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Flat knot 6.111

Min(phi) over symmetries of the knot is: [-3,-3,-2,2,3,3,0,0,3,2,3,0,4,3,4,2,1,2,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.111']
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+118t^5+105t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.111']
2-strand cable arrow polynomial of the knot is: 960K12K2*3 - 2624K1*2K2*2 - 320K1*2K2K4 + 2304K1*2K2 - 288K12K3*2 - 192K1*2K3K5 - 32K1*2K5*2 - 2224K1*2 + 192K1K22K3K4 - 576K1K22K3 + 192K1K2*2K4K5 - 640K1K22K5 - 192K1K2K3K4 - 192K1K2K3K6 + 3200K1K2K3 - 64K1K2K4K5 - 64K1K2K5K6 + 704K1K3K4 + 608K1K4K5 + 80K1K5K6 - 32K24K4*2 + 128K2*4K4 - 976K24 + 128K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 384K22K3*2 - 352K2*2K4*2 + 1216K2*2K4 - 320K22K5*2 - 48K2*2K6*2 - 1520K2*2 + 1136K2K3K5 + 288K2K4K6 + 112K2K5K7 + 16K2K6K8 + 48K3*2K6 - 1072K32 - 552K4*2 - 480K5*2 - 80K6*2 - 2K8**2 + 1912
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.111']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73622', 'vk6.73635', 'vk6.74416', 'vk6.75032', 'vk6.75576', 'vk6.75600', 'vk6.76601', 'vk6.76959', 'vk6.78539', 'vk6.78567', 'vk6.79451', 'vk6.79868', 'vk6.80234', 'vk6.80895', 'vk6.83697', 'vk6.84867', 'vk6.85672', 'vk6.87625', 'vk6.88436', 'vk6.89286']
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: [-3,-3,-2,2,3,3,0,0,3,2,3,0,4,3,4,2,1,2,1,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.111']

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+118t^5+105t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.111']

2-strand cable arrow polynomial of the knot is: 960*K1**2*K2**3 - 2624*K1**2*K2**2 - 320*K1**2*K2*K4 + 2304*K1**2*K2 - 288*K1**2*K3**2 - 192*K1**2*K3*K5 - 32*K1**2*K5**2 - 2224*K1**2 + 192*K1*K2**2*K3*K4 - 576*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 640*K1*K2**2*K5 - 192*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 3200*K1*K2*K3 - 64*K1*K2*K4*K5 - 64*K1*K2*K5*K6 + 704*K1*K3*K4 + 608*K1*K4*K5 + 80*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 976*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 384*K2**2*K3**2 - 352*K2**2*K4**2 + 1216*K2**2*K4 - 320*K2**2*K5**2 - 48*K2**2*K6**2 - 1520*K2**2 + 1136*K2*K3*K5 + 288*K2*K4*K6 + 112*K2*K5*K7 + 16*K2*K6*K8 + 48*K3**2*K6 - 1072*K3**2 - 552*K4**2 - 480*K5**2 - 80*K6**2 - 2*K8**2 + 1912

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.111']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73622', 'vk6.73635', 'vk6.74416', 'vk6.75032', 'vk6.75576', 'vk6.75600', 'vk6.76601', 'vk6.76959', 'vk6.78539', 'vk6.78567', 'vk6.79451', 'vk6.79868', 'vk6.80234', 'vk6.80895', 'vk6.83697', 'vk6.84867', 'vk6.85672', 'vk6.87625', 'vk6.88436', 'vk6.89286']

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	O1O2O3O4O5O6U3U2U6U1U5U4
R3 orbit	{'O1O2O3O4O5O6U3U2U6U1U5U4'}
R3 orbit length	1
Gauss code of -K	Same
Gauss code of K*	O1O2O3O4O5O6U4U2U1U6U5U3
Gauss code of -K*	O1O2O3O4O5O6U4U2U1U6U5U3
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 -2 -3 -3 3 3 2],[ 2 0 -1 -1 4 3 2],[ 3 1 0 0 4 3 2],[ 3 1 0 0 3 2 1],[-3 -4 -4 -3 0 0 0],[-3 -3 -3 -2 0 0 0],[-2 -2 -2 -1 0 0 0]]
Primitive based matrix	[[ 0 3 3 2 -2 -3 -3],[-3 0 0 0 -3 -2 -3],[-3 0 0 0 -4 -3 -4],[-2 0 0 0 -2 -1 -2],[ 2 3 4 2 0 -1 -1],[ 3 2 3 1 1 0 0],[ 3 3 4 2 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,-2,2,3,3,0,0,3,2,3,0,4,3,4,2,1,2,1,1,0]
Phi over symmetry	[-3,-3,-2,2,3,3,0,0,3,2,3,0,4,3,4,2,1,2,1,1,0]
Phi of -K	[-3,-3,-2,2,3,3,0,0,3,2,3,0,4,3,4,2,1,2,1,1,0]
Phi of K*	[-3,-3,-2,2,3,3,0,1,1,2,3,1,2,3,4,2,3,4,0,0,0]
Phi of -K*	[-3,-3,-2,2,3,3,0,1,1,2,3,1,2,3,4,2,3,4,0,0,0]
Symmetry type of based matrix	r
u-polynomial	0
Normalized Jones-Krushkal polynomial	8z^2+25z+19
Enhanced Jones-Krushkal polynomial	8w^3z^2+25w^2z+19w
Inner characteristic polynomial	t^6+74t^4+15t^2
Outer characteristic polynomial	t^7+118t^5+105t^3+4t
Flat arrow polynomial	4K12K2 - 4K1K3 + K4
2-strand cable arrow polynomial	960K12K2*3 - 2624K1*2K2*2 - 320K1*2K2K4 + 2304K1*2K2 - 288K12K3*2 - 192K1*2K3K5 - 32K1*2K5*2 - 2224K1*2 + 192K1K22K3K4 - 576K1K22K3 + 192K1K2*2K4K5 - 640K1K22K5 - 192K1K2K3K4 - 192K1K2K3K6 + 3200K1K2K3 - 64K1K2K4K5 - 64K1K2K5K6 + 704K1K3K4 + 608K1K4K5 + 80K1K5K6 - 32K24K4*2 + 128K2*4K4 - 976K24 + 128K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 384K22K3*2 - 352K2*2K4*2 + 1216K2*2K4 - 320K22K5*2 - 48K2*2K6*2 - 1520K2*2 + 1136K2K3K5 + 288K2K4K6 + 112K2K5K7 + 16K2K6K8 + 48K3*2K6 - 1072K32 - 552K4*2 - 480K5*2 - 80K6*2 - 2K8**2 + 1912
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}]]
If K is slice	False

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