4 + 256*K1**3*K2*K3 - 256*K1**3*K3 - 1536*K1**2*K2**4 + 2816*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 8000*K1**2*K2**2 - 192*K1**2*K2*K4 + 6976*K1**2*K2 - 512*K1**2*K3**2 - 4456*K1**2 + 3776*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 - 768*K1*K2**2*K5 + 128*K1*K2*K3**3 - 256*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 7040*K1*K2*K3 + 576*K1*K3*K4 + 32*K1*K4*K5 + 16*K1*K5*K6 - 704*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 512*K2**4*K4 - 3792*K2**4 + 384*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 2080*K2**2*K3**2 - 144*K2**2*K4**2 + 2416*K2**2*K4 - 128*K2**2*K5**2 - 48*K2**2*K6**2 - 1400*K2**2 + 960*K2*K3*K5 + 112*K2*K4*K6 + 16*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 32*K3**2*K6 - 1664*K3**2 - 296*K4**2 - 88*K5**2 - 24*K6**2 - 2*K8**2 + 3384

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1262']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70417', 'vk6.70429', 'vk6.70444', 'vk6.70455', 'vk6.70537', 'vk6.70618', 'vk6.70773', 'vk6.70856', 'vk6.70887', 'vk6.70902', 'vk6.70910', 'vk6.71016', 'vk6.71126', 'vk6.71255', 'vk6.71851', 'vk6.72295', 'vk6.76663', 'vk6.77642', 'vk6.87960', 'vk6.89216']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is r.

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	O1O2O3O4U5U3U4O6O5U1U2U6
R3 orbit	{'O1O2O3O4U5U3U4O6O5U1U2U6'}
R3 orbit length	1
Gauss code of -K	Same
Gauss code of K*	O1O2O3U4U1O5O6O4U5U6U2U3
Gauss code of -K*	O1O2O3U4U1O5O6O4U5U6U2U3
Diagrammatic symmetry type	r
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 0 0 2 -1 1],[ 2 0 1 0 2 0 1],[ 0 -1 0 0 2 -2 0],[ 0 0 0 0 1 -1 -1],[-2 -2 -2 -1 0 -2 -1],[ 1 0 2 1 2 0 1],[-1 -1 0 1 1 -1 0]]
Primitive based matrix	[[ 0 2 1 0 0 -1 -2],[-2 0 -1 -1 -2 -2 -2],[-1 1 0 1 0 -1 -1],[ 0 1 -1 0 0 -1 0],[ 0 2 0 0 0 -2 -1],[ 1 2 1 1 2 0 0],[ 2 2 1 0 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,0,1,2,1,1,2,2,2,-1,0,1,1,0,1,0,2,1,0]
Phi over symmetry	[-2,-1,0,0,1,2,0,0,1,1,2,1,2,1,2,0,-1,1,0,2,1]
Phi of -K	[-2,-1,0,0,1,2,1,1,2,2,2,-1,0,1,1,0,1,0,2,1,0]
Phi of K*	[-2,-1,0,0,1,2,0,0,1,1,2,1,2,1,2,0,-1,1,0,2,1]
Phi of -K*	[-2,-1,0,0,1,2,0,0,1,1,2,1,2,1,2,0,-1,1,0,2,1]
Symmetry type of based matrix	r
u-polynomial	0
Normalized Jones-Krushkal polynomial	5z^2+22z+25
Enhanced Jones-Krushkal polynomial	-2w^4z^2+7w^3z^2-4w^3z+26w^2z+25w
Inner characteristic polynomial	t^6+23t^4+35t^2+4
Outer characteristic polynomial	t^7+33t^5+97t^3+16t
Flat arrow polynomial	8K13 + 4K1*2K2 - 8K12 - 4K1K2 - 4K1K3 - 4K1 + 4*K2 + K4 + 4
2-strand cable arrow polynomial	-128K14K2*2 + 128K1*4K2 - 864K14 + 256K1*3K2K3 - 256K1*3K3 - 1536K12K2*4 + 2816K1*2K2*3 - 128K1*2K2*2K3*2 + 128K1*2K2*2K4 - 8000K12K2*2 - 192K1*2K2K4 + 6976K1*2K2 - 512K12K3*2 - 4456K1*2 + 3776K1K23K3 + 192K1K2*2K3K4 - 1792K1K22K3 - 768K1K2*2K5 + 128K1K2K33 - 256K1K2K3K4 - 192K1K2K3K6 + 7040K1K2K3 + 576K1K3K4 + 32K1K4K5 + 16K1K5K6 - 704K2*6 - 384K2*4K3*2 - 32K2*4K4*2 + 512K2*4K4 - 3792K24 + 384K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 2080K22K3*2 - 144K2*2K4*2 + 2416K2*2K4 - 128K22K5*2 - 48K2*2K6*2 - 1400K2*2 + 960K2K3K5 + 112K2K4K6 + 16K2K5K7 + 16K2K6K8 - 32K3*4 + 32K3*2K6 - 1664K32 - 296K4*2 - 88K5*2 - 24K6*2 - 2K8**2 + 3384
Genus of based matrix	1
Fillings of based matrix	[[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}]]
If K is slice	False

Contact