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Flat knot 6.1252

Min(phi) over symmetries of the knot is: [-2,-2,-1,1,2,2,-1,0,1,2,2,1,1,2,2,1,2,2,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1252']
Arrow polynomial of the knot is: 8K13 - 8K1*2 - 8K1K2 - 4K1K3 - 2K1 + 6K2 + 2K3 + 2*K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1252']
Outer characteristic polynomial of the knot is: t^7+49t^5+45t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1252']
2-strand cable arrow polynomial of the knot is: -832K14 + 512K1*3K2K3 + 64K1*3K3K4 - 576K1*3K3 + 256K12K2*3 - 128K1*2K2*2K3*2 - 3264K1*2K2*2 + 192K1*2K2K32 + 64K1*2K2K3K5 - 320K12K2K4 + 6304K1*2K2 - 960K12K3*2 - 128K1*2K3K5 - 32K1*2K4*2 - 32K1*2K5*2 - 5592K1*2 + 640K1K23K3 - 1344K1K2*2K3 - 320K1K2*2K5 + 128K1K2K33 - 320K1K2K3K4 - 192K1K2K3K6 + 6864K1K2K3 - 64K1K2K5K6 + 1408K1K3K4 + 320K1K4K5 + 128K1K5K6 + 16K1K6K7 - 64K26 + 128K2*4K4 - 992K24 + 64K2*3K3K5 - 64K2*3K6 - 1216K22K3*2 - 64K2*2K3K7 - 64K2*2K4*2 + 1568K2*2K4 - 64K22K5*2 - 48K2*2K6*2 - 4212K2*2 + 1040K2K3K5 + 208K2K4K6 + 96K2K5K7 + 32K2K6K8 - 32K3*4 + 32K3*2K6 - 2336K32 - 732K4*2 - 296K5*2 - 100K6*2 - 16K7*2 - 4K8**2 + 4342
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1252']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71359', 'vk6.71413', 'vk6.71880', 'vk6.71939', 'vk6.74319', 'vk6.74966', 'vk6.76535', 'vk6.76944', 'vk6.77019', 'vk6.77074', 'vk6.77396', 'vk6.79365', 'vk6.79791', 'vk6.80827', 'vk6.81277', 'vk6.81475', 'vk6.83850', 'vk6.87075', 'vk6.88040', 'vk6.89556']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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: [-2,-2,-1,1,2,2,-1,0,1,2,2,1,1,2,2,1,2,2,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.1252']

Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 4*K1*K3 - 2*K1 + 6*K2 + 2*K3 + 2*K4 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1252']

Outer characteristic polynomial of the knot is: t^7+49t^5+45t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1252']

2-strand cable arrow polynomial of the knot is: -832*K1**4 + 512*K1**3*K2*K3 + 64*K1**3*K3*K4 - 576*K1**3*K3 + 256*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 3264*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 6304*K1**2*K2 - 960*K1**2*K3**2 - 128*K1**2*K3*K5 - 32*K1**2*K4**2 - 32*K1**2*K5**2 - 5592*K1**2 + 640*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 320*K1*K2**2*K5 + 128*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 6864*K1*K2*K3 - 64*K1*K2*K5*K6 + 1408*K1*K3*K4 + 320*K1*K4*K5 + 128*K1*K5*K6 + 16*K1*K6*K7 - 64*K2**6 + 128*K2**4*K4 - 992*K2**4 + 64*K2**3*K3*K5 - 64*K2**3*K6 - 1216*K2**2*K3**2 - 64*K2**2*K3*K7 - 64*K2**2*K4**2 + 1568*K2**2*K4 - 64*K2**2*K5**2 - 48*K2**2*K6**2 - 4212*K2**2 + 1040*K2*K3*K5 + 208*K2*K4*K6 + 96*K2*K5*K7 + 32*K2*K6*K8 - 32*K3**4 + 32*K3**2*K6 - 2336*K3**2 - 732*K4**2 - 296*K5**2 - 100*K6**2 - 16*K7**2 - 4*K8**2 + 4342

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1252']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71359', 'vk6.71413', 'vk6.71880', 'vk6.71939', 'vk6.74319', 'vk6.74966', 'vk6.76535', 'vk6.76944', 'vk6.77019', 'vk6.77074', 'vk6.77396', 'vk6.79365', 'vk6.79791', 'vk6.80827', 'vk6.81277', 'vk6.81475', 'vk6.83850', 'vk6.87075', 'vk6.88040', 'vk6.89556']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is r.

