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Flat knot 6.163

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,1,2,1,4,4,0,0,1,1,0,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.163']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 4K12 - 2K1K2 - 2K1K3 - 2K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.80', '6.163', '6.227', '6.447']
Outer characteristic polynomial of the knot is: t^7+100t^5+84t^3+12t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.163']
2-strand cable arrow polynomial of the knot is: -192K12K2*4 - 128K1*2K2*3K4 + 1760K12K2*3 + 256K1*2K2*2K4 - 6352K12K2*2 + 128K1*2K2K42 - 704K1*2K2K4 + 6424K1*2K2 - 192K12K4*2 - 4888K1*2 + 736K1K23K3 + 160K1K2*2K3K4 - 1696K1K22K3 - 416K1K2*2K5 - 448K1K2K3K4 - 32K1K2K3K6 + 6048K1K2K3 - 32K1K2K4K5 + 968K1K3K4 + 248K1K4K5 - 32K2*6 - 32K2*4K4*2 + 352K2*4K4 - 2256K24 + 64K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 - 464K22K3*2 - 408K2*2K4*2 + 2456K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 2952K2*2 + 456K2K3K5 + 160K2K4K6 - 1560K3*2 - 734K4*2 - 112K5*2 - 16K6**2 + 3500
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.163']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17007', 'vk6.17249', 'vk6.20526', 'vk6.21922', 'vk6.23413', 'vk6.23720', 'vk6.27976', 'vk6.29447', 'vk6.35483', 'vk6.35933', 'vk6.39380', 'vk6.41567', 'vk6.42914', 'vk6.43214', 'vk6.45951', 'vk6.47632', 'vk6.55182', 'vk6.55425', 'vk6.57392', 'vk6.58562', 'vk6.59562', 'vk6.59897', 'vk6.62050', 'vk6.63042', 'vk6.64980', 'vk6.65189', 'vk6.66937', 'vk6.67792', 'vk6.68269', 'vk6.68424', 'vk6.69543', 'vk6.70245']
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,-1,1,2,3,1,2,1,4,4,0,0,1,1,0,2,2,-1,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.163']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.80', '6.163', '6.227', '6.447']

Outer characteristic polynomial of the knot is: t^7+100t^5+84t^3+12t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.163']

2-strand cable arrow polynomial of the knot is: -192*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 1760*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 6352*K1**2*K2**2 + 128*K1**2*K2*K4**2 - 704*K1**2*K2*K4 + 6424*K1**2*K2 - 192*K1**2*K4**2 - 4888*K1**2 + 736*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1696*K1*K2**2*K3 - 416*K1*K2**2*K5 - 448*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6048*K1*K2*K3 - 32*K1*K2*K4*K5 + 968*K1*K3*K4 + 248*K1*K4*K5 - 32*K2**6 - 32*K2**4*K4**2 + 352*K2**4*K4 - 2256*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 464*K2**2*K3**2 - 408*K2**2*K4**2 + 2456*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 2952*K2**2 + 456*K2*K3*K5 + 160*K2*K4*K6 - 1560*K3**2 - 734*K4**2 - 112*K5**2 - 16*K6**2 + 3500

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.163']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17007', 'vk6.17249', 'vk6.20526', 'vk6.21922', 'vk6.23413', 'vk6.23720', 'vk6.27976', 'vk6.29447', 'vk6.35483', 'vk6.35933', 'vk6.39380', 'vk6.41567', 'vk6.42914', 'vk6.43214', 'vk6.45951', 'vk6.47632', 'vk6.55182', 'vk6.55425', 'vk6.57392', 'vk6.58562', 'vk6.59562', 'vk6.59897', 'vk6.62050', 'vk6.63042', 'vk6.64980', 'vk6.65189', 'vk6.66937', 'vk6.67792', 'vk6.68269', 'vk6.68424', 'vk6.69543', 'vk6.70245']

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	O1O2O3O4O5U1O6U4U3U6U5U2
R3 orbit	{'O1O2O3O4O5U1O6U4U3U6U5U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U1U6U3U2O6U5
Gauss code of K*	O1O2O3O4O5U6U5U2U1U4O6U3
Gauss code of -K*	O1O2O3O4O5U3O6U2U5U4U1U6
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 -4 1 -1 -1 3 2],[ 4 0 4 2 1 3 2],[-1 -4 0 -2 -2 2 2],[ 1 -2 2 0 0 3 2],[ 1 -1 2 0 0 2 1],[-3 -3 -2 -3 -2 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix	[[ 0 3 2 1 -1 -1 -4],[-3 0 0 -2 -2 -3 -3],[-2 0 0 -2 -1 -2 -2],[-1 2 2 0 -2 -2 -4],[ 1 2 1 2 0 0 -1],[ 1 3 2 2 0 0 -2],[ 4 3 2 4 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,1,1,4,0,2,2,3,3,2,1,2,2,2,2,4,0,1,2]
Phi over symmetry	[-4,-1,-1,1,2,3,1,2,1,4,4,0,0,1,1,0,2,2,-1,0,1]
Phi of -K	[-4,-1,-1,1,2,3,1,2,1,4,4,0,0,1,1,0,2,2,-1,0,1]
Phi of K*	[-3,-2,-1,1,1,4,1,0,1,2,4,-1,1,2,4,0,0,1,0,1,2]
Phi of -K*	[-4,-1,-1,1,2,3,1,2,4,2,3,0,2,1,2,2,2,3,2,2,0]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial	8z^2+25z+19
Enhanced Jones-Krushkal polynomial	-2w^4z^2+10w^3z^2-4w^3z+29w^2z+19w
Inner characteristic polynomial	t^6+68t^4+35t^2
Outer characteristic polynomial	t^7+100t^5+84t^3+12t
Flat arrow polynomial	4K13 + 4K1*2K2 - 4K12 - 2K1K2 - 2K1K3 - 2K1 + K2 + 2
2-strand cable arrow polynomial	-192K12K2*4 - 128K1*2K2*3K4 + 1760K12K2*3 + 256K1*2K2*2K4 - 6352K12K2*2 + 128K1*2K2K42 - 704K1*2K2K4 + 6424K1*2K2 - 192K12K4*2 - 4888K1*2 + 736K1K23K3 + 160K1K2*2K3K4 - 1696K1K22K3 - 416K1K2*2K5 - 448K1K2K3K4 - 32K1K2K3K6 + 6048K1K2K3 - 32K1K2K4K5 + 968K1K3K4 + 248K1K4K5 - 32K2*6 - 32K2*4K4*2 + 352K2*4K4 - 2256K24 + 64K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 - 464K22K3*2 - 408K2*2K4*2 + 2456K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 2952K2*2 + 456K2K3K5 + 160K2K4K6 - 1560K3*2 - 734K4*2 - 112K5*2 - 16K6**2 + 3500
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice	False

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