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Flat knot 6.489

Min(phi) over symmetries of the knot is: [-4,-1,0,0,2,3,0,2,3,4,3,1,1,1,1,0,1,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.489']
Arrow polynomial of the knot is: 4K12K2 - 2K12 - 2K1K2 + K1 - 2K2**2 - K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.150', '6.343', '6.489']
Outer characteristic polynomial of the knot is: t^7+85t^5+41t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.489']
2-strand cable arrow polynomial of the knot is: -192K14 + 96K1*3K3K4 - 1152K1*2K2*2 + 64K1*2K2K42 - 1280K1*2K2K4 + 2816K1*2K2 - 448K12K3*2 - 512K1*2K4*2 - 32K1*2K4K6 - 3560K1*2 - 160K1K22K5 + 64K1K2K3K4*2 - 352K1K2K3K4 - 32K1K2K3K6 + 3304K1K2K3 - 96K1K2K4K5 - 32K1K2K4K7 + 2400K1K3K4 + 544K1K4K5 + 40K1K5K6 - 32K2*4K4*2 + 128K2*4K4 - 488K24 - 32K2*2K3*2 + 32K2*2K4*3 - 440K2*2K4*2 + 1712K2*2K4 - 2746K22 - 32K2K32K4 + 336K2K3K5 + 288K2K4K6 - 32K32K4*2 + 16K3*2K6 - 1576K32 + 16K3K4K7 - 8K44 - 1374K4*2 - 176K5*2 - 38K6**2 + 2964
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.489']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71583', 'vk6.71705', 'vk6.72124', 'vk6.72322', 'vk6.73476', 'vk6.74115', 'vk6.74127', 'vk6.74685', 'vk6.74698', 'vk6.75233', 'vk6.75484', 'vk6.76153', 'vk6.76168', 'vk6.77203', 'vk6.77311', 'vk6.77514', 'vk6.77663', 'vk6.78445', 'vk6.79116', 'vk6.79128', 'vk6.80031', 'vk6.80181', 'vk6.80623', 'vk6.80632', 'vk6.83739', 'vk6.83862', 'vk6.85057', 'vk6.85343', 'vk6.86667', 'vk6.86977', 'vk6.87409', 'vk6.89541']
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,-1,0,0,2,3,0,2,3,4,3,1,1,1,1,0,1,1,1,2,2]

Flat knots (up to 7 crossings) with same phi are :['6.489']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.150', '6.343', '6.489']

Outer characteristic polynomial of the knot is: t^7+85t^5+41t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.489']

2-strand cable arrow polynomial of the knot is: -192*K1**4 + 96*K1**3*K3*K4 - 1152*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 1280*K1**2*K2*K4 + 2816*K1**2*K2 - 448*K1**2*K3**2 - 512*K1**2*K4**2 - 32*K1**2*K4*K6 - 3560*K1**2 - 160*K1*K2**2*K5 + 64*K1*K2*K3*K4**2 - 352*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3304*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2400*K1*K3*K4 + 544*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 488*K2**4 - 32*K2**2*K3**2 + 32*K2**2*K4**3 - 440*K2**2*K4**2 + 1712*K2**2*K4 - 2746*K2**2 - 32*K2*K3**2*K4 + 336*K2*K3*K5 + 288*K2*K4*K6 - 32*K3**2*K4**2 + 16*K3**2*K6 - 1576*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 1374*K4**2 - 176*K5**2 - 38*K6**2 + 2964

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.489']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71583', 'vk6.71705', 'vk6.72124', 'vk6.72322', 'vk6.73476', 'vk6.74115', 'vk6.74127', 'vk6.74685', 'vk6.74698', 'vk6.75233', 'vk6.75484', 'vk6.76153', 'vk6.76168', 'vk6.77203', 'vk6.77311', 'vk6.77514', 'vk6.77663', 'vk6.78445', 'vk6.79116', 'vk6.79128', 'vk6.80031', 'vk6.80181', 'vk6.80623', 'vk6.80632', 'vk6.83739', 'vk6.83862', 'vk6.85057', 'vk6.85343', 'vk6.86667', 'vk6.86977', 'vk6.87409', 'vk6.89541']

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	O1O2O3O4O5U1U4U3O6U2U5U6
R3 orbit	{'O1O2O3O4O5U1U4U3O6U2U5U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U1U4O6U3U2U5
Gauss code of K*	O1O2O3U4O5O6O4U1U5U3U2U6
Gauss code of -K*	O1O2O3U1O4O5O6U2U5U4U3U6
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 0 0 3 2],[ 4 0 3 2 1 4 2],[ 1 -3 0 0 0 3 2],[ 0 -2 0 0 0 2 1],[ 0 -1 0 0 0 1 1],[-3 -4 -3 -2 -1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix	[[ 0 3 2 0 0 -1 -4],[-3 0 1 -1 -2 -3 -4],[-2 -1 0 -1 -1 -2 -2],[ 0 1 1 0 0 0 -1],[ 0 2 1 0 0 0 -2],[ 1 3 2 0 0 0 -3],[ 4 4 2 1 2 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,0,0,1,4,-1,1,2,3,4,1,1,2,2,0,0,1,0,2,3]
Phi over symmetry	[-4,-1,0,0,2,3,0,2,3,4,3,1,1,1,1,0,1,1,1,2,2]
Phi of -K	[-4,-1,0,0,2,3,0,2,3,4,3,1,1,1,1,0,1,1,1,2,2]
Phi of K*	[-3,-2,0,0,1,4,2,1,2,1,3,1,1,1,4,0,1,2,1,3,0]
Phi of -K*	[-4,-1,0,0,2,3,3,1,2,2,4,0,0,2,3,0,1,1,1,2,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial	6z^2+23z+23
Enhanced Jones-Krushkal polynomial	6w^3z^2+23w^2z+23w
Inner characteristic polynomial	t^6+55t^4+12t^2
Outer characteristic polynomial	t^7+85t^5+41t^3+4t
Flat arrow polynomial	4K12K2 - 2K12 - 2K1K2 + K1 - 2K2**2 - K2 + K3 + 2
2-strand cable arrow polynomial	-192K14 + 96K1*3K3K4 - 1152K1*2K2*2 + 64K1*2K2K42 - 1280K1*2K2K4 + 2816K1*2K2 - 448K12K3*2 - 512K1*2K4*2 - 32K1*2K4K6 - 3560K1*2 - 160K1K22K5 + 64K1K2K3K4*2 - 352K1K2K3K4 - 32K1K2K3K6 + 3304K1K2K3 - 96K1K2K4K5 - 32K1K2K4K7 + 2400K1K3K4 + 544K1K4K5 + 40K1K5K6 - 32K2*4K4*2 + 128K2*4K4 - 488K24 - 32K2*2K3*2 + 32K2*2K4*3 - 440K2*2K4*2 + 1712K2*2K4 - 2746K22 - 32K2K32K4 + 336K2K3K5 + 288K2K4K6 - 32K32K4*2 + 16K3*2K6 - 1576K32 + 16K3K4K7 - 8K44 - 1374K4*2 - 176K5*2 - 38K6**2 + 2964
Genus of based matrix	1
Fillings of based matrix	[[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}], [{4, 6}, {1, 5}, {2, 3}]]
If K is slice	False

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