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

Flat knot 6.116

Min(phi) over symmetries of the knot is: [-3,0,1,2,1,3,3,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.116', '6.458', '6.1180', '7.28052']
Arrow polynomial of the knot is: 12K13 + 4K1*2K2 - 12K12 - 10K1K2 - 2K1K3 - 4K1 - 2K22 + 5K2 + 2*K3 + K4 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.116']
Outer characteristic polynomial of the knot is: t^5+35t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.116', '6.458']
2-strand cable arrow polynomial of the knot is: -64K16 + 960K1*4K2 - 3936K14 + 576K1*3K2K3 - 1088K1*3K3 - 128K12K2*4 + 640K1*2K2*3 + 384K1*2K2*2K4 - 7472K12K2*2 + 128K1*2K2K32 + 64K1*2K2K42 - 1216K1*2K2K4 + 13024K1*2K2 - 1472K12K3*2 - 96K1*2K3K5 - 368K1*2K4*2 - 9300K1*2 + 672K1K23K3 + 128K1K2*2K3K4 - 2240K1K22K3 + 64K1K2*2K4K5 - 512K1K22K5 + 160K1K2K33 + 64K1K2K3K42 - 928K1K2K3K4 - 64K1K2K3K6 + 12800K1K2K3 + 3472K1K3K4 + 736K1K4K5 + 64K1K5K6 + 24K1K6K7 - 96K26 - 64K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 1488K24 + 64K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 + 64K22K3*2K4 - 1264K22K3*2 + 32K2*2K4*3 - 440K2*2K4*2 + 3240K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 7996K2*2 - 96K2K32K4 - 96K2K3K4K5 + 1544K2K3K5 - 32K2K42K6 + 304K2K4K6 + 24K2K5K7 + 8K2K6K8 - 256K3*4 - 112K3*2K4*2 + 184K3*2K6 - 4524K32 + 128K3K4K7 - 8K44 + 8K4*2K8 - 1902K42 - 560K5*2 - 148K6*2 - 48K7*2 - 2K8**2 + 8366
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.116']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11529', 'vk6.11860', 'vk6.12875', 'vk6.13182', 'vk6.20363', 'vk6.21705', 'vk6.27664', 'vk6.29209', 'vk6.31300', 'vk6.31695', 'vk6.32454', 'vk6.32869', 'vk6.39099', 'vk6.41353', 'vk6.45851', 'vk6.47514', 'vk6.52300', 'vk6.52564', 'vk6.53140', 'vk6.53444', 'vk6.57222', 'vk6.58445', 'vk6.61833', 'vk6.62965', 'vk6.63805', 'vk6.63937', 'vk6.64247', 'vk6.64443', 'vk6.66829', 'vk6.67698', 'vk6.69466', 'vk6.70189']
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: [-3,0,1,2,1,3,3,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.116', '6.458', '6.1180', '7.28052']

Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 12*K1**2 - 10*K1*K2 - 2*K1*K3 - 4*K1 - 2*K2**2 + 5*K2 + 2*K3 + K4 + 7

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.116']

Outer characteristic polynomial of the knot is: t^5+35t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.116', '6.458']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 960*K1**4*K2 - 3936*K1**4 + 576*K1**3*K2*K3 - 1088*K1**3*K3 - 128*K1**2*K2**4 + 640*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 7472*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1216*K1**2*K2*K4 + 13024*K1**2*K2 - 1472*K1**2*K3**2 - 96*K1**2*K3*K5 - 368*K1**2*K4**2 - 9300*K1**2 + 672*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 2240*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 512*K1*K2**2*K5 + 160*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 928*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 12800*K1*K2*K3 + 3472*K1*K3*K4 + 736*K1*K4*K5 + 64*K1*K5*K6 + 24*K1*K6*K7 - 96*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1488*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 1264*K2**2*K3**2 + 32*K2**2*K4**3 - 440*K2**2*K4**2 + 3240*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 7996*K2**2 - 96*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 1544*K2*K3*K5 - 32*K2*K4**2*K6 + 304*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 256*K3**4 - 112*K3**2*K4**2 + 184*K3**2*K6 - 4524*K3**2 + 128*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1902*K4**2 - 560*K5**2 - 148*K6**2 - 48*K7**2 - 2*K8**2 + 8366

