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Flat knot 6.1363

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,-1,1,1,2,2,0,1,1,2,0,0,-1,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1363']
Arrow polynomial of the knot is: 4K13 - 8K1*2 - 4K1K2 - K1 + 4K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863']
Outer characteristic polynomial of the knot is: t^7+44t^5+38t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1363']
2-strand cable arrow polynomial of the knot is: -320K16 - 320K1*4K2*2 + 1856K1*4K2 - 4496K14 + 544K1*3K2K3 - 736K1*3K3 - 192K12K2*4 + 864K1*2K2*3 + 192K1*2K2*2K4 - 5440K12K2*2 - 416K1*2K2K4 + 8328K1*2K2 - 368K12K3*2 - 48K1*2K4*2 - 2748K1*2 + 320K1K23K3 - 832K1K2*2K3 - 96K1K2*2K5 - 128K1K2K3K4 + 4584K1K2K3 + 400K1K3K4 + 48K1K4K5 - 32K2*6 + 64K2*4K4 - 784K24 - 192K2*2K3*2 - 48K2*2K4*2 + 752K2*2K4 - 2670K22 + 112K2K3K5 + 16K2K4K6 - 900K3*2 - 144K4*2 - 16K5*2 - 2K6**2 + 2846
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1363']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4145', 'vk6.4178', 'vk6.5383', 'vk6.5416', 'vk6.7513', 'vk6.7538', 'vk6.9014', 'vk6.9047', 'vk6.12419', 'vk6.12450', 'vk6.13349', 'vk6.13568', 'vk6.13601', 'vk6.14270', 'vk6.14719', 'vk6.14726', 'vk6.15201', 'vk6.15873', 'vk6.15882', 'vk6.30832', 'vk6.30863', 'vk6.32016', 'vk6.32047', 'vk6.33065', 'vk6.33098', 'vk6.33852', 'vk6.34311', 'vk6.48489', 'vk6.50274', 'vk6.53521', 'vk6.53949', 'vk6.54263']
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: [-2,-2,1,1,1,1,-1,1,1,2,2,0,1,1,2,0,0,-1,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1363']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.315', '6.337', '6.389', '6.418', '6.599', '6.675', '6.686', '6.688', '6.746', '6.747', '6.809', '6.1034', '6.1128', '6.1133', '6.1334', '6.1363', '6.1489', '6.1539', '6.1564', '6.1821', '6.1863']

Outer characteristic polynomial of the knot is: t^7+44t^5+38t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1363']

2-strand cable arrow polynomial of the knot is: -320*K1**6 - 320*K1**4*K2**2 + 1856*K1**4*K2 - 4496*K1**4 + 544*K1**3*K2*K3 - 736*K1**3*K3 - 192*K1**2*K2**4 + 864*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5440*K1**2*K2**2 - 416*K1**2*K2*K4 + 8328*K1**2*K2 - 368*K1**2*K3**2 - 48*K1**2*K4**2 - 2748*K1**2 + 320*K1*K2**3*K3 - 832*K1*K2**2*K3 - 96*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 4584*K1*K2*K3 + 400*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 784*K2**4 - 192*K2**2*K3**2 - 48*K2**2*K4**2 + 752*K2**2*K4 - 2670*K2**2 + 112*K2*K3*K5 + 16*K2*K4*K6 - 900*K3**2 - 144*K4**2 - 16*K5**2 - 2*K6**2 + 2846

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1363']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4145', 'vk6.4178', 'vk6.5383', 'vk6.5416', 'vk6.7513', 'vk6.7538', 'vk6.9014', 'vk6.9047', 'vk6.12419', 'vk6.12450', 'vk6.13349', 'vk6.13568', 'vk6.13601', 'vk6.14270', 'vk6.14719', 'vk6.14726', 'vk6.15201', 'vk6.15873', 'vk6.15882', 'vk6.30832', 'vk6.30863', 'vk6.32016', 'vk6.32047', 'vk6.33065', 'vk6.33098', 'vk6.33852', 'vk6.34311', 'vk6.48489', 'vk6.50274', 'vk6.53521', 'vk6.53949', 'vk6.54263']

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	O1O2O3U2O4O5U3U6U5O6U1U4
R3 orbit	{'O1O2O3U2O4O5U3U6U5O6U1U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U3O5U6U5U1O6O4U2
Gauss code of K*	O1O2O3U2O4O5U4U6U1O6U5U3
Gauss code of -K*	O1O2O3U1U4O5U3U5U6O4O6U2
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 -1 -1 -1 2 2 -1],[ 1 0 -1 0 2 2 0],[ 1 1 0 1 1 1 1],[ 1 0 -1 0 2 1 0],[-2 -2 -1 -2 0 1 -3],[-2 -2 -1 -1 -1 0 -2],[ 1 0 -1 0 3 2 0]]
Primitive based matrix	[[ 0 2 2 -1 -1 -1 -1],[-2 0 1 -1 -2 -2 -3],[-2 -1 0 -1 -1 -2 -2],[ 1 1 1 0 1 1 1],[ 1 2 1 -1 0 0 0],[ 1 2 2 -1 0 0 0],[ 1 3 2 -1 0 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,1,1,1,1,-1,1,2,2,3,1,1,2,2,-1,-1,-1,0,0,0]
Phi over symmetry	[-2,-2,1,1,1,1,-1,1,1,2,2,0,1,1,2,0,0,-1,0,-1,-1]
Phi of -K	[-1,-1,-1,-1,2,2,-1,-1,-1,2,2,0,0,0,1,0,1,1,1,2,-1]
Phi of K*	[-2,-2,1,1,1,1,-1,1,1,2,2,0,1,1,2,0,0,-1,0,-1,-1]
Phi of -K*	[-1,-1,-1,-1,2,2,-1,0,0,1,2,1,1,1,1,0,2,2,2,3,-1]
Symmetry type of based matrix	c
u-polynomial	-2t^2+4t
Normalized Jones-Krushkal polynomial	2z^2+19z+31
Enhanced Jones-Krushkal polynomial	2w^3z^2+19w^2z+31w
Inner characteristic polynomial	t^6+32t^4+22t^2+1
Outer characteristic polynomial	t^7+44t^5+38t^3+5t
Flat arrow polynomial	4K13 - 8K1*2 - 4K1K2 - K1 + 4K2 + K3 + 5
2-strand cable arrow polynomial	-320K16 - 320K1*4K2*2 + 1856K1*4K2 - 4496K14 + 544K1*3K2K3 - 736K1*3K3 - 192K12K2*4 + 864K1*2K2*3 + 192K1*2K2*2K4 - 5440K12K2*2 - 416K1*2K2K4 + 8328K1*2K2 - 368K12K3*2 - 48K1*2K4*2 - 2748K1*2 + 320K1K23K3 - 832K1K2*2K3 - 96K1K2*2K5 - 128K1K2K3K4 + 4584K1K2K3 + 400K1K3K4 + 48K1K4K5 - 32K2*6 + 64K2*4K4 - 784K24 - 192K2*2K3*2 - 48K2*2K4*2 + 752K2*2K4 - 2670K22 + 112K2K3K5 + 16K2K4K6 - 900K3*2 - 144K4*2 - 16K5*2 - 2K6**2 + 2846
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {4}, {2, 3}]]
If K is slice	False

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