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Flat knot 6.1654

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,0,2,3,3,0,2,2,1,0,0,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1654']
Arrow polynomial of the knot is: -14K12 - 8K1K2 + 4K1 + 7K2 + 4K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1591', '6.1654']
Outer characteristic polynomial of the knot is: t^7+39t^5+48t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.948', '6.1654']
2-strand cable arrow polynomial of the knot is: -128K16 - 192K1*4K2*2 + 1696K1*4K2 - 6016K14 + 992K1*3K2K3 + 128K1*3K3K4 - 1440K1*3K3 + 32K13K4K5 + 256K1*2K2*2K4 - 6432K12K2*2 - 992K1*2K2K4 + 13608K1*2K2 - 1408K12K3*2 - 432K1*2K4*2 - 64K1*2K5*2 - 8212K1*2 - 1248K1K22K3 - 224K1K2*2K5 - 512K1K2K3K4 + 10656K1K2K3 + 3016K1K3K4 + 752K1K4K5 + 56K1K5K6 - 408K24 - 272K2*2K3*2 - 64K2*2K4*2 + 1968K2*2K4 - 7632K22 + 736K2K3K5 + 64K2K4K6 - 3916K3*2 - 1674K4*2 - 440K5*2 - 32K6**2 + 7776
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1654']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3656', 'vk6.3751', 'vk6.3946', 'vk6.4041', 'vk6.4497', 'vk6.4594', 'vk6.5879', 'vk6.6008', 'vk6.7147', 'vk6.7326', 'vk6.7417', 'vk6.7936', 'vk6.8057', 'vk6.9366', 'vk6.17908', 'vk6.18005', 'vk6.18746', 'vk6.24447', 'vk6.24865', 'vk6.25328', 'vk6.37485', 'vk6.43882', 'vk6.44212', 'vk6.44517', 'vk6.48296', 'vk6.48359', 'vk6.50085', 'vk6.50199', 'vk6.50581', 'vk6.50646', 'vk6.55853', 'vk6.60720']
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,-2,0,1,1,2,-1,0,2,3,3,0,2,2,1,0,0,1,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.1654']

Arrow polynomial of the knot is: -14*K1**2 - 8*K1*K2 + 4*K1 + 7*K2 + 4*K3 + 8

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1591', '6.1654']

Outer characteristic polynomial of the knot is: t^7+39t^5+48t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.948', '6.1654']

2-strand cable arrow polynomial of the knot is: -128*K1**6 - 192*K1**4*K2**2 + 1696*K1**4*K2 - 6016*K1**4 + 992*K1**3*K2*K3 + 128*K1**3*K3*K4 - 1440*K1**3*K3 + 32*K1**3*K4*K5 + 256*K1**2*K2**2*K4 - 6432*K1**2*K2**2 - 992*K1**2*K2*K4 + 13608*K1**2*K2 - 1408*K1**2*K3**2 - 432*K1**2*K4**2 - 64*K1**2*K5**2 - 8212*K1**2 - 1248*K1*K2**2*K3 - 224*K1*K2**2*K5 - 512*K1*K2*K3*K4 + 10656*K1*K2*K3 + 3016*K1*K3*K4 + 752*K1*K4*K5 + 56*K1*K5*K6 - 408*K2**4 - 272*K2**2*K3**2 - 64*K2**2*K4**2 + 1968*K2**2*K4 - 7632*K2**2 + 736*K2*K3*K5 + 64*K2*K4*K6 - 3916*K3**2 - 1674*K4**2 - 440*K5**2 - 32*K6**2 + 7776

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1654']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3656', 'vk6.3751', 'vk6.3946', 'vk6.4041', 'vk6.4497', 'vk6.4594', 'vk6.5879', 'vk6.6008', 'vk6.7147', 'vk6.7326', 'vk6.7417', 'vk6.7936', 'vk6.8057', 'vk6.9366', 'vk6.17908', 'vk6.18005', 'vk6.18746', 'vk6.24447', 'vk6.24865', 'vk6.25328', 'vk6.37485', 'vk6.43882', 'vk6.44212', 'vk6.44517', 'vk6.48296', 'vk6.48359', 'vk6.50085', 'vk6.50199', 'vk6.50581', 'vk6.50646', 'vk6.55853', 'vk6.60720']

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	O1O2O3U1O4U5U4U3O5O6U2U6
R3 orbit	{'O1O2O3U1O4U5U4U3O5O6U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U2O4O5U1U6U5O6U3
Gauss code of K*	O1O2O3U1U4O5O4U6U5U3O6U2
Gauss code of -K*	O1O2O3U2O4U1U5U4O6O5U6U3
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 2 1 -2 1],[ 2 0 2 1 0 1 1],[ 0 -2 0 1 1 -2 1],[-2 -1 -1 0 0 -3 0],[-1 0 -1 0 0 -1 0],[ 2 -1 2 3 1 0 1],[-1 -1 -1 0 0 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -2 -2],[-2 0 0 0 -1 -1 -3],[-1 0 0 0 -1 0 -1],[-1 0 0 0 -1 -1 -1],[ 0 1 1 1 0 -2 -2],[ 2 1 0 1 2 0 1],[ 2 3 1 1 2 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,2,2,0,0,1,1,3,0,1,0,1,1,1,1,2,2,-1]
Phi over symmetry	[-2,-2,0,1,1,2,-1,0,2,3,3,0,2,2,1,0,0,1,0,1,1]
Phi of -K	[-2,-2,0,1,1,2,-1,0,2,3,3,0,2,2,1,0,0,1,0,1,1]
Phi of K*	[-2,-1,-1,0,2,2,1,1,1,1,3,0,0,2,2,0,2,3,0,0,-1]
Phi of -K*	[-2,-2,0,1,1,2,-1,2,1,1,3,2,0,1,1,1,1,1,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	21z+43
Enhanced Jones-Krushkal polynomial	21w^2z+43w
Inner characteristic polynomial	t^6+25t^4+27t^2+4
Outer characteristic polynomial	t^7+39t^5+48t^3+8t
Flat arrow polynomial	-14K12 - 8K1K2 + 4K1 + 7K2 + 4K3 + 8
2-strand cable arrow polynomial	-128K16 - 192K1*4K2*2 + 1696K1*4K2 - 6016K14 + 992K1*3K2K3 + 128K1*3K3K4 - 1440K1*3K3 + 32K13K4K5 + 256K1*2K2*2K4 - 6432K12K2*2 - 992K1*2K2K4 + 13608K1*2K2 - 1408K12K3*2 - 432K1*2K4*2 - 64K1*2K5*2 - 8212K1*2 - 1248K1K22K3 - 224K1K2*2K5 - 512K1K2K3K4 + 10656K1K2K3 + 3016K1K3K4 + 752K1K4K5 + 56K1K5K6 - 408K24 - 272K2*2K3*2 - 64K2*2K4*2 + 1968K2*2K4 - 7632K22 + 736K2K3K5 + 64K2K4K6 - 3916K3*2 - 1674K4*2 - 440K5*2 - 32K6**2 + 7776
Genus of based matrix	1
Fillings of based matrix	[[{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {1, 3}]]
If K is slice	False

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