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Flat knot 6.1284

Min(phi) over symmetries of the knot is: [-3,0,0,0,1,2,0,2,2,1,3,0,1,0,0,0,1,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1284']
Arrow polynomial of the knot is: 4K13 - 6K1*2 - 6K1K2 + 3K2 + 2*K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.311', '6.528', '6.536', '6.817', '6.982', '6.984', '6.1284']
Outer characteristic polynomial of the knot is: t^7+47t^5+56t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.885', '6.1284']
2-strand cable arrow polynomial of the knot is: -128K14K2*2 + 256K1*4K2 - 2096K14 + 448K1*3K2K3 - 768K1*3K3 + 320K12K2*3 - 3184K1*2K2*2 - 544K1*2K2K4 + 6464K1*2K2 - 912K12K3*2 - 4392K1*2 + 96K1K23K3 - 96K1K2*2K3 - 160K1K2K3K4 + 5416K1K2K3 + 968K1K3K4 + 64K1K4K5 - 32K2*6 + 96K2*4K4 - 472K24 - 208K2*2K3*2 - 120K2*2K4*2 + 664K2*2K4 - 3388K22 + 360K2K3K5 + 80K2K4K6 - 32K3*4 + 80K3*2K6 - 1820K32 - 418K4*2 - 132K5*2 - 52K6**2 + 3640
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1284']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17105', 'vk6.17348', 'vk6.20569', 'vk6.21977', 'vk6.23494', 'vk6.23833', 'vk6.28034', 'vk6.29492', 'vk6.35647', 'vk6.36086', 'vk6.39444', 'vk6.41643', 'vk6.43009', 'vk6.43321', 'vk6.46032', 'vk6.47698', 'vk6.55256', 'vk6.55508', 'vk6.57450', 'vk6.58617', 'vk6.59660', 'vk6.60008', 'vk6.62125', 'vk6.63090', 'vk6.65064', 'vk6.65257', 'vk6.66985', 'vk6.67849', 'vk6.68323', 'vk6.68473', 'vk6.69603', 'vk6.70295']
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: [-3,0,0,0,1,2,0,2,2,1,3,0,1,0,0,0,1,1,1,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1284']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.311', '6.528', '6.536', '6.817', '6.982', '6.984', '6.1284']

Outer characteristic polynomial of the knot is: t^7+47t^5+56t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.885', '6.1284']

2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 256*K1**4*K2 - 2096*K1**4 + 448*K1**3*K2*K3 - 768*K1**3*K3 + 320*K1**2*K2**3 - 3184*K1**2*K2**2 - 544*K1**2*K2*K4 + 6464*K1**2*K2 - 912*K1**2*K3**2 - 4392*K1**2 + 96*K1*K2**3*K3 - 96*K1*K2**2*K3 - 160*K1*K2*K3*K4 + 5416*K1*K2*K3 + 968*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 472*K2**4 - 208*K2**2*K3**2 - 120*K2**2*K4**2 + 664*K2**2*K4 - 3388*K2**2 + 360*K2*K3*K5 + 80*K2*K4*K6 - 32*K3**4 + 80*K3**2*K6 - 1820*K3**2 - 418*K4**2 - 132*K5**2 - 52*K6**2 + 3640

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1284']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17105', 'vk6.17348', 'vk6.20569', 'vk6.21977', 'vk6.23494', 'vk6.23833', 'vk6.28034', 'vk6.29492', 'vk6.35647', 'vk6.36086', 'vk6.39444', 'vk6.41643', 'vk6.43009', 'vk6.43321', 'vk6.46032', 'vk6.47698', 'vk6.55256', 'vk6.55508', 'vk6.57450', 'vk6.58617', 'vk6.59660', 'vk6.60008', 'vk6.62125', 'vk6.63090', 'vk6.65064', 'vk6.65257', 'vk6.66985', 'vk6.67849', 'vk6.68323', 'vk6.68473', 'vk6.69603', 'vk6.70295']

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	O1O2O3U1O4O5U6U3O6U4U2U5
R3 orbit	{'O1O2O3U1O4O5U6U3O6U4U2U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U2U5O6U1U6O4O5U3
Gauss code of K*	O1O2O3U4U2U5O4U1U3O6O5U6
Gauss code of -K*	O1O2O3U4O5O4U1U3O6U5U2U6
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 -2 0 0 0 3 -1],[ 2 0 2 1 1 2 1],[ 0 -2 0 0 1 3 -1],[ 0 -1 0 0 0 1 0],[ 0 -1 -1 0 0 1 0],[-3 -2 -3 -1 -1 0 -3],[ 1 -1 1 0 0 3 0]]
Primitive based matrix	[[ 0 3 0 0 0 -1 -2],[-3 0 -1 -1 -3 -3 -2],[ 0 1 0 0 0 0 -1],[ 0 1 0 0 -1 0 -1],[ 0 3 0 1 0 -1 -2],[ 1 3 0 0 1 0 -1],[ 2 2 1 1 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,0,0,0,1,2,1,1,3,3,2,0,0,0,1,1,0,1,1,2,1]
Phi over symmetry	[-3,0,0,0,1,2,0,2,2,1,3,0,1,0,0,0,1,1,1,1,0]
Phi of -K	[-2,-1,0,0,0,3,0,0,1,1,3,0,1,1,1,-1,0,0,0,2,2]
Phi of K*	[-3,0,0,0,1,2,0,2,2,1,3,0,1,0,0,0,1,1,1,1,0]
Phi of -K*	[-2,-1,0,0,0,3,1,1,1,2,2,0,0,1,3,0,-1,1,0,1,3]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	z^2+18z+33
Enhanced Jones-Krushkal polynomial	w^3z^2+18w^2z+33w
Inner characteristic polynomial	t^6+33t^4+35t^2+4
Outer characteristic polynomial	t^7+47t^5+56t^3+8t
Flat arrow polynomial	4K13 - 6K1*2 - 6K1K2 + 3K2 + 2*K3 + 4
2-strand cable arrow polynomial	-128K14K2*2 + 256K1*4K2 - 2096K14 + 448K1*3K2K3 - 768K1*3K3 + 320K12K2*3 - 3184K1*2K2*2 - 544K1*2K2K4 + 6464K1*2K2 - 912K12K3*2 - 4392K1*2 + 96K1K23K3 - 96K1K2*2K3 - 160K1K2K3K4 + 5416K1K2K3 + 968K1K3K4 + 64K1K4K5 - 32K2*6 + 96K2*4K4 - 472K24 - 208K2*2K3*2 - 120K2*2K4*2 + 664K2*2K4 - 3388K22 + 360K2K3K5 + 80K2K4K6 - 32K3*4 + 80K3*2K6 - 1820K32 - 418K4*2 - 132K5*2 - 52K6**2 + 3640
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{6}, {1, 5}, {2, 4}, {3}]]
If K is slice	False

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