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Flat knot 6.1930

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,1,1,1,2,1,1,2,1,0,-1,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1930']
Arrow polynomial of the knot is: 4K12K2 - 12K12 - 8K1K2 - 4K1K3 + 4K1 + 6K2 + 4K3 + K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1930']
Outer characteristic polynomial of the knot is: t^7+28t^5+54t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1930']
2-strand cable arrow polynomial of the knot is: -2016K14 + 2048K1*3K2K3 + 64K1*3K3K4 - 960K1*3K3 + 128K12K2*3 - 640K1*2K2*2K3*2 - 10048K1*2K2*2 + 256K1*2K2K32 + 64K1*2K2K3K5 - 2144K12K2K4 + 12816K1*2K2 - 1632K12K3*2 - 224K1*2K4*2 - 32K1*2K4K6 - 9216K1*2 + 2624K1K23K3 + 256K1K2*2K3K4 - 2048K1K22K3 + 64K1K2*2K4K5 - 1280K1K22K5 + 512K1K2K33 - 448K1K2K3K4 + 13840K1K2K3 - 256K1K2K4K5 - 64K1K2K4K7 + 2544K1K3K4 + 464K1K4K5 + 96K1K5K6 + 16K1K6K7 - 32K24K4*2 + 128K2*4K4 - 2624K24 + 64K2*3K4K6 - 96K2*3K6 - 1984K22K3*2 - 352K2*2K4*2 - 32K2*2K4K8 + 3488K2*2K4 - 128K22K5*2 - 16K2*2K6*2 - 6752K2*2 + 1456K2K3K5 + 336K2K4K6 + 96K2K5K7 + 16K2K6K8 - 128K3*4 - 3752K3*2 - 1212K4*2 - 328K5*2 - 80K6*2 - 16K7*2 - 2K8**2 + 7164
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1930']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71643', 'vk6.71662', 'vk6.71822', 'vk6.72244', 'vk6.72265', 'vk6.72374', 'vk6.72382', 'vk6.77259', 'vk6.77276', 'vk6.77357', 'vk6.77380', 'vk6.77607', 'vk6.77701', 'vk6.77723', 'vk6.81409', 'vk6.81443', 'vk6.86955', 'vk6.87164', 'vk6.88001', 'vk6.89549']
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,-2,1,1,1,1,0,1,1,1,2,1,1,2,1,0,-1,0,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1930']

Arrow polynomial of the knot is: 4*K1**2*K2 - 12*K1**2 - 8*K1*K2 - 4*K1*K3 + 4*K1 + 6*K2 + 4*K3 + K4 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1930']

Outer characteristic polynomial of the knot is: t^7+28t^5+54t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1930']

2-strand cable arrow polynomial of the knot is: -2016*K1**4 + 2048*K1**3*K2*K3 + 64*K1**3*K3*K4 - 960*K1**3*K3 + 128*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 - 10048*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 64*K1**2*K2*K3*K5 - 2144*K1**2*K2*K4 + 12816*K1**2*K2 - 1632*K1**2*K3**2 - 224*K1**2*K4**2 - 32*K1**2*K4*K6 - 9216*K1**2 + 2624*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 1280*K1*K2**2*K5 + 512*K1*K2*K3**3 - 448*K1*K2*K3*K4 + 13840*K1*K2*K3 - 256*K1*K2*K4*K5 - 64*K1*K2*K4*K7 + 2544*K1*K3*K4 + 464*K1*K4*K5 + 96*K1*K5*K6 + 16*K1*K6*K7 - 32*K2**4*K4**2 + 128*K2**4*K4 - 2624*K2**4 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 1984*K2**2*K3**2 - 352*K2**2*K4**2 - 32*K2**2*K4*K8 + 3488*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 6752*K2**2 + 1456*K2*K3*K5 + 336*K2*K4*K6 + 96*K2*K5*K7 + 16*K2*K6*K8 - 128*K3**4 - 3752*K3**2 - 1212*K4**2 - 328*K5**2 - 80*K6**2 - 16*K7**2 - 2*K8**2 + 7164

