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

Flat knot 6.504

Min(phi) over symmetries of the knot is: [-4,-1,-1,0,3,3,0,1,3,3,4,0,1,1,1,0,1,2,2,3,0]
Flat knots (up to 7 crossings) with same phi are :['6.504']
Arrow polynomial of the knot is: -2K12 - 4K1K2 - 2K1K3 + 2K1 + 2K2 + 2K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.180', '6.263', '6.295', '6.317', '6.350', '6.473', '6.504']
Outer characteristic polynomial of the knot is: t^7+108t^5+166t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.504']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 480K1*4K2 - 1296K14 + 512K1*3K2K3 - 320K1*3K3 + 160K12K2*3 - 3024K1*2K2*2 - 544K1*2K2K4 + 4248K1*2K2 - 272K12K3*2 - 64K1*2K4*2 - 2708K1*2 + 64K1K23K3 + 128K1K2*2K3K4 - 768K1K22K3 + 64K1K2*2K4K5 - 352K1K22K5 - 96K1K2K3K4 - 64K1K2K3K6 + 4008K1K2K3 - 32K1K2K4K5 + 1032K1K3K4 + 344K1K4K5 + 16K1K5K6 - 72K24 - 160K2*2K3*2 - 240K2*2K4*2 + 1080K2*2K4 - 112K22K5*2 - 8K2*2K6*2 - 2688K2*2 + 536K2K3K5 + 144K2K4K6 + 48K2K5K7 + 8K2K6K8 + 8K3*2K6 - 1312K32 - 720K4*2 - 264K5*2 - 24K6*2 - 4K7*2 - 2K8**2 + 2520
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.504']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16929', 'vk6.17171', 'vk6.20221', 'vk6.21515', 'vk6.23325', 'vk6.23619', 'vk6.27421', 'vk6.29031', 'vk6.35366', 'vk6.35788', 'vk6.38834', 'vk6.41027', 'vk6.42843', 'vk6.43122', 'vk6.45599', 'vk6.47358', 'vk6.55083', 'vk6.55334', 'vk6.57060', 'vk6.58184', 'vk6.59478', 'vk6.59768', 'vk6.61581', 'vk6.62753', 'vk6.64928', 'vk6.65135', 'vk6.66682', 'vk6.67520', 'vk6.68223', 'vk6.68365', 'vk6.69333', 'vk6.70084']
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,-1,-1,0,3,3,0,1,3,3,4,0,1,1,1,0,1,2,2,3,0]

Flat knots (up to 7 crossings) with same phi are :['6.504']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.180', '6.263', '6.295', '6.317', '6.350', '6.473', '6.504']

Outer characteristic polynomial of the knot is: t^7+108t^5+166t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.504']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 480*K1**4*K2 - 1296*K1**4 + 512*K1**3*K2*K3 - 320*K1**3*K3 + 160*K1**2*K2**3 - 3024*K1**2*K2**2 - 544*K1**2*K2*K4 + 4248*K1**2*K2 - 272*K1**2*K3**2 - 64*K1**2*K4**2 - 2708*K1**2 + 64*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 - 96*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 4008*K1*K2*K3 - 32*K1*K2*K4*K5 + 1032*K1*K3*K4 + 344*K1*K4*K5 + 16*K1*K5*K6 - 72*K2**4 - 160*K2**2*K3**2 - 240*K2**2*K4**2 + 1080*K2**2*K4 - 112*K2**2*K5**2 - 8*K2**2*K6**2 - 2688*K2**2 + 536*K2*K3*K5 + 144*K2*K4*K6 + 48*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 1312*K3**2 - 720*K4**2 - 264*K5**2 - 24*K6**2 - 4*K7**2 - 2*K8**2 + 2520

