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Flat knot 6.1912

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,0,0,1,2,1,2,0,0,0,0,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1912']
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+22t^5+87t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1912', '6.1913']
2-strand cable arrow polynomial of the knot is: -512K14 + 512K1*3K2K3 - 128K1*2K2*2K3*2 - 2880K1*2K2*2 - 480K1*2K2K4 + 2768K1*2K2 - 1024K12K3*2 - 2176K1*2 + 1088K1K23K3 + 256K1K2*2K3K4 - 1088K1K22K3 - 576K1K2*2K5 + 512K1K2K33 - 64K1K2K3K4 - 192K1K2K3K6 + 5776K1K2K3 - 192K1K2K4K5 + 1440K1K3K4 + 48K1K5K6 - 32K24K4*2 + 128K2*4K4 - 1392K24 + 64K2*3K4K6 - 64K2*3K6 - 2432K22K3*2 - 256K2*2K4*2 - 32K2*2K4K8 + 1472K2*2K4 - 16K22K6*2 - 1856K2*2 - 352K2K32K4 + 1568K2K3K5 + 288K2K4K6 + 16K2K6K8 - 512K3*4 + 464K3*2K6 - 1664K32 - 448K4*2 - 160K5*2 - 112K6*2 - 2K8**2 + 2240
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1912']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4883', 'vk6.5228', 'vk6.6478', 'vk6.6898', 'vk6.8445', 'vk6.8864', 'vk6.9782', 'vk6.10075', 'vk6.20829', 'vk6.22227', 'vk6.29790', 'vk6.39889', 'vk6.46442', 'vk6.47996', 'vk6.48841', 'vk6.49112', 'vk6.51364', 'vk6.51577', 'vk6.63283', 'vk6.67124']
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,-2,1,1,1,1,0,0,0,1,2,1,2,0,0,0,0,0,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1912']

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+22t^5+87t^3+10t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1912', '6.1913']

2-strand cable arrow polynomial of the knot is: -512*K1**4 + 512*K1**3*K2*K3 - 128*K1**2*K2**2*K3**2 - 2880*K1**2*K2**2 - 480*K1**2*K2*K4 + 2768*K1**2*K2 - 1024*K1**2*K3**2 - 2176*K1**2 + 1088*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 576*K1*K2**2*K5 + 512*K1*K2*K3**3 - 64*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 5776*K1*K2*K3 - 192*K1*K2*K4*K5 + 1440*K1*K3*K4 + 48*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1392*K2**4 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 2432*K2**2*K3**2 - 256*K2**2*K4**2 - 32*K2**2*K4*K8 + 1472*K2**2*K4 - 16*K2**2*K6**2 - 1856*K2**2 - 352*K2*K3**2*K4 + 1568*K2*K3*K5 + 288*K2*K4*K6 + 16*K2*K6*K8 - 512*K3**4 + 464*K3**2*K6 - 1664*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 :['6.1912']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4883', 'vk6.5228', 'vk6.6478', 'vk6.6898', 'vk6.8445', 'vk6.8864', 'vk6.9782', 'vk6.10075', 'vk6.20829', 'vk6.22227', 'vk6.29790', 'vk6.39889', 'vk6.46442', 'vk6.47996', 'vk6.48841', 'vk6.49112', 'vk6.51364', 'vk6.51577', 'vk6.63283', 'vk6.67124']

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	O1O2O3U1U3U2O4O5O6U4U6U5
R3 orbit	{'O1O2O3U1U3U2O4O5O6U4U6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U5U6O5O4O6U2U1U3
Gauss code of K*	O1O2O3U4U5U6O4O6O5U1U3U2
Gauss code of -K*	O1O2O3U2U1U3O4O5O6U5U4U6
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 -2 1 1],[ 2 0 2 1 0 0 0],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0],[ 2 0 0 0 0 2 1],[-1 0 0 0 -2 0 0],[-1 0 0 0 -1 0 0]]
Primitive based matrix	[[ 0 1 1 1 1 -2 -2],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[-1 0 0 0 0 -1 0],[-1 0 0 0 0 -2 0],[ 2 0 0 1 2 0 0],[ 2 1 2 0 0 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,2,2,0,0,0,0,1,0,0,0,2,0,1,0,2,0,0]
Phi over symmetry	[-2,-2,1,1,1,1,0,0,0,1,2,1,2,0,0,0,0,0,0,0,0]
Phi of -K	[-2,-2,1,1,1,1,0,1,2,3,3,3,3,1,2,0,0,0,0,0,0]
Phi of K*	[-1,-1,-1,-1,2,2,0,0,0,1,3,0,0,2,3,0,3,1,3,2,0]
Phi of -K*	[-2,-2,1,1,1,1,0,0,0,1,2,1,2,0,0,0,0,0,0,0,0]
Symmetry type of based matrix	c
u-polynomial	2t^2-4t
Normalized Jones-Krushkal polynomial	9z^2+30z+25
Enhanced Jones-Krushkal polynomial	9w^3z^2+30w^2z+25w
Inner characteristic polynomial	t^6+10t^4+25t^2
Outer characteristic polynomial	t^7+22t^5+87t^3+10t
Flat arrow polynomial	4K12K2 - 4K1K3 + K4
2-strand cable arrow polynomial	-512K14 + 512K1*3K2K3 - 128K1*2K2*2K3*2 - 2880K1*2K2*2 - 480K1*2K2K4 + 2768K1*2K2 - 1024K12K3*2 - 2176K1*2 + 1088K1K23K3 + 256K1K2*2K3K4 - 1088K1K22K3 - 576K1K2*2K5 + 512K1K2K33 - 64K1K2K3K4 - 192K1K2K3K6 + 5776K1K2K3 - 192K1K2K4K5 + 1440K1K3K4 + 48K1K5K6 - 32K24K4*2 + 128K2*4K4 - 1392K24 + 64K2*3K4K6 - 64K2*3K6 - 2432K22K3*2 - 256K2*2K4*2 - 32K2*2K4K8 + 1472K2*2K4 - 16K22K6*2 - 1856K2*2 - 352K2K32K4 + 1568K2K3K5 + 288K2K4K6 + 16K2K6K8 - 512K3*4 + 464K3*2K6 - 1664K32 - 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}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice	False

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