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1930']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71643', 'vk6.71662', 'vk6.71822', 'vk6.72244', 'vk6.72265', 'vk6.72374', 'vk6.72382', 'vk6.77259', 'vk6.77276', 'vk6.77357', 'vk6.77380', 'vk6.77607', 'vk6.77701', 'vk6.77723', 'vk6.81409', 'vk6.81443', 'vk6.86955', 'vk6.87164', 'vk6.88001', 'vk6.89549']

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	O1O2O3U1U4U2O5O6O4U5U3U6
R3 orbit	{'O1O2O3U1U4U2O5O6O4U5U3U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U1U5O6O4O5U2U6U3
Gauss code of K*	O1O2O3U4U5U2O4O6O5U1U3U6
Gauss code of -K*	O1O2O3U4U1U3O5O4O6U2U5U6
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 1 1 1 -2 1],[ 2 0 1 2 1 0 1],[-1 -1 0 0 -1 -1 0],[-1 -2 0 0 0 -1 1],[-1 -1 1 0 0 -2 0],[ 2 0 1 1 2 0 1],[-1 -1 0 -1 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 1 -2 -2],[-1 0 1 0 0 -1 -2],[-1 -1 0 0 0 -1 -1],[-1 0 0 0 1 -2 -1],[-1 0 0 -1 0 -1 -1],[ 2 1 1 2 1 0 0],[ 2 2 1 1 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,2,2,-1,0,0,1,2,0,0,1,1,-1,2,1,1,1,0]
Phi over symmetry	[-2,-2,1,1,1,1,0,1,1,1,2,1,1,2,1,0,-1,0,0,-1,0]
Phi of -K	[-2,-2,1,1,1,1,0,1,2,2,2,2,1,2,2,0,-1,0,0,-1,0]
Phi of K*	[-1,-1,-1,-1,2,2,-1,0,0,2,2,0,0,1,2,-1,2,2,2,1,0]
Phi of -K*	[-2,-2,1,1,1,1,0,1,1,1,2,1,1,2,1,0,-1,0,0,-1,0]
Symmetry type of based matrix	c
u-polynomial	2t^2-4t
Normalized Jones-Krushkal polynomial	5z^2+26z+33
Enhanced Jones-Krushkal polynomial	5w^3z^2+26w^2z+33w
Inner characteristic polynomial	t^6+16t^4+28t^2
Outer characteristic polynomial	t^7+28t^5+54t^3+4t
Flat arrow polynomial	4K12K2 - 12K12 - 8K1K2 - 4K1K3 + 4K1 + 6K2 + 4K3 + K4 + 6
2-strand cable arrow polynomial	-2016K14 + 2048K1*3K2K3 + 64K1*3K3K4 - 960K1*3K3 + 128K12K2*3 - 640K1*2K2*2K3*2 - 10048K1*2K2*2 + 256K1*2K2K32 + 64K1*2K2K3K5 - 2144K12K2K4 + 12816K1*2K2 - 1632K12K3*2 - 224K1*2K4*2 - 32K1*2K4K6 - 9216K1*2 + 2624K1K23K3 + 256K1K2*2K3K4 - 2048K1K22K3 + 64K1K2*2K4K5 - 1280K1K22K5 + 512K1K2K33 - 448K1K2K3K4 + 13840K1K2K3 - 256K1K2K4K5 - 64K1K2K4K7 + 2544K1K3K4 + 464K1K4K5 + 96K1K5K6 + 16K1K6K7 - 32K24K4*2 + 128K2*4K4 - 2624K24 + 64K2*3K4K6 - 96K2*3K6 - 1984K22K3*2 - 352K2*2K4*2 - 32K2*2K4K8 + 3488K2*2K4 - 128K22K5*2 - 16K2*2K6*2 - 6752K2*2 + 1456K2K3K5 + 336K2K4K6 + 96K2K5K7 + 16K2K6K8 - 128K3*4 - 3752K3*2 - 1212K4*2 - 328K5*2 - 80K6*2 - 16K7*2 - 2K8**2 + 7164
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice	False

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