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Flat knot 6.343

Min(phi) over symmetries of the knot is: [-4,-2,-1,2,2,3,0,2,2,3,5,1,1,1,2,1,2,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.343']
Arrow polynomial of the knot is: 4K12K2 - 2K12 - 2K1K2 + K1 - 2K2**2 - K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.150', '6.343', '6.489']
Outer characteristic polynomial of the knot is: t^7+101t^5+64t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.343']
2-strand cable arrow polynomial of the knot is: -288K14 + 96K1*3K3K4 - 1088K1*2K2*2 + 32K1*2K2K42 - 992K1*2K2K4 + 3440K1*2K2 - 320K12K3*2 - 192K1*2K3K5 - 384K1*2K4*2 - 4224K1*2 - 576K1K22K3 - 32K1K2*2K5 + 160K1K2K3K4*2 - 576K1K2K3K4 + 3824K1K2K3 - 32K1K2K4K5 - 32K1K2K4K7 + 2600K1K3K4 + 760K1K4K5 - 32K24K4*2 + 128K2*4K4 - 488K24 + 96K2*2K3*2K4 - 256K22K3*2 + 32K2*2K4*3 - 504K2*2K4*2 + 2080K2*2K4 - 3554K22 - 96K2K32K4 - 64K2K3K4K5 + 616K2K3K5 + 352K2K4K6 - 160K32K4*2 + 8K3*2K6 - 2024K32 + 64K3K4K7 - 8K44 - 1670K4*2 - 360K5*2 - 54K6**2 + 3716
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.343']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71585', 'vk6.71709', 'vk6.72128', 'vk6.72324', 'vk6.73488', 'vk6.74111', 'vk6.74128', 'vk6.74680', 'vk6.74697', 'vk6.75243', 'vk6.75494', 'vk6.76149', 'vk6.76169', 'vk6.77201', 'vk6.77307', 'vk6.77511', 'vk6.77661', 'vk6.78451', 'vk6.79113', 'vk6.79129', 'vk6.80035', 'vk6.80185', 'vk6.80621', 'vk6.80633', 'vk6.83735', 'vk6.83861', 'vk6.85065', 'vk6.85341', 'vk6.86665', 'vk6.86981', 'vk6.87423', 'vk6.89543']
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: [-4,-2,-1,2,2,3,0,2,2,3,5,1,1,1,2,1,2,3,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.343']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.150', '6.343', '6.489']

Outer characteristic polynomial of the knot is: t^7+101t^5+64t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.343']

2-strand cable arrow polynomial of the knot is: -288*K1**4 + 96*K1**3*K3*K4 - 1088*K1**2*K2**2 + 32*K1**2*K2*K4**2 - 992*K1**2*K2*K4 + 3440*K1**2*K2 - 320*K1**2*K3**2 - 192*K1**2*K3*K5 - 384*K1**2*K4**2 - 4224*K1**2 - 576*K1*K2**2*K3 - 32*K1*K2**2*K5 + 160*K1*K2*K3*K4**2 - 576*K1*K2*K3*K4 + 3824*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2600*K1*K3*K4 + 760*K1*K4*K5 - 32*K2**4*K4**2 + 128*K2**4*K4 - 488*K2**4 + 96*K2**2*K3**2*K4 - 256*K2**2*K3**2 + 32*K2**2*K4**3 - 504*K2**2*K4**2 + 2080*K2**2*K4 - 3554*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 616*K2*K3*K5 + 352*K2*K4*K6 - 160*K3**2*K4**2 + 8*K3**2*K6 - 2024*K3**2 + 64*K3*K4*K7 - 8*K4**4 - 1670*K4**2 - 360*K5**2 - 54*K6**2 + 3716