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.116']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11529', 'vk6.11860', 'vk6.12875', 'vk6.13182', 'vk6.20363', 'vk6.21705', 'vk6.27664', 'vk6.29209', 'vk6.31300', 'vk6.31695', 'vk6.32454', 'vk6.32869', 'vk6.39099', 'vk6.41353', 'vk6.45851', 'vk6.47514', 'vk6.52300', 'vk6.52564', 'vk6.53140', 'vk6.53444', 'vk6.57222', 'vk6.58445', 'vk6.61833', 'vk6.62965', 'vk6.63805', 'vk6.63937', 'vk6.64247', 'vk6.64443', 'vk6.66829', 'vk6.67698', 'vk6.69466', 'vk6.70189']

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	O1O2O3O4O5O6U3U5U1U6U4U2
R3 orbit	{'O1O2O3O4O5O6U3U5U1U6U4U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U5U3U1U6U2U4
Gauss code of K*	O1O2O3O4O5O6U3U6U1U5U2U4
Gauss code of -K*	O1O2O3O4O5O6U3U5U2U6U1U4
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 -3 1 -3 2 0 3],[ 3 0 3 -1 3 1 3],[-1 -3 0 -3 1 0 2],[ 3 1 3 0 3 1 2],[-2 -3 -1 -3 0 -1 1],[ 0 -1 0 -1 1 0 1],[-3 -3 -2 -2 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 0 -3],[-2 0 -1 -1 -3],[-1 1 0 0 -3],[ 0 1 0 0 -1],[ 3 3 3 1 0]]
If based matrix primitive	False
Phi of primitive based matrix	[-2,-1,0,3,1,1,3,0,3,1]
Phi over symmetry	[-3,0,1,2,1,3,3,0,1,1]
Phi of -K	[-3,0,1,2,2,1,2,1,1,0]
Phi of K*	[-2,-1,0,3,0,1,2,1,1,2]
Phi of -K*	[-3,0,1,2,1,3,3,0,1,1]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	4z^2+25z+35
Enhanced Jones-Krushkal polynomial	4w^3z^2+25w^2z+35w
Inner characteristic polynomial	t^4+21t^2+4
Outer characteristic polynomial	t^5+35t^3+7t
Flat arrow polynomial	12K13 + 4K1*2K2 - 12K12 - 10K1K2 - 2K1K3 - 4K1 - 2K22 + 5K2 + 2*K3 + K4 + 7
2-strand cable arrow polynomial	-64K16 + 960K1*4K2 - 3936K14 + 576K1*3K2K3 - 1088K1*3K3 - 128K12K2*4 + 640K1*2K2*3 + 384K1*2K2*2K4 - 7472K12K2*2 + 128K1*2K2K32 + 64K1*2K2K42 - 1216K1*2K2K4 + 13024K1*2K2 - 1472K12K3*2 - 96K1*2K3K5 - 368K1*2K4*2 - 9300K1*2 + 672K1K23K3 + 128K1K2*2K3K4 - 2240K1K22K3 + 64K1K2*2K4K5 - 512K1K22K5 + 160K1K2K33 + 64K1K2K3K42 - 928K1K2K3K4 - 64K1K2K3K6 + 12800K1K2K3 + 3472K1K3K4 + 736K1K4K5 + 64K1K5K6 + 24K1K6K7 - 96K26 - 64K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 1488K24 + 64K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 + 64K22K3*2K4 - 1264K22K3*2 + 32K2*2K4*3 - 440K2*2K4*2 + 3240K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 7996K2*2 - 96K2K32K4 - 96K2K3K4K5 + 1544K2K3K5 - 32K2K42K6 + 304K2K4K6 + 24K2K5K7 + 8K2K6K8 - 256K3*4 - 112K3*2K4*2 + 184K3*2K6 - 4524K32 + 128K3K4K7 - 8K44 + 8K4*2K8 - 1902K42 - 560K5*2 - 148K6*2 - 48K7*2 - 2K8**2 + 8366
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}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice	False

Contact