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

Flat knot 6.1691

Min(phi) over symmetries of the knot is: [-2,-1,0,0,1,2,-1,1,2,1,1,0,0,0,1,1,1,0,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.1691']
Arrow polynomial of the knot is: 4K13 - 12K1*2 - 4K1K2 - K1 + 6K2 + K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1345', '6.1510', '6.1565', '6.1691', '6.1812']
Outer characteristic polynomial of the knot is: t^7+43t^5+160t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1691']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 416K1*4K2 - 2304K14 + 416K1*3K2K3 - 384K1*3K3 - 320K12K2*4 + 768K1*2K2*3 + 64K1*2K2*2K4 - 6432K12K2*2 + 32K1*2K2K32 - 416K1*2K2K4 + 9648K1*2K2 - 256K12K3*2 - 5940K1*2 + 512K1K23K3 - 928K1K2*2K3 - 320K1K2*2K5 - 32K1K2K3K4 + 6824K1K2K3 + 592K1K3K4 + 48K1K4K5 - 32K2*6 + 64K2*4K4 - 1344K24 - 32K2*3K6 - 240K22K3*2 - 16K2*2K4*2 + 1304K2*2K4 - 4038K22 + 216K2K3K5 + 16K2K4K6 - 1732K3*2 - 328K4*2 - 40K5*2 - 2K6**2 + 4406
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1691']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16503', 'vk6.16596', 'vk6.18095', 'vk6.18431', 'vk6.22934', 'vk6.23031', 'vk6.24542', 'vk6.24959', 'vk6.34905', 'vk6.35012', 'vk6.36685', 'vk6.37107', 'vk6.42472', 'vk6.42585', 'vk6.43961', 'vk6.44276', 'vk6.54730', 'vk6.54827', 'vk6.55919', 'vk6.56211', 'vk6.59194', 'vk6.59259', 'vk6.60446', 'vk6.60806', 'vk6.64744', 'vk6.64801', 'vk6.65569', 'vk6.65879', 'vk6.68038', 'vk6.68103', 'vk6.68647', 'vk6.68860']
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,-1,0,0,1,2,-1,1,2,1,1,0,0,0,1,1,1,0,1,2,1]

Flat knots (up to 7 crossings) with same phi are :['6.1691']

Arrow polynomial of the knot is: 4*K1**3 - 12*K1**2 - 4*K1*K2 - K1 + 6*K2 + K3 + 7

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1345', '6.1510', '6.1565', '6.1691', '6.1812']

Outer characteristic polynomial of the knot is: t^7+43t^5+160t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1691']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 416*K1**4*K2 - 2304*K1**4 + 416*K1**3*K2*K3 - 384*K1**3*K3 - 320*K1**2*K2**4 + 768*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 6432*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 416*K1**2*K2*K4 + 9648*K1**2*K2 - 256*K1**2*K3**2 - 5940*K1**2 + 512*K1*K2**3*K3 - 928*K1*K2**2*K3 - 320*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 6824*K1*K2*K3 + 592*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1344*K2**4 - 32*K2**3*K6 - 240*K2**2*K3**2 - 16*K2**2*K4**2 + 1304*K2**2*K4 - 4038*K2**2 + 216*K2*K3*K5 + 16*K2*K4*K6 - 1732*K3**2 - 328*K4**2 - 40*K5**2 - 2*K6**2 + 4406

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1691']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16503', 'vk6.16596', 'vk6.18095', 'vk6.18431', 'vk6.22934', 'vk6.23031', 'vk6.24542', 'vk6.24959', 'vk6.34905', 'vk6.35012', 'vk6.36685', 'vk6.37107', 'vk6.42472', 'vk6.42585', 'vk6.43961', 'vk6.44276', 'vk6.54730', 'vk6.54827', 'vk6.55919', 'vk6.56211', 'vk6.59194', 'vk6.59259', 'vk6.60446', 'vk6.60806', 'vk6.64744', 'vk6.64801', 'vk6.65569', 'vk6.65879', 'vk6.68038', 'vk6.68103', 'vk6.68647', 'vk6.68860']

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	O1O2O3U4O5U3U1U6O4O6U2U5
R3 orbit	{'O1O2O3U4O5U3U1U6O4O6U2U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U2O5O6U5U3U1O4U6
Gauss code of K*	O1O2O3U4U3O5O6U2U5U1O4U6
Gauss code of -K*	O1O2O3U4O5U3U6U2O4O6U1U5
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 -1 0 0 -2 2 1],[ 1 0 0 0 0 2 2],[ 0 0 0 1 -2 1 1],[ 0 0 -1 0 0 0 1],[ 2 0 2 0 0 3 2],[-2 -2 -1 0 -3 0 -2],[-1 -2 -1 -1 -2 2 0]]
Primitive based matrix	[[ 0 2 1 0 0 -1 -2],[-2 0 -2 0 -1 -2 -3],[-1 2 0 -1 -1 -2 -2],[ 0 0 1 0 -1 0 0],[ 0 1 1 1 0 0 -2],[ 1 2 2 0 0 0 0],[ 2 3 2 0 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,0,1,2,2,0,1,2,3,1,1,2,2,1,0,0,0,2,0]
Phi over symmetry	[-2,-1,0,0,1,2,-1,1,2,1,1,0,0,0,1,1,1,0,1,2,1]
Phi of -K	[-2,-1,0,0,1,2,1,0,2,1,1,1,1,0,1,-1,0,1,0,2,-1]
Phi of K*	[-2,-1,0,0,1,2,-1,1,2,1,1,0,0,0,1,1,1,0,1,2,1]
Phi of -K*	[-2,-1,0,0,1,2,0,0,2,2,3,0,0,2,2,-1,1,0,1,1,2]
Symmetry type of based matrix	c
u-polynomial	0
Normalized Jones-Krushkal polynomial	2z^2+21z+35
Enhanced Jones-Krushkal polynomial	2w^3z^2+21w^2z+35w
Inner characteristic polynomial	t^6+33t^4+116t^2+4
Outer characteristic polynomial	t^7+43t^5+160t^3+9t
Flat arrow polynomial	4K13 - 12K1*2 - 4K1K2 - K1 + 6K2 + K3 + 7
2-strand cable arrow polynomial	-192K14K2*2 + 416K1*4K2 - 2304K14 + 416K1*3K2K3 - 384K1*3K3 - 320K12K2*4 + 768K1*2K2*3 + 64K1*2K2*2K4 - 6432K12K2*2 + 32K1*2K2K32 - 416K1*2K2K4 + 9648K1*2K2 - 256K12K3*2 - 5940K1*2 + 512K1K23K3 - 928K1K2*2K3 - 320K1K2*2K5 - 32K1K2K3K4 + 6824K1K2K3 + 592K1K3K4 + 48K1K4K5 - 32K2*6 + 64K2*4K4 - 1344K24 - 32K2*3K6 - 240K22K3*2 - 16K2*2K4*2 + 1304K2*2K4 - 4038K22 + 216K2K3K5 + 16K2K4K6 - 1732K3*2 - 328K4*2 - 40K5*2 - 2K6**2 + 4406
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice	False

Contact