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

Flat knot 6.48

Min(phi) over symmetries of the knot is: [-5,0,0,1,1,3,1,2,4,5,3,0,1,1,1,1,1,2,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.48']
Arrow polynomial of the knot is: -2K1K4 + K3 + K5 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.4', '6.12', '6.15', '6.48']
Outer characteristic polynomial of the knot is: t^7+108t^5+107t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.48']
2-strand cable arrow polynomial of the knot is: -256K14 + 128K1*3K2K3 - 64K1*3K3 - 864K12K2*2 + 2176K1*2K2 - 864K12K3*2 - 32K1*2K5*2 - 3400K1*2 + 128K1K23K3 - 160K1K2*2K3 - 384K1K2*2K5 + 384K1K2K33 - 448K1K2K3K4 - 128K1K2K3K6 + 4320K1K2K3 - 96K1K2K4K5 - 64K1K3*2K5 + 1392K1K3K4 + 680K1K4K5 + 136K1K5K6 + 16K1K7K8 - 2K102 + 8K10K2K8 - 416K24 + 64K2*3K3K5 - 1248K2*2K3*2 - 96K2*2K3K7 - 32K2*2K4*2 - 32K2*2K4K8 + 1120K2*2K4 - 128K22K5*2 - 8K2*2K8*2 - 3388K2*2 - 192K2K32K4 + 2200K2K3K5 + 232K2K4K6 + 128K2K5K7 + 40K2K6K8 - 256K34 + 240K3*2K6 - 2392K32 + 64K3K4K7 + 24K3K5K8 + 24K4*2K8 - 1120K42 - 952K5*2 - 154K6*2 - 56K7*2 - 44K8**2 + 3770
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.48']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20035', 'vk6.20115', 'vk6.21313', 'vk6.21391', 'vk6.27090', 'vk6.27199', 'vk6.28789', 'vk6.28879', 'vk6.38479', 'vk6.38609', 'vk6.40672', 'vk6.40793', 'vk6.45365', 'vk6.45481', 'vk6.47128', 'vk6.47215', 'vk6.56840', 'vk6.56928', 'vk6.57976', 'vk6.58060', 'vk6.61358', 'vk6.61474', 'vk6.62530', 'vk6.62620', 'vk6.66554', 'vk6.66636', 'vk6.67343', 'vk6.67419', 'vk6.69198', 'vk6.69272', 'vk6.69947', 'vk6.70009']
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: [-5,0,0,1,1,3,1,2,4,5,3,0,1,1,1,1,1,2,0,2,2]

Flat knots (up to 7 crossings) with same phi are :['6.48']

Arrow polynomial of the knot is: -2*K1*K4 + K3 + K5 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.4', '6.12', '6.15', '6.48']

Outer characteristic polynomial of the knot is: t^7+108t^5+107t^3+14t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.48']

2-strand cable arrow polynomial of the knot is: -256*K1**4 + 128*K1**3*K2*K3 - 64*K1**3*K3 - 864*K1**2*K2**2 + 2176*K1**2*K2 - 864*K1**2*K3**2 - 32*K1**2*K5**2 - 3400*K1**2 + 128*K1*K2**3*K3 - 160*K1*K2**2*K3 - 384*K1*K2**2*K5 + 384*K1*K2*K3**3 - 448*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 4320*K1*K2*K3 - 96*K1*K2*K4*K5 - 64*K1*K3**2*K5 + 1392*K1*K3*K4 + 680*K1*K4*K5 + 136*K1*K5*K6 + 16*K1*K7*K8 - 2*K10**2 + 8*K10*K2*K8 - 416*K2**4 + 64*K2**3*K3*K5 - 1248*K2**2*K3**2 - 96*K2**2*K3*K7 - 32*K2**2*K4**2 - 32*K2**2*K4*K8 + 1120*K2**2*K4 - 128*K2**2*K5**2 - 8*K2**2*K8**2 - 3388*K2**2 - 192*K2*K3**2*K4 + 2200*K2*K3*K5 + 232*K2*K4*K6 + 128*K2*K5*K7 + 40*K2*K6*K8 - 256*K3**4 + 240*K3**2*K6 - 2392*K3**2 + 64*K3*K4*K7 + 24*K3*K5*K8 + 24*K4**2*K8 - 1120*K4**2 - 952*K5**2 - 154*K6**2 - 56*K7**2 - 44*K8**2 + 3770

