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Flat knot 6.139

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,-1,1,2,2,0,1,1,1,1,1,2,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.139']
Arrow polynomial of the knot is: 4K12K2 - 4K1K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']
Outer characteristic polynomial of the knot is: t^7+40t^5+96t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.138', '6.139']
2-strand cable arrow polynomial of the knot is: -512K14 + 512K1*3K2K3 - 128K1*2K2*2K3*2 - 2240K1*2K2*2 - 384K1*2K2K4 + 2048K1*2K2 - 2048K12K3*2 - 2032K1*2 + 832K1K23K3 + 256K1K2*2K3K4 - 640K1K22K3 - 128K1K2*2K5 + 512K1K2K33 - 64K1K2K3K4 - 384K1K2K3K6 + 5808K1K2K3 + 1728K1K3K4 + 48K1K5K6 - 32K24K4*2 + 128K2*4K4 - 976K24 + 64K2*3K4K6 - 64K2*3K6 - 2624K22K3*2 - 256K2*2K4*2 + 960K2*2K4 - 48K22K6*2 - 1760K2*2 - 192K2K32K4 + 1504K2K3K5 + 288K2K4K6 + 16K2K6K8 - 640K3*4 + 448K3*2K6 - 1520K32 - 448K4*2 - 160K5*2 - 112K6*2 - 2K8**2 + 2240
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.139']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4806', 'vk6.5146', 'vk6.6367', 'vk6.6801', 'vk6.8327', 'vk6.8769', 'vk6.9700', 'vk6.10008', 'vk6.21108', 'vk6.22538', 'vk6.28554', 'vk6.42204', 'vk6.46830', 'vk6.48067', 'vk6.48822', 'vk6.49867', 'vk6.50832', 'vk6.51516', 'vk6.58987', 'vk6.69823']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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,-1,-1,1,1,2,-1,-1,1,2,2,0,1,1,1,1,1,2,0,2,2]

Flat knots (up to 7 crossings) with same phi are :['6.139']

Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1*K3 + K4

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']

Outer characteristic polynomial of the knot is: t^7+40t^5+96t^3+11t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.138', '6.139']

2-strand cable arrow polynomial of the knot is: -512*K1**4 + 512*K1**3*K2*K3 - 128*K1**2*K2**2*K3**2 - 2240*K1**2*K2**2 - 384*K1**2*K2*K4 + 2048*K1**2*K2 - 2048*K1**2*K3**2 - 2032*K1**2 + 832*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 640*K1*K2**2*K3 - 128*K1*K2**2*K5 + 512*K1*K2*K3**3 - 64*K1*K2*K3*K4 - 384*K1*K2*K3*K6 + 5808*K1*K2*K3 + 1728*K1*K3*K4 + 48*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 976*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 2624*K2**2*K3**2 - 256*K2**2*K4**2 + 960*K2**2*K4 - 48*K2**2*K6**2 - 1760*K2**2 - 192*K2*K3**2*K4 + 1504*K2*K3*K5 + 288*K2*K4*K6 + 16*K2*K6*K8 - 640*K3**4 + 448*K3**2*K6 - 1520*K3**2 - 448*K4**2 - 160*K5**2 - 112*K6**2 - 2*K8**2 + 2240

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.139']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4806', 'vk6.5146', 'vk6.6367', 'vk6.6801', 'vk6.8327', 'vk6.8769', 'vk6.9700', 'vk6.10008', 'vk6.21108', 'vk6.22538', 'vk6.28554', 'vk6.42204', 'vk6.46830', 'vk6.48067', 'vk6.48822', 'vk6.49867', 'vk6.50832', 'vk6.51516', 'vk6.58987', 'vk6.69823']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is r.

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	O1O2O3O4O5O6U4U6U5U2U1U3
R3 orbit	{'O1O2O3O4O5O6U4U6U5U2U1U3'}
R3 orbit length	1
Gauss code of -K	Same
Gauss code of K*	O1O2O3O4O5O6U5U4U6U1U3U2
Gauss code of -K*	O1O2O3O4O5O6U5U4U6U1U3U2
Diagrammatic symmetry type	r
Flat genus of the diagram	3
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -1 -1 2 -2 1 1],[ 1 0 0 2 -2 1 1],[ 1 0 0 1 -2 1 1],[-2 -2 -1 0 -2 1 1],[ 2 2 2 2 0 2 1],[-1 -1 -1 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 2 1 1 -1 -1 -2],[-2 0 1 1 -1 -2 -2],[-1 -1 0 0 -1 -1 -1],[-1 -1 0 0 -1 -1 -2],[ 1 1 1 1 0 0 -2],[ 1 2 1 1 0 0 -2],[ 2 2 1 2 2 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,1,1,2,-1,-1,1,2,2,0,1,1,1,1,1,2,0,2,2]
Phi over symmetry	[-2,-1,-1,1,1,2,-1,-1,1,2,2,0,1,1,1,1,1,2,0,2,2]
Phi of -K	[-2,-1,-1,1,1,2,-1,-1,1,2,2,0,1,1,1,1,1,2,0,2,2]
Phi of K*	[-2,-1,-1,1,1,2,2,2,1,2,2,0,1,1,1,1,1,2,0,-1,-1]
Phi of -K*	[-2,-1,-1,1,1,2,2,2,1,2,2,0,1,1,1,1,1,2,0,-1,-1]
Symmetry type of based matrix	r
u-polynomial	0
Normalized Jones-Krushkal polynomial	9z^2+30z+25
Enhanced Jones-Krushkal polynomial	9w^3z^2+30w^2z+25w
Inner characteristic polynomial	t^6+28t^4+34t^2+1
Outer characteristic polynomial	t^7+40t^5+96t^3+11t
Flat arrow polynomial	4K12K2 - 4K1K3 + K4
2-strand cable arrow polynomial	-512K14 + 512K1*3K2K3 - 128K1*2K2*2K3*2 - 2240K1*2K2*2 - 384K1*2K2K4 + 2048K1*2K2 - 2048K12K3*2 - 2032K1*2 + 832K1K23K3 + 256K1K2*2K3K4 - 640K1K22K3 - 128K1K2*2K5 + 512K1K2K33 - 64K1K2K3K4 - 384K1K2K3K6 + 5808K1K2K3 + 1728K1K3K4 + 48K1K5K6 - 32K24K4*2 + 128K2*4K4 - 976K24 + 64K2*3K4K6 - 64K2*3K6 - 2624K22K3*2 - 256K2*2K4*2 + 960K2*2K4 - 48K22K6*2 - 1760K2*2 - 192K2K32K4 + 1504K2K3K5 + 288K2K4K6 + 16K2K6K8 - 640K3*4 + 448K3*2K6 - 1520K32 - 448K4*2 - 160K5*2 - 112K6*2 - 2K8**2 + 2240
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}]]
If K is slice	False

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