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

Flat knot 6.1142

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,1,2,2,0,1,0,1,1,0,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1142']
Arrow polynomial of the knot is: -8K14 + 4K1*3 + 8K1*2K2 - 2K1K2 - 2K1K3 - 2*K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.950', '6.1142']
Outer characteristic polynomial of the knot is: t^7+60t^5+75t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1142']
2-strand cable arrow polynomial of the knot is: -464K14 - 768K1*2K2*6 + 896K1*2K2*5 - 2880K1*2K2*4 + 3968K1*2K2*3 - 7280K1*2K2*2 - 96K1*2K2K4 + 5624K1*2K2 - 48K12K3*2 - 3540K1*2 + 1152K1K25K3 - 640K1K2*4K3 - 256K1K2*4K5 + 3584K1K2*3K3 + 96K1K2*2K3K4 - 1792K1K22K3 - 288K1K2*2K5 - 32K1K2K3K4 + 4872K1K2K3 + 296K1K3K4 - 128K28 + 256K2*6K4 - 1440K26 - 128K2*5K6 - 320K24K3*2 - 64K2*4K4*2 + 1184K2*4K4 - 3216K24 + 96K2*3K3K5 + 64K2*3K4K6 - 96K2*3K6 - 1408K22K3*2 - 200K2*2K4*2 + 1880K2*2K4 - 8K22K6*2 - 460K2*2 - 32K2K32K4 + 328K2K3K5 + 80K2K4K6 + 8K32K6 - 1060K32 - 218K4*2 - 16K5*2 - 12K6**2 + 2504
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1142']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11415', 'vk6.11700', 'vk6.12715', 'vk6.13070', 'vk6.20266', 'vk6.21583', 'vk6.27528', 'vk6.29104', 'vk6.31152', 'vk6.31483', 'vk6.32304', 'vk6.32743', 'vk6.38929', 'vk6.41149', 'vk6.45686', 'vk6.47405', 'vk6.52170', 'vk6.52403', 'vk6.52989', 'vk6.53314', 'vk6.57091', 'vk6.58254', 'vk6.61661', 'vk6.62829', 'vk6.63744', 'vk6.63846', 'vk6.64164', 'vk6.64358', 'vk6.66730', 'vk6.67599', 'vk6.69378', 'vk6.70114']
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,-1,0,1,1,2,1,1,1,2,2,0,1,0,1,1,0,1,-1,-1,1]

Flat knots (up to 7 crossings) with same phi are :['6.1142']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.950', '6.1142']

Outer characteristic polynomial of the knot is: t^7+60t^5+75t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1142']

2-strand cable arrow polynomial of the knot is: -464*K1**4 - 768*K1**2*K2**6 + 896*K1**2*K2**5 - 2880*K1**2*K2**4 + 3968*K1**2*K2**3 - 7280*K1**2*K2**2 - 96*K1**2*K2*K4 + 5624*K1**2*K2 - 48*K1**2*K3**2 - 3540*K1**2 + 1152*K1*K2**5*K3 - 640*K1*K2**4*K3 - 256*K1*K2**4*K5 + 3584*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1792*K1*K2**2*K3 - 288*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 4872*K1*K2*K3 + 296*K1*K3*K4 - 128*K2**8 + 256*K2**6*K4 - 1440*K2**6 - 128*K2**5*K6 - 320*K2**4*K3**2 - 64*K2**4*K4**2 + 1184*K2**4*K4 - 3216*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1408*K2**2*K3**2 - 200*K2**2*K4**2 + 1880*K2**2*K4 - 8*K2**2*K6**2 - 460*K2**2 - 32*K2*K3**2*K4 + 328*K2*K3*K5 + 80*K2*K4*K6 + 8*K3**2*K6 - 1060*K3**2 - 218*K4**2 - 16*K5**2 - 12*K6**2 + 2504

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1142']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11415', 'vk6.11700', 'vk6.12715', 'vk6.13070', 'vk6.20266', 'vk6.21583', 'vk6.27528', 'vk6.29104', 'vk6.31152', 'vk6.31483', 'vk6.32304', 'vk6.32743', 'vk6.38929', 'vk6.41149', 'vk6.45686', 'vk6.47405', 'vk6.52170', 'vk6.52403', 'vk6.52989', 'vk6.53314', 'vk6.57091', 'vk6.58254', 'vk6.61661', 'vk6.62829', 'vk6.63744', 'vk6.63846', 'vk6.64164', 'vk6.64358', 'vk6.66730', 'vk6.67599', 'vk6.69378', 'vk6.70114']

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	O1O2O3O4U1U2U5O6O5U3U4U6
R3 orbit	{'O1O2O3O4U1U2U5O6O5U3U4U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U1U2O6O5U6U3U4
Gauss code of K*	O1O2O3U4U3O5O6O4U1U2U5U6
Gauss code of -K*	O1O2O3U4U1O4O5O6U2U3U5U6
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 0 2 1 1],[ 3 0 1 2 3 3 2],[ 1 -1 0 1 2 1 2],[ 0 -2 -1 0 1 0 1],[-2 -3 -2 -1 0 -2 0],[-1 -3 -1 0 2 0 1],[-1 -2 -2 -1 0 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 0 -2 -1 -2 -3],[-1 0 0 -1 -1 -2 -2],[-1 2 1 0 0 -1 -3],[ 0 1 1 0 0 -1 -2],[ 1 2 2 1 1 0 -1],[ 3 3 2 3 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,0,2,1,2,3,1,1,2,2,0,1,3,1,2,1]
Phi over symmetry	[-3,-1,0,1,1,2,1,1,1,2,2,0,1,0,1,1,0,1,-1,-1,1]
Phi of -K	[-3,-1,0,1,1,2,1,1,1,2,2,0,1,0,1,1,0,1,-1,-1,1]
Phi of K*	[-2,-1,-1,0,1,3,-1,1,1,1,2,1,1,1,1,0,0,2,0,1,1]
Phi of -K*	[-3,-1,0,1,1,2,1,2,2,3,3,1,2,1,2,1,0,1,-1,0,2]
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^4z^2+8w^3z^2-4w^3z+21w^2z+19w
Inner characteristic polynomial	t^6+44t^4+26t^2+1
Outer characteristic polynomial	t^7+60t^5+75t^3+8t
Flat arrow polynomial	-8K14 + 4K1*3 + 8K1*2K2 - 2K1K2 - 2K1K3 - 2*K1 + K2 + 2
2-strand cable arrow polynomial	-464K14 - 768K1*2K2*6 + 896K1*2K2*5 - 2880K1*2K2*4 + 3968K1*2K2*3 - 7280K1*2K2*2 - 96K1*2K2K4 + 5624K1*2K2 - 48K12K3*2 - 3540K1*2 + 1152K1K25K3 - 640K1K2*4K3 - 256K1K2*4K5 + 3584K1K2*3K3 + 96K1K2*2K3K4 - 1792K1K22K3 - 288K1K2*2K5 - 32K1K2K3K4 + 4872K1K2K3 + 296K1K3K4 - 128K28 + 256K2*6K4 - 1440K26 - 128K2*5K6 - 320K24K3*2 - 64K2*4K4*2 + 1184K2*4K4 - 3216K24 + 96K2*3K3K5 + 64K2*3K4K6 - 96K2*3K6 - 1408K22K3*2 - 200K2*2K4*2 + 1880K2*2K4 - 8K22K6*2 - 460K2*2 - 32K2K32K4 + 328K2K3K5 + 80K2K4K6 + 8K32K6 - 1060K32 - 218K4*2 - 16K5*2 - 12K6**2 + 2504
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