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

Flat knot 6.323

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,2,2,2,4,2,1,1,2,0,1,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.323']
Arrow polynomial of the knot is: -6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']
Outer characteristic polynomial of the knot is: t^7+63t^5+37t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.323']
2-strand cable arrow polynomial of the knot is: -80K14 + 32K1*3K2K3 - 128K1*3K3 - 1568K12K2*2 + 32K1*2K2K32 - 224K1*2K2K4 + 2656K1*2K2 - 144K12K3*2 - 48K1*2K4*2 - 2712K1*2 + 96K1K23K3 - 192K1K2*2K3 - 64K1K2*2K5 - 32K1K2K3K4 + 2808K1K2K3 + 560K1K3K4 + 88K1K4K5 - 392K2*4 - 96K2*2K3*2 - 8K2*2K4*2 + 616K2*2K4 - 1846K22 + 104K2K3K5 + 8K2K4K6 - 1036K3*2 - 370K4*2 - 52K5*2 - 2K6**2 + 1992
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.323']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71454', 'vk6.71484', 'vk6.71487', 'vk6.71505', 'vk6.71531', 'vk6.71543', 'vk6.71548', 'vk6.71980', 'vk6.72006', 'vk6.72035', 'vk6.72057', 'vk6.72522', 'vk6.72544', 'vk6.72643', 'vk6.72665', 'vk6.72915', 'vk6.72953', 'vk6.73115', 'vk6.73139', 'vk6.77083', 'vk6.77108', 'vk6.77112', 'vk6.77131', 'vk6.77148', 'vk6.77162', 'vk6.77165', 'vk6.77453', 'vk6.77458', 'vk6.81298', 'vk6.81424', 'vk6.81545', 'vk6.81553', 'vk6.85468', 'vk6.85475', 'vk6.86883', 'vk6.86897', 'vk6.87255', 'vk6.87735', 'vk6.89353', 'vk6.89509']
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,-2,0,1,2,2,-1,2,2,2,4,2,1,1,2,0,1,1,1,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.323']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']

Outer characteristic polynomial of the knot is: t^7+63t^5+37t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.323']

2-strand cable arrow polynomial of the knot is: -80*K1**4 + 32*K1**3*K2*K3 - 128*K1**3*K3 - 1568*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 224*K1**2*K2*K4 + 2656*K1**2*K2 - 144*K1**2*K3**2 - 48*K1**2*K4**2 - 2712*K1**2 + 96*K1*K2**3*K3 - 192*K1*K2**2*K3 - 64*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 2808*K1*K2*K3 + 560*K1*K3*K4 + 88*K1*K4*K5 - 392*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 616*K2**2*K4 - 1846*K2**2 + 104*K2*K3*K5 + 8*K2*K4*K6 - 1036*K3**2 - 370*K4**2 - 52*K5**2 - 2*K6**2 + 1992

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.323']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71454', 'vk6.71484', 'vk6.71487', 'vk6.71505', 'vk6.71531', 'vk6.71543', 'vk6.71548', 'vk6.71980', 'vk6.72006', 'vk6.72035', 'vk6.72057', 'vk6.72522', 'vk6.72544', 'vk6.72643', 'vk6.72665', 'vk6.72915', 'vk6.72953', 'vk6.73115', 'vk6.73139', 'vk6.77083', 'vk6.77108', 'vk6.77112', 'vk6.77131', 'vk6.77148', 'vk6.77162', 'vk6.77165', 'vk6.77453', 'vk6.77458', 'vk6.81298', 'vk6.81424', 'vk6.81545', 'vk6.81553', 'vk6.85468', 'vk6.85475', 'vk6.86883', 'vk6.86897', 'vk6.87255', 'vk6.87735', 'vk6.89353', 'vk6.89509']

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	O1O2O3O4O5U2U4O6U5U1U6U3
R3 orbit	{'O1O2O3O4O5U2U4O6U5U1U6U3', 'O1O2O3O4U1O5U4O6U2U5U6U3'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U3U6U5U1O6U2U4
Gauss code of K*	O1O2O3O4U2U5U4U6U1O5O6U3
Gauss code of -K*	O1O2O3O4U2O5O6U4U5U1U6U3
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 -2 -3 2 0 1 2],[ 2 0 -2 3 0 2 2],[ 3 2 0 3 1 2 1],[-2 -3 -3 0 -1 0 1],[ 0 0 -1 1 0 1 1],[-1 -2 -2 0 -1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 2 1 0 -2 -3],[-2 0 1 0 -1 -3 -3],[-2 -1 0 -1 -1 -2 -1],[-1 0 1 0 -1 -2 -2],[ 0 1 1 1 0 0 -1],[ 2 3 2 2 0 0 -2],[ 3 3 1 2 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,2,3,-1,0,1,3,3,1,1,2,1,1,2,2,0,1,2]
Phi over symmetry	[-3,-2,0,1,2,2,-1,2,2,2,4,2,1,1,2,0,1,1,1,0,-1]
Phi of -K	[-3,-2,0,1,2,2,-1,2,2,2,4,2,1,1,2,0,1,1,1,0,-1]
Phi of K*	[-2,-2,-1,0,2,3,-1,0,1,2,4,1,1,1,2,0,1,2,2,2,-1]
Phi of -K*	[-3,-2,0,1,2,2,2,1,2,1,3,0,2,2,3,1,1,1,1,0,-1]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	4w^3z^2+17w^2z+19w
Inner characteristic polynomial	t^6+41t^4+8t^2
Outer characteristic polynomial	t^7+63t^5+37t^3+3t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-80K14 + 32K1*3K2K3 - 128K1*3K3 - 1568K12K2*2 + 32K1*2K2K32 - 224K1*2K2K4 + 2656K1*2K2 - 144K12K3*2 - 48K1*2K4*2 - 2712K1*2 + 96K1K23K3 - 192K1K2*2K3 - 64K1K2*2K5 - 32K1K2K3K4 + 2808K1K2K3 + 560K1K3K4 + 88K1K4K5 - 392K2*4 - 96K2*2K3*2 - 8K2*2K4*2 + 616K2*2K4 - 1846K22 + 104K2K3K5 + 8K2K4K6 - 1036K3*2 - 370K4*2 - 52K5*2 - 2K6**2 + 1992
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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}], [{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}, {2, 5}, {4}, {3}, {1}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{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}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

Contact