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.48']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20035', 'vk6.20115', 'vk6.21313', 'vk6.21391', 'vk6.27090', 'vk6.27199', 'vk6.28789', 'vk6.28879', 'vk6.38479', 'vk6.38609', 'vk6.40672', 'vk6.40793', 'vk6.45365', 'vk6.45481', 'vk6.47128', 'vk6.47215', 'vk6.56840', 'vk6.56928', 'vk6.57976', 'vk6.58060', 'vk6.61358', 'vk6.61474', 'vk6.62530', 'vk6.62620', 'vk6.66554', 'vk6.66636', 'vk6.67343', 'vk6.67419', 'vk6.69198', 'vk6.69272', 'vk6.69947', 'vk6.70009']

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	O1O2O3O4O5O6U1U5U4U6U3U2
R3 orbit	{'O1O2O3O4O5O6U1U5U4U6U3U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U5U4U1U3U2U6
Gauss code of K*	O1O2O3O4O5O6U1U6U5U3U2U4
Gauss code of -K*	O1O2O3O4O5O6U3U5U4U2U1U6
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 1 1 0 0 3],[ 5 0 5 4 2 1 3],[-1 -5 0 0 -1 -1 2],[-1 -4 0 0 -1 -1 2],[ 0 -2 1 1 0 0 2],[ 0 -1 1 1 0 0 1],[-3 -3 -2 -2 -2 -1 0]]
Primitive based matrix	[[ 0 3 1 1 0 0 -5],[-3 0 -2 -2 -1 -2 -3],[-1 2 0 0 -1 -1 -4],[-1 2 0 0 -1 -1 -5],[ 0 1 1 1 0 0 -1],[ 0 2 1 1 0 0 -2],[ 5 3 4 5 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,0,0,5,2,2,1,2,3,0,1,1,4,1,1,5,0,1,2]
Phi over symmetry	[-5,0,0,1,1,3,1,2,4,5,3,0,1,1,1,1,1,2,0,2,2]
Phi of -K	[-5,0,0,1,1,3,3,4,1,2,5,0,0,0,1,0,0,2,0,0,0]
Phi of K*	[-3,-1,-1,0,0,5,0,0,1,2,5,0,0,0,1,0,0,2,0,3,4]
Phi of -K*	[-5,0,0,1,1,3,1,2,4,5,3,0,1,1,1,1,1,2,0,2,2]
Symmetry type of based matrix	c
u-polynomial	t^5-t^3-2t
Normalized Jones-Krushkal polynomial	8z^2+29z+27
Enhanced Jones-Krushkal polynomial	8w^3z^2+29w^2z+27w
Inner characteristic polynomial	t^6+72t^4+21t^2+1
Outer characteristic polynomial	t^7+108t^5+107t^3+14t
Flat arrow polynomial	-2K1K4 + K3 + K5 + 1
2-strand cable arrow polynomial	-256K14 + 128K1*3K2K3 - 64K1*3K3 - 864K12K2*2 + 2176K1*2K2 - 864K12K3*2 - 32K1*2K5*2 - 3400K1*2 + 128K1K23K3 - 160K1K2*2K3 - 384K1K2*2K5 + 384K1K2K33 - 448K1K2K3K4 - 128K1K2K3K6 + 4320K1K2K3 - 96K1K2K4K5 - 64K1K3*2K5 + 1392K1K3K4 + 680K1K4K5 + 136K1K5K6 + 16K1K7K8 - 2K102 + 8K10K2K8 - 416K24 + 64K2*3K3K5 - 1248K2*2K3*2 - 96K2*2K3K7 - 32K2*2K4*2 - 32K2*2K4K8 + 1120K2*2K4 - 128K22K5*2 - 8K2*2K8*2 - 3388K2*2 - 192K2K32K4 + 2200K2K3K5 + 232K2K4K6 + 128K2K5K7 + 40K2K6K8 - 256K34 + 240K3*2K6 - 2392K32 + 64K3K4K7 + 24K3K5K8 + 24K4*2K8 - 1120K42 - 952K5*2 - 154K6*2 - 56K7*2 - 44K8**2 + 3770
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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

Contact