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Flat knot 6.483

Min(phi) over symmetries of the knot is: [-4,-1,-1,2,2,2,0,2,2,4,4,1,1,1,2,1,2,2,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.483']
Arrow polynomial of the knot is: 4K13 - 4K1*2 - 4K1K2 - 2K1K3 - K1 + 3K2 + K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.156', '6.476', '6.483']
Outer characteristic polynomial of the knot is: t^7+81t^5+86t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.483']
2-strand cable arrow polynomial of the knot is: -240K14 + 192K1*3K2K3 + 128K1*2K2*3 - 192K1*2K2*2K3*2 - 1264K1*2K2*2 + 1472K1*2K2 - 608K12K3*2 - 32K1*2K5*2 - 1608K1*2 + 416K1K23K3 + 416K1K2K33 + 2192K1K2K3 + 352K1K3K4 + 168K1K4K5 + 48K1K5K6 + 24K1K7K8 - 32K26 + 64K2*4K4 - 608K24 + 32K2*3K3K5 - 944K2*2K3*2 - 48K2*2K4*2 + 448K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 1050K2*2 + 464K2K3K5 + 24K2K4K6 + 64K2K5K7 + 16K2K6K8 - 208K3*4 + 32K3*2K6 - 636K32 + 16K3K4K7 - 298K42 - 180K5*2 - 30K6*2 - 40K7*2 - 18K8**2 + 1538
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.483']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71347', 'vk6.71398', 'vk6.71414', 'vk6.71857', 'vk6.71879', 'vk6.71921', 'vk6.71940', 'vk6.74315', 'vk6.74342', 'vk6.74962', 'vk6.74985', 'vk6.76531', 'vk6.76554', 'vk6.76940', 'vk6.76992', 'vk6.77018', 'vk6.77055', 'vk6.77075', 'vk6.77384', 'vk6.79369', 'vk6.79795', 'vk6.79816', 'vk6.80830', 'vk6.80846', 'vk6.81272', 'vk6.81469', 'vk6.81473', 'vk6.83843', 'vk6.87068', 'vk6.87074', 'vk6.88038', 'vk6.89565']
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: [-4,-1,-1,2,2,2,0,2,2,4,4,1,1,1,2,1,2,2,-1,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.483']

Arrow polynomial of the knot is: 4*K1**3 - 4*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 3*K2 + K3 + K4 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.156', '6.476', '6.483']

Outer characteristic polynomial of the knot is: t^7+81t^5+86t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.483']

2-strand cable arrow polynomial of the knot is: -240*K1**4 + 192*K1**3*K2*K3 + 128*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 1264*K1**2*K2**2 + 1472*K1**2*K2 - 608*K1**2*K3**2 - 32*K1**2*K5**2 - 1608*K1**2 + 416*K1*K2**3*K3 + 416*K1*K2*K3**3 + 2192*K1*K2*K3 + 352*K1*K3*K4 + 168*K1*K4*K5 + 48*K1*K5*K6 + 24*K1*K7*K8 - 32*K2**6 + 64*K2**4*K4 - 608*K2**4 + 32*K2**3*K3*K5 - 944*K2**2*K3**2 - 48*K2**2*K4**2 + 448*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 1050*K2**2 + 464*K2*K3*K5 + 24*K2*K4*K6 + 64*K2*K5*K7 + 16*K2*K6*K8 - 208*K3**4 + 32*K3**2*K6 - 636*K3**2 + 16*K3*K4*K7 - 298*K4**2 - 180*K5**2 - 30*K6**2 - 40*K7**2 - 18*K8**2 + 1538

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.483']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71347', 'vk6.71398', 'vk6.71414', 'vk6.71857', 'vk6.71879', 'vk6.71921', 'vk6.71940', 'vk6.74315', 'vk6.74342', 'vk6.74962', 'vk6.74985', 'vk6.76531', 'vk6.76554', 'vk6.76940', 'vk6.76992', 'vk6.77018', 'vk6.77055', 'vk6.77075', 'vk6.77384', 'vk6.79369', 'vk6.79795', 'vk6.79816', 'vk6.80830', 'vk6.80846', 'vk6.81272', 'vk6.81469', 'vk6.81473', 'vk6.83843', 'vk6.87068', 'vk6.87074', 'vk6.88038', 'vk6.89565']

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	O1O2O3O4O5U1U3U5O6U2U4U6
R3 orbit	{'O1O2O3O4O5U1U3U5O6U2U4U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U2U4O6U1U3U5
Gauss code of K*	O1O2O3U4O5O6O4U1U5U2U6U3
Gauss code of -K*	O1O2O3U1O4O5O6U4U2U5U3U6
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 -4 -1 -1 2 2 2],[ 4 0 3 1 4 2 2],[ 1 -3 0 -1 2 1 2],[ 1 -1 1 0 2 1 1],[-2 -4 -2 -2 0 0 1],[-2 -2 -1 -1 0 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix	[[ 0 2 2 2 -1 -1 -4],[-2 0 1 0 -2 -2 -4],[-2 -1 0 0 -1 -2 -2],[-2 0 0 0 -1 -1 -2],[ 1 2 1 1 0 1 -1],[ 1 2 2 1 -1 0 -3],[ 4 4 2 2 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-2,1,1,4,-1,0,2,2,4,0,1,2,2,1,1,2,-1,1,3]
Phi over symmetry	[-4,-1,-1,2,2,2,0,2,2,4,4,1,1,1,2,1,2,2,-1,0,0]
Phi of -K	[-4,-1,-1,2,2,2,0,2,2,4,4,1,1,1,2,1,2,2,-1,0,0]
Phi of K*	[-2,-2,-2,1,1,4,-1,0,1,2,4,0,1,1,2,2,2,4,-1,0,2]
Phi of -K*	[-4,-1,-1,2,2,2,1,3,2,2,4,1,1,1,2,1,2,2,0,0,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-3t^2+2t
Normalized Jones-Krushkal polynomial	11z+23
Enhanced Jones-Krushkal polynomial	-2w^3z+13w^2z+23w
Inner characteristic polynomial	t^6+51t^4+15t^2
Outer characteristic polynomial	t^7+81t^5+86t^3
Flat arrow polynomial	4K13 - 4K1*2 - 4K1K2 - 2K1K3 - K1 + 3K2 + K3 + K4 + 3
2-strand cable arrow polynomial	-240K14 + 192K1*3K2K3 + 128K1*2K2*3 - 192K1*2K2*2K3*2 - 1264K1*2K2*2 + 1472K1*2K2 - 608K12K3*2 - 32K1*2K5*2 - 1608K1*2 + 416K1K23K3 + 416K1K2K33 + 2192K1K2K3 + 352K1K3K4 + 168K1K4K5 + 48K1K5K6 + 24K1K7K8 - 32K26 + 64K2*4K4 - 608K24 + 32K2*3K3K5 - 944K2*2K3*2 - 48K2*2K4*2 + 448K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 1050K2*2 + 464K2K3K5 + 24K2K4K6 + 64K2K5K7 + 16K2K6K8 - 208K3*4 + 32K3*2K6 - 636K32 + 16K3K4K7 - 298K42 - 180K5*2 - 30K6*2 - 40K7*2 - 18K8**2 + 1538
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice	False

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