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

Flat knot 6.5

Min(phi) over symmetries of the knot is: [-5,-3,0,2,2,4,1,2,3,5,4,1,2,4,3,1,2,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.5']
Arrow polynomial of the knot is: -8K14 - 8K1*3K2 + 12K13 + 4K1*2K2 + 4K12K3 + 4K12 - 4K1K2 - 4K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.5']
Outer characteristic polynomial of the knot is: t^7+157t^5+172t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.5']
2-strand cable arrow polynomial of the knot is: -32K14 - 64K1*3K3 - 512K12K2*6 + 1024K1*2K2*5 - 3840K1*2K2*4 - 128K1*2K2*3K4 + 3808K12K2*3 - 4736K1*2K2*2 - 160K1*2K2K4 + 2592K1*2K2 - 32K12K3*2 - 1920K1*2 + 2688K1K25K3 + 256K1K2*4K3K4 - 1664K1K24K3 - 384K1K2*4K5 - 128K1K2*3K3K4 + 4032K1K23K3 + 96K1K2*2K3K4 - 992K1K22K3 + 32K1K2*2K4K5 - 160K1K22K5 - 96K1K2K3K4 + 3176K1K2K3 + 192K1K3K4 + 32K1K4K5 - 128K2*8 - 512K2*6K3*2 - 128K2*6K4*2 + 512K2*6K4 - 3040K26 + 256K2*5K3K5 + 128K2*5K4K6 - 128K2*5K6 - 1408K24K3*2 - 352K2*4K4*2 + 2208K2*4K4 - 32K24K6*2 - 1824K2*4 + 352K2*3K3K5 + 64K2*3K4K6 - 32K2*3K6 - 1152K22K3*2 - 256K2*2K4*2 + 1312K2*2K4 - 32K22K5*2 + 360K2*2 + 304K2K3K5 + 48K2K4K6 + 8K2K5K7 - 840K32 - 192K4*2 - 56K5*2 - 8K6**2 + 1486
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.5']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73972', 'vk6.73974', 'vk6.74483', 'vk6.74487', 'vk6.75935', 'vk6.75943', 'vk6.76701', 'vk6.76705', 'vk6.78937', 'vk6.78941', 'vk6.79475', 'vk6.79482', 'vk6.80463', 'vk6.80467', 'vk6.80945', 'vk6.80947', 'vk6.82982', 'vk6.83036', 'vk6.83744', 'vk6.83746', 'vk6.83915', 'vk6.85527', 'vk6.85529', 'vk6.85693', 'vk6.85778', 'vk6.85809', 'vk6.85813', 'vk6.86713', 'vk6.86715', 'vk6.87820', 'vk6.89573', 'vk6.89951']
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: [-5,-3,0,2,2,4,1,2,3,5,4,1,2,4,3,1,2,2,0,1,2]

Flat knots (up to 7 crossings) with same phi are :['6.5']

Arrow polynomial of the knot is: -8*K1**4 - 8*K1**3*K2 + 12*K1**3 + 4*K1**2*K2 + 4*K1**2*K3 + 4*K1**2 - 4*K1*K2 - 4*K1 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.5']

Outer characteristic polynomial of the knot is: t^7+157t^5+172t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.5']

2-strand cable arrow polynomial of the knot is: -32*K1**4 - 64*K1**3*K3 - 512*K1**2*K2**6 + 1024*K1**2*K2**5 - 3840*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3808*K1**2*K2**3 - 4736*K1**2*K2**2 - 160*K1**2*K2*K4 + 2592*K1**2*K2 - 32*K1**2*K3**2 - 1920*K1**2 + 2688*K1*K2**5*K3 + 256*K1*K2**4*K3*K4 - 1664*K1*K2**4*K3 - 384*K1*K2**4*K5 - 128*K1*K2**3*K3*K4 + 4032*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 992*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 160*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 3176*K1*K2*K3 + 192*K1*K3*K4 + 32*K1*K4*K5 - 128*K2**8 - 512*K2**6*K3**2 - 128*K2**6*K4**2 + 512*K2**6*K4 - 3040*K2**6 + 256*K2**5*K3*K5 + 128*K2**5*K4*K6 - 128*K2**5*K6 - 1408*K2**4*K3**2 - 352*K2**4*K4**2 + 2208*K2**4*K4 - 32*K2**4*K6**2 - 1824*K2**4 + 352*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1152*K2**2*K3**2 - 256*K2**2*K4**2 + 1312*K2**2*K4 - 32*K2**2*K5**2 + 360*K2**2 + 304*K2*K3*K5 + 48*K2*K4*K6 + 8*K2*K5*K7 - 840*K3**2 - 192*K4**2 - 56*K5**2 - 8*K6**2 + 1486

