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

Flat knot 6.462

Min(phi) over symmetries of the knot is: [-3,-2,0,1,1,3,0,3,1,2,3,2,0,1,2,1,0,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.462']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 4K12 - 8K1K2 - 2K1K3 - 2K1 - 2K22 + K2 + 2K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.462']
Outer characteristic polynomial of the knot is: t^7+76t^5+96t^3+15t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.462']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 96K1*4K2 - 240K14 + 128K1*3K2*3K3 + 640K13K2K3 - 64K1*3K3 - 1088K12K2*4 - 128K1*2K2*3K4 + 2112K12K2*3 - 320K1*2K2*2K3*2 + 160K1*2K2*2K4 - 7616K12K2*2 + 64K1*2K2K32 - 1216K1*2K2K4 + 5448K1*2K2 - 512K12K3*2 - 80K1*2K4*2 - 3884K1*2 - 384K1K24K3 - 128K1K2*4K5 + 2528K1K2*3K3 + 704K1K2*2K3K4 - 1632K1K22K3 + 96K1K2*2K4K5 - 352K1K22K5 + 128K1K2K33 - 416K1K2K3K4 - 64K1K2K3K6 + 7520K1K2K3 - 128K1K2K4K5 - 32K1K2K4K7 + 1816K1K3K4 + 248K1K4K5 + 8K1K5K6 - 192K2*6 - 128K2*4K3*2 - 32K2*4K4*2 + 576K2*4K4 - 2368K24 + 160K2*3K3K5 + 32K2*3K4K6 + 128K2*2K3*2K4 - 1584K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 608K2*2K4*2 - 32K2*2K4K8 + 2216K2*2K4 - 80K22K5*2 - 8K2*2K6*2 - 2136K2*2 - 96K2K32K4 - 32K2K3K4K5 + 792K2K3K5 + 248K2K4K6 + 24K2K5K7 + 8K2K6K8 - 16K34 - 32K3*2K4*2 + 16K3*2K6 - 2124K32 + 24K3K4K7 - 8K44 + 8K4*2K8 - 1030K42 - 172K5*2 - 16K6*2 - 4K7*2 - 2K8**2 + 3310
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.462']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4719', 'vk6.5036', 'vk6.6244', 'vk6.6696', 'vk6.8220', 'vk6.8660', 'vk6.9602', 'vk6.9929', 'vk6.20292', 'vk6.21627', 'vk6.27588', 'vk6.29142', 'vk6.39006', 'vk6.41256', 'vk6.45774', 'vk6.47453', 'vk6.48759', 'vk6.48962', 'vk6.49560', 'vk6.49774', 'vk6.50773', 'vk6.50981', 'vk6.51254', 'vk6.51459', 'vk6.57151', 'vk6.58337', 'vk6.61777', 'vk6.62898', 'vk6.66772', 'vk6.67650', 'vk6.69420', 'vk6.70144']
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,-2,0,1,1,3,0,3,1,2,3,2,0,1,2,1,0,2,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.462']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.462']

Outer characteristic polynomial of the knot is: t^7+76t^5+96t^3+15t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.462']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 240*K1**4 + 128*K1**3*K2**3*K3 + 640*K1**3*K2*K3 - 64*K1**3*K3 - 1088*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2112*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 + 160*K1**2*K2**2*K4 - 7616*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 1216*K1**2*K2*K4 + 5448*K1**2*K2 - 512*K1**2*K3**2 - 80*K1**2*K4**2 - 3884*K1**2 - 384*K1*K2**4*K3 - 128*K1*K2**4*K5 + 2528*K1*K2**3*K3 + 704*K1*K2**2*K3*K4 - 1632*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 352*K1*K2**2*K5 + 128*K1*K2*K3**3 - 416*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7520*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1816*K1*K3*K4 + 248*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 576*K2**4*K4 - 2368*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 + 128*K2**2*K3**2*K4 - 1584*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 608*K2**2*K4**2 - 32*K2**2*K4*K8 + 2216*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 2136*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 792*K2*K3*K5 + 248*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 16*K3**2*K6 - 2124*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1030*K4**2 - 172*K5**2 - 16*K6**2 - 4*K7**2 - 2*K8**2 + 3310

