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Flat knot 6.41

Min(phi) over symmetries of the knot is: [-5,-1,0,2,2,2,1,4,2,3,5,2,1,2,2,1,1,1,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.41']
Arrow polynomial of the knot is: 4K12K3 - 2K12 - 6K1K2 - 2K1K4 + 2K1 + K2 + 2*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.19', '6.22', '6.41']
Outer characteristic polynomial of the knot is: t^7+111t^5+172t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.41']
2-strand cable arrow polynomial of the knot is: -16K14 - 1120K1*2K2*2 + 600K1*2K2 - 16K12K3*2 - 1516K1*2 + 1376K1K23K3 + 3232K1K2K3 + 304K1K3K4 + 64K1K4K5 + 16K1K5K6 - 768K24K3*2 - 32K2*4K6*2 - 984K2*4 + 768K2*3K3K5 + 64K2*3K4K6 + 32K2*3K6K8 - 2224K2*2K3*2 - 56K2*2K4*2 + 160K2*2K4 - 256K22K5*2 - 64K2*2K6*2 - 8K2*2K8*2 - 636K2*2 + 1240K2K3K5 + 112K2K4K6 + 32K2K5K7 + 24K2K6K8 - 64K3*2K4*2 - 1432K3*2 + 48K3K4K7 - 218K42 - 260K5*2 - 52K6*2 - 8K7*2 - 4K8**2 + 1660
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.41']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70502', 'vk6.70506', 'vk6.70565', 'vk6.70569', 'vk6.70710', 'vk6.70718', 'vk6.70811', 'vk6.70817', 'vk6.70979', 'vk6.70985', 'vk6.71061', 'vk6.71069', 'vk6.71200', 'vk6.71208', 'vk6.71275', 'vk6.71279', 'vk6.71744', 'vk6.72163', 'vk6.74073', 'vk6.74144', 'vk6.74642', 'vk6.74711', 'vk6.76197', 'vk6.76230', 'vk6.77545', 'vk6.79077', 'vk6.79153', 'vk6.80650', 'vk6.81256', 'vk6.87006', 'vk6.87936', 'vk6.89139']
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,-1,0,2,2,2,1,4,2,3,5,2,1,2,2,1,1,1,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.41']

Arrow polynomial of the knot is: 4*K1**2*K3 - 2*K1**2 - 6*K1*K2 - 2*K1*K4 + 2*K1 + K2 + 2*K3 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.19', '6.22', '6.41']

Outer characteristic polynomial of the knot is: t^7+111t^5+172t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.41']

2-strand cable arrow polynomial of the knot is: -16*K1**4 - 1120*K1**2*K2**2 + 600*K1**2*K2 - 16*K1**2*K3**2 - 1516*K1**2 + 1376*K1*K2**3*K3 + 3232*K1*K2*K3 + 304*K1*K3*K4 + 64*K1*K4*K5 + 16*K1*K5*K6 - 768*K2**4*K3**2 - 32*K2**4*K6**2 - 984*K2**4 + 768*K2**3*K3*K5 + 64*K2**3*K4*K6 + 32*K2**3*K6*K8 - 2224*K2**2*K3**2 - 56*K2**2*K4**2 + 160*K2**2*K4 - 256*K2**2*K5**2 - 64*K2**2*K6**2 - 8*K2**2*K8**2 - 636*K2**2 + 1240*K2*K3*K5 + 112*K2*K4*K6 + 32*K2*K5*K7 + 24*K2*K6*K8 - 64*K3**2*K4**2 - 1432*K3**2 + 48*K3*K4*K7 - 218*K4**2 - 260*K5**2 - 52*K6**2 - 8*K7**2 - 4*K8**2 + 1660

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.41']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70502', 'vk6.70506', 'vk6.70565', 'vk6.70569', 'vk6.70710', 'vk6.70718', 'vk6.70811', 'vk6.70817', 'vk6.70979', 'vk6.70985', 'vk6.71061', 'vk6.71069', 'vk6.71200', 'vk6.71208', 'vk6.71275', 'vk6.71279', 'vk6.71744', 'vk6.72163', 'vk6.74073', 'vk6.74144', 'vk6.74642', 'vk6.74711', 'vk6.76197', 'vk6.76230', 'vk6.77545', 'vk6.79077', 'vk6.79153', 'vk6.80650', 'vk6.81256', 'vk6.87006', 'vk6.87936', 'vk6.89139']

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	O1O2O3O4O5O6U1U4U6U5U2U3
R3 orbit	{'O1O2O3O4O5O6U1U4U6U5U2U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U4U5U2U1U3U6
Gauss code of K*	O1O2O3O4O5O6U1U5U6U2U4U3
Gauss code of -K*	O1O2O3O4O5O6U4U3U5U1U2U6
Diagrammatic symmetry type	c
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -5 0 2 -1 2 2],[ 5 0 4 5 1 3 2],[ 0 -4 0 1 -2 1 1],[-2 -5 -1 0 -2 1 1],[ 1 -1 2 2 0 2 1],[-2 -3 -1 -1 -2 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 2 2 2 0 -1 -5],[-2 0 1 1 -1 -2 -5],[-2 -1 0 0 -1 -1 -2],[-2 -1 0 0 -1 -2 -3],[ 0 1 1 1 0 -2 -4],[ 1 2 1 2 2 0 -1],[ 5 5 2 3 4 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-2,0,1,5,-1,-1,1,2,5,0,1,1,2,1,2,3,2,4,1]
Phi over symmetry	[-5,-1,0,2,2,2,1,4,2,3,5,2,1,2,2,1,1,1,0,-1,-1]
Phi of -K	[-5,-1,0,2,2,2,3,1,2,4,5,-1,1,1,2,1,1,1,-1,-1,0]
Phi of K*	[-2,-2,-2,0,1,5,-1,0,1,1,4,1,1,1,2,1,2,5,-1,1,3]
Phi of -K*	[-5,-1,0,2,2,2,1,4,2,3,5,2,1,2,2,1,1,1,0,-1,-1]
Symmetry type of based matrix	c
u-polynomial	t^5-3t^2+t
Normalized Jones-Krushkal polynomial	3z+7
Enhanced Jones-Krushkal polynomial	4w^4z-16w^3z+4w^3+15w^2z+3w
Inner characteristic polynomial	t^6+73t^4+35t^2
Outer characteristic polynomial	t^7+111t^5+172t^3
Flat arrow polynomial	4K12K3 - 2K12 - 6K1K2 - 2K1K4 + 2K1 + K2 + 2*K3 + 2
2-strand cable arrow polynomial	-16K14 - 1120K1*2K2*2 + 600K1*2K2 - 16K12K3*2 - 1516K1*2 + 1376K1K23K3 + 3232K1K2K3 + 304K1K3K4 + 64K1K4K5 + 16K1K5K6 - 768K24K3*2 - 32K2*4K6*2 - 984K2*4 + 768K2*3K3K5 + 64K2*3K4K6 + 32K2*3K6K8 - 2224K2*2K3*2 - 56K2*2K4*2 + 160K2*2K4 - 256K22K5*2 - 64K2*2K6*2 - 8K2*2K8*2 - 636K2*2 + 1240K2K3K5 + 112K2K4K6 + 32K2K5K7 + 24K2K6K8 - 64K3*2K4*2 - 1432K3*2 + 48K3K4K7 - 218K42 - 260K5*2 - 52K6*2 - 8K7*2 - 4K8**2 + 1660
Genus of based matrix	1
Fillings of based matrix	[[{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}]]
If K is slice	False

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