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.5']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73972', 'vk6.73974', 'vk6.74483', 'vk6.74487', 'vk6.75935', 'vk6.75943', 'vk6.76701', 'vk6.76705', 'vk6.78937', 'vk6.78941', 'vk6.79475', 'vk6.79482', 'vk6.80463', 'vk6.80467', 'vk6.80945', 'vk6.80947', 'vk6.82982', 'vk6.83036', 'vk6.83744', 'vk6.83746', 'vk6.83915', 'vk6.85527', 'vk6.85529', 'vk6.85693', 'vk6.85778', 'vk6.85809', 'vk6.85813', 'vk6.86713', 'vk6.86715', 'vk6.87820', 'vk6.89573', 'vk6.89951']

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	O1O2O3O4O5O6U1U2U4U5U6U3
R3 orbit	{'O1O2O3O4O5O6U1U2U4U5U6U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U4U1U2U3U5U6
Gauss code of K*	O1O2O3O4O5O6U1U2U6U3U4U5
Gauss code of -K*	O1O2O3O4O5O6U2U3U4U1U5U6
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 -5 -3 2 0 2 4],[ 5 0 1 5 2 3 4],[ 3 -1 0 4 1 2 3],[-2 -5 -4 0 -2 0 2],[ 0 -2 -1 2 0 1 2],[-2 -3 -2 0 -1 0 1],[-4 -4 -3 -2 -2 -1 0]]
Primitive based matrix	[[ 0 4 2 2 0 -3 -5],[-4 0 -1 -2 -2 -3 -4],[-2 1 0 0 -1 -2 -3],[-2 2 0 0 -2 -4 -5],[ 0 2 1 2 0 -1 -2],[ 3 3 2 4 1 0 -1],[ 5 4 3 5 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,-2,0,3,5,1,2,2,3,4,0,1,2,3,2,4,5,1,2,1]
Phi over symmetry	[-5,-3,0,2,2,4,1,2,3,5,4,1,2,4,3,1,2,2,0,1,2]
Phi of -K	[-5,-3,0,2,2,4,1,3,2,4,5,2,1,3,4,0,1,2,0,0,1]
Phi of K*	[-4,-2,-2,0,3,5,0,1,2,4,5,0,0,1,2,1,3,4,2,3,1]
Phi of -K*	[-5,-3,0,2,2,4,1,2,3,5,4,1,2,4,3,1,2,2,0,1,2]
Symmetry type of based matrix	c
u-polynomial	t^5-t^4+t^3-2t^2
Normalized Jones-Krushkal polynomial	2z^2+7z+7
Enhanced Jones-Krushkal polynomial	-6w^4z^2+8w^3z^2-10w^3z+17w^2z+7w
Inner characteristic polynomial	t^6+99t^4+20t^2
Outer characteristic polynomial	t^7+157t^5+172t^3+6t
Flat arrow polynomial	-8K14 - 8K1*3K2 + 12K13 + 4K1*2K2 + 4K12K3 + 4K12 - 4K1K2 - 4K1 + 1
2-strand cable arrow polynomial	-32K14 - 64K1*3K3 - 512K12K2*6 + 1024K1*2K2*5 - 3840K1*2K2*4 - 128K1*2K2*3K4 + 3808K12K2*3 - 4736K1*2K2*2 - 160K1*2K2K4 + 2592K1*2K2 - 32K12K3*2 - 1920K1*2 + 2688K1K25K3 + 256K1K2*4K3K4 - 1664K1K24K3 - 384K1K2*4K5 - 128K1K2*3K3K4 + 4032K1K23K3 + 96K1K2*2K3K4 - 992K1K22K3 + 32K1K2*2K4K5 - 160K1K22K5 - 96K1K2K3K4 + 3176K1K2K3 + 192K1K3K4 + 32K1K4K5 - 128K2*8 - 512K2*6K3*2 - 128K2*6K4*2 + 512K2*6K4 - 3040K26 + 256K2*5K3K5 + 128K2*5K4K6 - 128K2*5K6 - 1408K24K3*2 - 352K2*4K4*2 + 2208K2*4K4 - 32K24K6*2 - 1824K2*4 + 352K2*3K3K5 + 64K2*3K4K6 - 32K2*3K6 - 1152K22K3*2 - 256K2*2K4*2 + 1312K2*2K4 - 32K22K5*2 + 360K2*2 + 304K2K3K5 + 48K2K4K6 + 8K2K5K7 - 840K32 - 192K4*2 - 56K5*2 - 8K6**2 + 1486
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}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}]]
If K is slice	False

Contact