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.462']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4719', 'vk6.5036', 'vk6.6244', 'vk6.6696', 'vk6.8220', 'vk6.8660', 'vk6.9602', 'vk6.9929', 'vk6.20292', 'vk6.21627', 'vk6.27588', 'vk6.29142', 'vk6.39006', 'vk6.41256', 'vk6.45774', 'vk6.47453', 'vk6.48759', 'vk6.48962', 'vk6.49560', 'vk6.49774', 'vk6.50773', 'vk6.50981', 'vk6.51254', 'vk6.51459', 'vk6.57151', 'vk6.58337', 'vk6.61777', 'vk6.62898', 'vk6.66772', 'vk6.67650', 'vk6.69420', 'vk6.70144']

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	O1O2O3O4O5U6U4O6U1U2U5U3
R3 orbit	{'O1O2O3O4O5U6U4O6U1U2U5U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U1U4U5O6U2U6
Gauss code of K*	O1O2O3O4U1U2U4U5U3O6O5U6
Gauss code of -K*	O1O2O3O4U5O6O5U2U6U1U3U4
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 2 0 3 -1],[ 3 0 1 3 1 3 2],[ 1 -1 0 2 1 2 0],[-2 -3 -2 0 0 1 -3],[ 0 -1 -1 0 0 0 0],[-3 -3 -2 -1 0 0 -3],[ 1 -2 0 3 0 3 0]]
Primitive based matrix	[[ 0 3 2 0 -1 -1 -3],[-3 0 -1 0 -2 -3 -3],[-2 1 0 0 -2 -3 -3],[ 0 0 0 0 -1 0 -1],[ 1 2 2 1 0 0 -1],[ 1 3 3 0 0 0 -2],[ 3 3 3 1 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,0,1,1,3,1,0,2,3,3,0,2,3,3,1,0,1,0,1,2]
Phi over symmetry	[-3,-2,0,1,1,3,0,3,1,2,3,2,0,1,2,1,0,2,0,0,1]
Phi of -K	[-3,-1,-1,0,2,3,0,1,2,2,3,0,1,0,1,0,1,2,2,3,0]
Phi of K*	[-3,-2,0,1,1,3,0,3,1,2,3,2,0,1,2,1,0,2,0,0,1]
Phi of -K*	[-3,-1,-1,0,2,3,1,2,1,3,3,0,1,2,2,0,3,3,0,0,1]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	4w^3z^2-8w^3z+25w^2z+19w
Inner characteristic polynomial	t^6+52t^4+51t^2+4
Outer characteristic polynomial	t^7+76t^5+96t^3+15t
Flat arrow polynomial	8K13 + 4K1*2K2 - 4K12 - 8K1K2 - 2K1K3 - 2K1 - 2K22 + K2 + 2K3 + K4 + 3
2-strand cable arrow polynomial	-256K14K2*2 + 96K1*4K2 - 240K14 + 128K1*3K2*3K3 + 640K13K2K3 - 64K1*3K3 - 1088K12K2*4 - 128K1*2K2*3K4 + 2112K12K2*3 - 320K1*2K2*2K3*2 + 160K1*2K2*2K4 - 7616K12K2*2 + 64K1*2K2K32 - 1216K1*2K2K4 + 5448K1*2K2 - 512K12K3*2 - 80K1*2K4*2 - 3884K1*2 - 384K1K24K3 - 128K1K2*4K5 + 2528K1K2*3K3 + 704K1K2*2K3K4 - 1632K1K22K3 + 96K1K2*2K4K5 - 352K1K22K5 + 128K1K2K33 - 416K1K2K3K4 - 64K1K2K3K6 + 7520K1K2K3 - 128K1K2K4K5 - 32K1K2K4K7 + 1816K1K3K4 + 248K1K4K5 + 8K1K5K6 - 192K2*6 - 128K2*4K3*2 - 32K2*4K4*2 + 576K2*4K4 - 2368K24 + 160K2*3K3K5 + 32K2*3K4K6 + 128K2*2K3*2K4 - 1584K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 608K2*2K4*2 - 32K2*2K4K8 + 2216K2*2K4 - 80K22K5*2 - 8K2*2K6*2 - 2136K2*2 - 96K2K32K4 - 32K2K3K4K5 + 792K2K3K5 + 248K2K4K6 + 24K2K5K7 + 8K2K6K8 - 16K34 - 32K3*2K4*2 + 16K3*2K6 - 2124K32 + 24K3K4K7 - 8K44 + 8K4*2K8 - 1030K42 - 172K5*2 - 16K6*2 - 4K7*2 - 2K8**2 + 3310
Genus of based matrix	2
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{3, 6}, {5}, {4}, {2}, {1}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{5, 6}, {4}, {3}, {2}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

Contact