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	O1O2O3O4U5U1U3O5O6U2U4U6
R3 orbit	{'O1O2O3O4U5U1U3O5O6U2U4U6'}
R3 orbit length	1
Gauss code of -K	Same
Gauss code of K*	O1O2O3U1U4O5O6O4U2U5U3U6
Gauss code of -K*	O1O2O3U1U4O5O6O4U2U5U3U6
Diagrammatic symmetry type	r
Flat genus of the diagram	3
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -2 -1 1 2 -2 2],[ 2 0 1 1 2 1 2],[ 1 -1 0 1 2 0 2],[-1 -1 -1 0 0 -1 1],[-2 -2 -2 0 0 -2 1],[ 2 -1 0 1 2 0 2],[-2 -2 -2 -1 -1 -2 0]]
Primitive based matrix	[[ 0 2 2 1 -1 -2 -2],[-2 0 1 0 -2 -2 -2],[-2 -1 0 -1 -2 -2 -2],[-1 0 1 0 -1 -1 -1],[ 1 2 2 1 0 0 -1],[ 2 2 2 1 0 0 -1],[ 2 2 2 1 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,1,2,2,-1,0,2,2,2,1,2,2,2,1,1,1,0,1,1]
Phi over symmetry	[-2,-2,-1,1,2,2,-1,0,1,2,2,1,1,2,2,1,2,2,0,1,1]
Phi of -K	[-2,-2,-1,1,2,2,-1,0,2,2,2,1,2,2,2,1,1,1,0,1,1]
Phi of K*	[-2,-2,-1,1,2,2,-1,0,1,2,2,1,1,2,2,1,2,2,0,1,1]
Phi of -K*	[-2,-2,-1,1,2,2,-1,0,1,2,2,1,1,2,2,1,2,2,0,1,1]
Symmetry type of based matrix	r
u-polynomial	0
Normalized Jones-Krushkal polynomial	2z^2+19z+31
Enhanced Jones-Krushkal polynomial	2w^3z^2+19w^2z+31w
Inner characteristic polynomial	t^6+31t^4+7t^2
Outer characteristic polynomial	t^7+49t^5+45t^3+4t
Flat arrow polynomial	8K13 - 8K1*2 - 8K1K2 - 4K1K3 - 2K1 + 6K2 + 2K3 + 2*K4 + 5
2-strand cable arrow polynomial	-832K14 + 512K1*3K2K3 + 64K1*3K3K4 - 576K1*3K3 + 256K12K2*3 - 128K1*2K2*2K3*2 - 3264K1*2K2*2 + 192K1*2K2K32 + 64K1*2K2K3K5 - 320K12K2K4 + 6304K1*2K2 - 960K12K3*2 - 128K1*2K3K5 - 32K1*2K4*2 - 32K1*2K5*2 - 5592K1*2 + 640K1K23K3 - 1344K1K2*2K3 - 320K1K2*2K5 + 128K1K2K33 - 320K1K2K3K4 - 192K1K2K3K6 + 6864K1K2K3 - 64K1K2K5K6 + 1408K1K3K4 + 320K1K4K5 + 128K1K5K6 + 16K1K6K7 - 64K26 + 128K2*4K4 - 992K24 + 64K2*3K3K5 - 64K2*3K6 - 1216K22K3*2 - 64K2*2K3K7 - 64K2*2K4*2 + 1568K2*2K4 - 64K22K5*2 - 48K2*2K6*2 - 4212K2*2 + 1040K2K3K5 + 208K2K4K6 + 96K2K5K7 + 32K2K6K8 - 32K3*4 + 32K3*2K6 - 2336K32 - 732K4*2 - 296K5*2 - 100K6*2 - 16K7*2 - 4K8**2 + 4342
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice	False

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