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Flat knot 6.659

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,1,2,2,4,1,1,0,1,1,1,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.659']
Arrow polynomial of the knot is: 4K13 - 8K1*2 - 2K1K2 - 2K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']
Outer characteristic polynomial of the knot is: t^7+58t^5+34t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.659']
2-strand cable arrow polynomial of the knot is: -64K16 + 672K1*4K2 - 3168K14 + 352K1*3K2K3 + 64K1*3K3K4 - 896K1*3K3 - 192K12K2*4 + 480K1*2K2*3 + 224K1*2K2*2K4 - 3520K12K2*2 - 640K1*2K2K4 + 7184K1*2K2 - 864K12K3*2 - 128K1*2K4*2 - 3772K1*2 + 128K1K23K3 - 160K1K2*2K3 - 32K1K2*2K5 - 64K1K2K3K4 + 4776K1K2K3 + 888K1K3K4 + 48K1K4K5 - 32K2*6 + 32K2*4K4 - 336K24 - 32K2*2K3*2 - 8K2*2K4*2 + 344K2*2K4 - 2904K22 + 8K2K3K5 - 1288K32 - 228K4*2 - 4K5**2 + 3130
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.659']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11519', 'vk6.11850', 'vk6.12869', 'vk6.13176', 'vk6.20358', 'vk6.21701', 'vk6.27662', 'vk6.29208', 'vk6.31294', 'vk6.31689', 'vk6.32452', 'vk6.32867', 'vk6.39100', 'vk6.41356', 'vk6.45856', 'vk6.47519', 'vk6.52302', 'vk6.52566', 'vk6.53146', 'vk6.53450', 'vk6.57217', 'vk6.58440', 'vk6.61831', 'vk6.62964', 'vk6.63811', 'vk6.63943', 'vk6.64257', 'vk6.64453', 'vk6.66830', 'vk6.67700', 'vk6.69470', 'vk6.70194']
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,-1,-1,1,2,2,-1,1,2,2,4,1,1,0,1,1,1,2,1,1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.659']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']

Outer characteristic polynomial of the knot is: t^7+58t^5+34t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.659']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 672*K1**4*K2 - 3168*K1**4 + 352*K1**3*K2*K3 + 64*K1**3*K3*K4 - 896*K1**3*K3 - 192*K1**2*K2**4 + 480*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 3520*K1**2*K2**2 - 640*K1**2*K2*K4 + 7184*K1**2*K2 - 864*K1**2*K3**2 - 128*K1**2*K4**2 - 3772*K1**2 + 128*K1*K2**3*K3 - 160*K1*K2**2*K3 - 32*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 4776*K1*K2*K3 + 888*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 336*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 344*K2**2*K4 - 2904*K2**2 + 8*K2*K3*K5 - 1288*K3**2 - 228*K4**2 - 4*K5**2 + 3130

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.659']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11519', 'vk6.11850', 'vk6.12869', 'vk6.13176', 'vk6.20358', 'vk6.21701', 'vk6.27662', 'vk6.29208', 'vk6.31294', 'vk6.31689', 'vk6.32452', 'vk6.32867', 'vk6.39100', 'vk6.41356', 'vk6.45856', 'vk6.47519', 'vk6.52302', 'vk6.52566', 'vk6.53146', 'vk6.53450', 'vk6.57217', 'vk6.58440', 'vk6.61831', 'vk6.62964', 'vk6.63811', 'vk6.63943', 'vk6.64257', 'vk6.64453', 'vk6.66830', 'vk6.67700', 'vk6.69470', 'vk6.70194']

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	O1O2O3O4U2O5U3U1O6U5U6U4
R3 orbit	{'O1O2O3O4U2O5U3U1O6U5U6U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U1U5U6O5U4U2O6U3
Gauss code of K*	O1O2O3U4U5U6U3O5U1O6O4U2
Gauss code of -K*	O1O2O3U2O4O5U3O6U1U5U6U4
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 -2 -1 3 1 1],[ 2 0 -1 1 4 2 1],[ 2 1 0 1 2 1 0],[ 1 -1 -1 0 2 1 1],[-3 -4 -2 -2 0 -1 1],[-1 -2 -1 -1 1 0 1],[-1 -1 0 -1 -1 -1 0]]
Primitive based matrix	[[ 0 3 1 1 -1 -2 -2],[-3 0 1 -1 -2 -2 -4],[-1 -1 0 -1 -1 0 -1],[-1 1 1 0 -1 -1 -2],[ 1 2 1 1 0 -1 -1],[ 2 2 0 1 1 0 1],[ 2 4 1 2 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,1,2,2,-1,1,2,2,4,1,1,0,1,1,1,2,1,1,-1]
Phi over symmetry	[-3,-1,-1,1,2,2,-1,1,2,2,4,1,1,0,1,1,1,2,1,1,-1]
Phi of -K	[-2,-2,-1,1,1,3,-1,0,2,3,3,0,1,2,1,1,1,2,-1,1,3]
Phi of K*	[-3,-1,-1,1,2,2,1,3,2,1,3,1,1,1,2,1,2,3,0,0,-1]
Phi of -K*	[-2,-2,-1,1,1,3,-1,1,1,2,4,1,0,1,2,1,1,2,-1,-1,1]
Symmetry type of based matrix	c
u-polynomial	-t^3+2t^2-t
Normalized Jones-Krushkal polynomial	2z^2+19z+31
Enhanced Jones-Krushkal polynomial	2w^3z^2+19w^2z+31w
Inner characteristic polynomial	t^6+38t^4+8t^2
Outer characteristic polynomial	t^7+58t^5+34t^3+4t
Flat arrow polynomial	4K13 - 8K1*2 - 2K1K2 - 2K1 + 4*K2 + 5
2-strand cable arrow polynomial	-64K16 + 672K1*4K2 - 3168K14 + 352K1*3K2K3 + 64K1*3K3K4 - 896K1*3K3 - 192K12K2*4 + 480K1*2K2*3 + 224K1*2K2*2K4 - 3520K12K2*2 - 640K1*2K2K4 + 7184K1*2K2 - 864K12K3*2 - 128K1*2K4*2 - 3772K1*2 + 128K1K23K3 - 160K1K2*2K3 - 32K1K2*2K5 - 64K1K2K3K4 + 4776K1K2K3 + 888K1K3K4 + 48K1K4K5 - 32K2*6 + 32K2*4K4 - 336K24 - 32K2*2K3*2 - 8K2*2K4*2 + 344K2*2K4 - 2904K22 + 8K2K3K5 - 1288K32 - 228K4*2 - 4K5**2 + 3130
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice	False

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