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.504']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16929', 'vk6.17171', 'vk6.20221', 'vk6.21515', 'vk6.23325', 'vk6.23619', 'vk6.27421', 'vk6.29031', 'vk6.35366', 'vk6.35788', 'vk6.38834', 'vk6.41027', 'vk6.42843', 'vk6.43122', 'vk6.45599', 'vk6.47358', 'vk6.55083', 'vk6.55334', 'vk6.57060', 'vk6.58184', 'vk6.59478', 'vk6.59768', 'vk6.61581', 'vk6.62753', 'vk6.64928', 'vk6.65135', 'vk6.66682', 'vk6.67520', 'vk6.68223', 'vk6.68365', 'vk6.69333', 'vk6.70084']

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	O1O2O3O4O5U1U6U3O6U2U5U4
R3 orbit	{'O1O2O3O4O5U1U6U3O6U2U5U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U1U4O6U3U6U5
Gauss code of K*	O1O2O3U2O4O5O6U1U4U3U6U5
Gauss code of -K*	O1O2O3U4O5O4O6U2U1U5U3U6
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 0 3 3 -1],[ 4 0 2 1 4 3 3],[ 1 -2 0 1 3 2 0],[ 0 -1 -1 0 1 0 0],[-3 -4 -3 -1 0 0 -3],[-3 -3 -2 0 0 0 -3],[ 1 -3 0 0 3 3 0]]
Primitive based matrix	[[ 0 3 3 0 -1 -1 -4],[-3 0 0 0 -2 -3 -3],[-3 0 0 -1 -3 -3 -4],[ 0 0 1 0 -1 0 -1],[ 1 2 3 1 0 0 -2],[ 1 3 3 0 0 0 -3],[ 4 3 4 1 2 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,0,1,1,4,0,0,2,3,3,1,3,3,4,1,0,1,0,2,3]
Phi over symmetry	[-4,-1,-1,0,3,3,0,1,3,3,4,0,1,1,1,0,1,2,2,3,0]
Phi of -K	[-4,-1,-1,0,3,3,0,1,3,3,4,0,1,1,1,0,1,2,2,3,0]
Phi of K*	[-3,-3,0,1,1,4,0,2,1,1,3,3,1,2,4,1,0,3,0,0,1]
Phi of -K*	[-4,-1,-1,0,3,3,2,3,1,3,4,0,1,2,3,0,3,3,0,1,0]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^3+2t
Normalized Jones-Krushkal polynomial	5z^2+22z+25
Enhanced Jones-Krushkal polynomial	5w^3z^2+22w^2z+25w
Inner characteristic polynomial	t^6+72t^4+96t^2
Outer characteristic polynomial	t^7+108t^5+166t^3+4t
Flat arrow polynomial	-2K12 - 4K1K2 - 2K1K3 + 2K1 + 2K2 + 2K3 + K4 + 2
2-strand cable arrow polynomial	-256K14K2*2 + 480K1*4K2 - 1296K14 + 512K1*3K2K3 - 320K1*3K3 + 160K12K2*3 - 3024K1*2K2*2 - 544K1*2K2K4 + 4248K1*2K2 - 272K12K3*2 - 64K1*2K4*2 - 2708K1*2 + 64K1K23K3 + 128K1K2*2K3K4 - 768K1K22K3 + 64K1K2*2K4K5 - 352K1K22K5 - 96K1K2K3K4 - 64K1K2K3K6 + 4008K1K2K3 - 32K1K2K4K5 + 1032K1K3K4 + 344K1K4K5 + 16K1K5K6 - 72K24 - 160K2*2K3*2 - 240K2*2K4*2 + 1080K2*2K4 - 112K22K5*2 - 8K2*2K6*2 - 2688K2*2 + 536K2K3K5 + 144K2K4K6 + 48K2K5K7 + 8K2K6K8 + 8K3*2K6 - 1312K32 - 720K4*2 - 264K5*2 - 24K6*2 - 4K7*2 - 2K8**2 + 2520
Genus of based matrix	1
Fillings of based matrix	[[{3, 6}, {2, 5}, {1, 4}]]
If K is slice	False

Contact