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

Flat knot 6.1578

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,2,1,0,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1578']
Arrow polynomial of the knot is: 4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845']
Outer characteristic polynomial of the knot is: t^7+24t^5+31t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1578']
2-strand cable arrow polynomial of the knot is: -128K16 + 768K1*4K2*3 - 1536K1*4K2*2 + 1600K1*4K2 - 2224K14 - 384K1*3K2*2K3 + 736K13K2K3 - 608K1*3K3 + 384K12K2*5 - 1984K1*2K2*4 + 3872K1*2K2*3 - 7072K1*2K2*2 + 64K1*2K2K32 - 672K1*2K2K4 + 6800K1*2K2 - 112K12K3*2 - 3628K1*2 - 384K1K24K3 + 1824K1K2*3K3 + 96K1K2*2K3K4 - 1056K1K22K3 - 32K1K2*2K5 - 160K1K2K3K4 + 4848K1K2K3 + 552K1K3K4 - 288K26 + 352K2*4K4 - 1864K24 - 32K2*3K6 - 624K22K3*2 - 112K2*2K4*2 + 960K2*2K4 - 1590K22 + 192K2K3K5 + 24K2K4K6 - 1116K3*2 - 282K4*2 - 16K5*2 - 2K6**2 + 2832
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1578']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14092', 'vk6.14099', 'vk6.14289', 'vk6.14302', 'vk6.15520', 'vk6.15531', 'vk6.16016', 'vk6.16021', 'vk6.16266', 'vk6.16271', 'vk6.16277', 'vk6.22578', 'vk6.22580', 'vk6.34048', 'vk6.34093', 'vk6.34107', 'vk6.34488', 'vk6.34528', 'vk6.34538', 'vk6.34553', 'vk6.34567', 'vk6.42258', 'vk6.54061', 'vk6.54066', 'vk6.54282', 'vk6.54511', 'vk6.54549', 'vk6.54565', 'vk6.59008', 'vk6.64511', 'vk6.64522', 'vk6.64621']
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,-1,0,1,1,1,0,1,0,1,2,1,1,1,2,1,0,0,-1,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1578']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845']

Outer characteristic polynomial of the knot is: t^7+24t^5+31t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1578']

2-strand cable arrow polynomial of the knot is: -128*K1**6 + 768*K1**4*K2**3 - 1536*K1**4*K2**2 + 1600*K1**4*K2 - 2224*K1**4 - 384*K1**3*K2**2*K3 + 736*K1**3*K2*K3 - 608*K1**3*K3 + 384*K1**2*K2**5 - 1984*K1**2*K2**4 + 3872*K1**2*K2**3 - 7072*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 6800*K1**2*K2 - 112*K1**2*K3**2 - 3628*K1**2 - 384*K1*K2**4*K3 + 1824*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 32*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4848*K1*K2*K3 + 552*K1*K3*K4 - 288*K2**6 + 352*K2**4*K4 - 1864*K2**4 - 32*K2**3*K6 - 624*K2**2*K3**2 - 112*K2**2*K4**2 + 960*K2**2*K4 - 1590*K2**2 + 192*K2*K3*K5 + 24*K2*K4*K6 - 1116*K3**2 - 282*K4**2 - 16*K5**2 - 2*K6**2 + 2832

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1578']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.14092', 'vk6.14099', 'vk6.14289', 'vk6.14302', 'vk6.15520', 'vk6.15531', 'vk6.16016', 'vk6.16021', 'vk6.16266', 'vk6.16271', 'vk6.16277', 'vk6.22578', 'vk6.22580', 'vk6.34048', 'vk6.34093', 'vk6.34107', 'vk6.34488', 'vk6.34528', 'vk6.34538', 'vk6.34553', 'vk6.34567', 'vk6.42258', 'vk6.54061', 'vk6.54066', 'vk6.54282', 'vk6.54511', 'vk6.54549', 'vk6.54565', 'vk6.59008', 'vk6.64511', 'vk6.64522', 'vk6.64621']

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	O1O2O3U4O5U2U3O6O4U6U1U5
R3 orbit	{'O1O2O3U4O5U2U3O6O4U6U1U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U3U5O6O5U1U2O4U6
Gauss code of K*	O1O2O3U2U4U5O6U3O4O5U1U6
Gauss code of -K*	O1O2O3U4U3O5O6U1O4U5U6U2
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 -1 1 0 2 -1],[ 1 0 0 2 0 2 -1],[ 1 0 0 1 0 1 -1],[-1 -2 -1 0 -1 0 -1],[ 0 0 0 1 0 1 -1],[-2 -2 -1 0 -1 0 0],[ 1 1 1 1 1 0 0]]
Primitive based matrix	[[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 0 -1 -2],[-1 0 0 -1 -1 -1 -2],[ 0 1 1 0 -1 0 0],[ 1 0 1 1 0 1 1],[ 1 1 1 0 -1 0 0],[ 1 2 2 0 -1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,2,1,0,0,-1,-1,0]
Phi over symmetry	[-2,-1,0,1,1,1,0,1,0,1,2,1,1,1,2,1,0,0,-1,-1,0]
Phi of -K	[-1,-1,-1,0,1,2,-1,-1,0,1,3,0,1,0,1,1,1,2,0,1,1]
Phi of K*	[-2,-1,0,1,1,1,1,1,1,2,3,0,0,1,1,1,1,0,0,-1,-1]
Phi of -K*	[-1,-1,-1,0,1,2,-1,0,0,1,1,1,1,1,0,0,2,2,1,1,0]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	2z^2+15z+23
Enhanced Jones-Krushkal polynomial	-2w^4z^2+4w^3z^2-6w^3z+21w^2z+23w
Inner characteristic polynomial	t^6+16t^4+10t^2
Outer characteristic polynomial	t^7+24t^5+31t^3+7t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-128K16 + 768K1*4K2*3 - 1536K1*4K2*2 + 1600K1*4K2 - 2224K14 - 384K1*3K2*2K3 + 736K13K2K3 - 608K1*3K3 + 384K12K2*5 - 1984K1*2K2*4 + 3872K1*2K2*3 - 7072K1*2K2*2 + 64K1*2K2K32 - 672K1*2K2K4 + 6800K1*2K2 - 112K12K3*2 - 3628K1*2 - 384K1K24K3 + 1824K1K2*3K3 + 96K1K2*2K3K4 - 1056K1K22K3 - 32K1K2*2K5 - 160K1K2K3K4 + 4848K1K2K3 + 552K1K3K4 - 288K26 + 352K2*4K4 - 1864K24 - 32K2*3K6 - 624K22K3*2 - 112K2*2K4*2 + 960K2*2K4 - 1590K22 + 192K2K3K5 + 24K2K4K6 - 1116K3*2 - 282K4*2 - 16K5*2 - 2K6**2 + 2832
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}], [{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}, {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}, {4}, {1, 3}, {2}]]
If K is slice	False

Contact