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.343']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71585', 'vk6.71709', 'vk6.72128', 'vk6.72324', 'vk6.73488', 'vk6.74111', 'vk6.74128', 'vk6.74680', 'vk6.74697', 'vk6.75243', 'vk6.75494', 'vk6.76149', 'vk6.76169', 'vk6.77201', 'vk6.77307', 'vk6.77511', 'vk6.77661', 'vk6.78451', 'vk6.79113', 'vk6.79129', 'vk6.80035', 'vk6.80185', 'vk6.80621', 'vk6.80633', 'vk6.83735', 'vk6.83861', 'vk6.85065', 'vk6.85341', 'vk6.86665', 'vk6.86981', 'vk6.87423', 'vk6.89543']

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	O1O2O3O4O5U3U2O6U1U4U6U5
R3 orbit	{'O1O2O3O4O5U3U2O6U1U4U6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U6U2U5O6U4U3
Gauss code of K*	O1O2O3O4U1U5U6U2U4O6O5U3
Gauss code of -K*	O1O2O3O4U2O5O6U1U3U6U5U4
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 -3 -2 -2 1 4 2],[ 3 0 0 0 3 5 2],[ 2 0 0 0 2 3 1],[ 2 0 0 0 1 2 1],[-1 -3 -2 -1 0 2 1],[-4 -5 -3 -2 -2 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 4 2 1 -2 -2 -3],[-4 0 0 -2 -2 -3 -5],[-2 0 0 -1 -1 -1 -2],[-1 2 1 0 -1 -2 -3],[ 2 2 1 1 0 0 0],[ 2 3 1 2 0 0 0],[ 3 5 2 3 0 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,-1,2,2,3,0,2,2,3,5,1,1,1,2,1,2,3,0,0,0]
Phi over symmetry	[-4,-2,-1,2,2,3,0,2,2,3,5,1,1,1,2,1,2,3,0,0,0]
Phi of -K	[-3,-2,-2,1,2,4,1,1,1,3,2,0,1,3,3,2,3,4,0,1,2]
Phi of K*	[-4,-2,-1,2,2,3,2,1,3,4,2,0,3,3,3,1,2,1,0,1,1]
Phi of -K*	[-3,-2,-2,1,2,4,0,0,3,2,5,0,1,1,2,2,1,3,1,2,0]
Symmetry type of based matrix	c
u-polynomial	-t^4+t^3+t^2-t
Normalized Jones-Krushkal polynomial	7z^2+26z+25
Enhanced Jones-Krushkal polynomial	7w^3z^2+26w^2z+25w
Inner characteristic polynomial	t^6+63t^4+11t^2
Outer characteristic polynomial	t^7+101t^5+64t^3+5t
Flat arrow polynomial	4K12K2 - 2K12 - 2K1K2 + K1 - 2K2**2 - K2 + K3 + 2
2-strand cable arrow polynomial	-288K14 + 96K1*3K3K4 - 1088K1*2K2*2 + 32K1*2K2K42 - 992K1*2K2K4 + 3440K1*2K2 - 320K12K3*2 - 192K1*2K3K5 - 384K1*2K4*2 - 4224K1*2 - 576K1K22K3 - 32K1K2*2K5 + 160K1K2K3K4*2 - 576K1K2K3K4 + 3824K1K2K3 - 32K1K2K4K5 - 32K1K2K4K7 + 2600K1K3K4 + 760K1K4K5 - 32K24K4*2 + 128K2*4K4 - 488K24 + 96K2*2K3*2K4 - 256K22K3*2 + 32K2*2K4*3 - 504K2*2K4*2 + 2080K2*2K4 - 3554K22 - 96K2K32K4 - 64K2K3K4K5 + 616K2K3K5 + 352K2K4K6 - 160K32K4*2 + 8K3*2K6 - 2024K32 + 64K3K4K7 - 8K44 - 1670K4*2 - 360K5*2 - 54K6**2 + 3716
Genus of based matrix	2
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{2, 6}, {5}, {4}, {3}, {1}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{4, 6}, {5}, {3}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {1, 5}, {4}, {3}, {2}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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