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Flat knot 6.275

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,0,3,3,2,4,2,1,1,1,0,1,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.275']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 8K12 - 8K1K2 + K1 - 2K2*2 + 2K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.275']
Outer characteristic polynomial of the knot is: t^7+75t^5+103t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.275']
2-strand cable arrow polynomial of the knot is: 32K14K2 - 320K14 - 352K1*3K3 - 128K12K2*4 + 608K1*2K2*3 + 160K1*2K2*2K4 - 3376K12K2*2 - 768K1*2K2K4 + 6280K1*2K2 - 224K12K3*2 - 64K1*2K4*2 - 6420K1*2 + 960K1K23K3 + 96K1K2*2K3K4 - 1888K1K22K3 - 320K1K2*2K5 - 544K1K2K3K4 - 32K1K2K3K6 + 7824K1K2K3 - 32K1K2K4K5 + 1896K1K3K4 + 208K1K4K5 + 16K1K5K6 - 32K26 - 128K2*4K3*2 - 32K2*4K4*2 + 224K2*4K4 - 1328K24 + 64K2*3K3K5 - 32K2*3K6 + 128K22K3*2K4 - 1440K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 368K2*2K4*2 + 2616K2*2K4 - 5166K22 - 96K2K32K4 - 32K2K3K4K5 + 1152K2K3K5 + 280K2K4K6 - 32K32K4*2 + 40K3*2K6 - 3028K32 + 16K3K4K7 - 8K44 - 1288K4*2 - 224K5*2 - 66K6**2 + 5166
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.275']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71460', 'vk6.71511', 'vk6.71514', 'vk6.71982', 'vk6.71986', 'vk6.72037', 'vk6.72041', 'vk6.73221', 'vk6.73234', 'vk6.73254', 'vk6.73265', 'vk6.73655', 'vk6.73675', 'vk6.75144', 'vk6.75162', 'vk6.77081', 'vk6.77129', 'vk6.77138', 'vk6.77422', 'vk6.77430', 'vk6.78079', 'vk6.78090', 'vk6.78114', 'vk6.78124', 'vk6.81291', 'vk6.81534', 'vk6.81544', 'vk6.85465', 'vk6.85471', 'vk6.86877', 'vk6.87729', 'vk6.89505']
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,-1,0,1,2,2,0,3,3,2,4,2,1,1,1,0,1,1,1,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.275']

Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 2*K2 + 3*K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.275']

Outer characteristic polynomial of the knot is: t^7+75t^5+103t^3+10t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.275']

2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 320*K1**4 - 352*K1**3*K3 - 128*K1**2*K2**4 + 608*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 3376*K1**2*K2**2 - 768*K1**2*K2*K4 + 6280*K1**2*K2 - 224*K1**2*K3**2 - 64*K1**2*K4**2 - 6420*K1**2 + 960*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1888*K1*K2**2*K3 - 320*K1*K2**2*K5 - 544*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7824*K1*K2*K3 - 32*K1*K2*K4*K5 + 1896*K1*K3*K4 + 208*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1328*K2**4 + 64*K2**3*K3*K5 - 32*K2**3*K6 + 128*K2**2*K3**2*K4 - 1440*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 368*K2**2*K4**2 + 2616*K2**2*K4 - 5166*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1152*K2*K3*K5 + 280*K2*K4*K6 - 32*K3**2*K4**2 + 40*K3**2*K6 - 3028*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 1288*K4**2 - 224*K5**2 - 66*K6**2 + 5166

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.275']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71460', 'vk6.71511', 'vk6.71514', 'vk6.71982', 'vk6.71986', 'vk6.72037', 'vk6.72041', 'vk6.73221', 'vk6.73234', 'vk6.73254', 'vk6.73265', 'vk6.73655', 'vk6.73675', 'vk6.75144', 'vk6.75162', 'vk6.77081', 'vk6.77129', 'vk6.77138', 'vk6.77422', 'vk6.77430', 'vk6.78079', 'vk6.78090', 'vk6.78114', 'vk6.78124', 'vk6.81291', 'vk6.81534', 'vk6.81544', 'vk6.85465', 'vk6.85471', 'vk6.86877', 'vk6.87729', 'vk6.89505']

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	O1O2O3O4O5U1U4O6U5U2U6U3
R3 orbit	{'O1O2O3O4O5U1U4O6U5U2U6U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U6U4U1O6U2U5
Gauss code of K*	O1O2O3O4U5U2U4U6U1O5O6U3
Gauss code of -K*	O1O2O3O4U2O5O6U4U5U1U3U6
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 -4 -1 2 0 1 2],[ 4 0 3 4 1 2 2],[ 1 -3 0 2 -1 1 2],[-2 -4 -2 0 -1 0 1],[ 0 -1 1 1 0 1 1],[-1 -2 -1 0 -1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 2 1 0 -1 -4],[-2 0 1 0 -1 -2 -4],[-2 -1 0 -1 -1 -2 -2],[-1 0 1 0 -1 -1 -2],[ 0 1 1 1 0 1 -1],[ 1 2 2 1 -1 0 -3],[ 4 4 2 2 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,1,4,-1,0,1,2,4,1,1,2,2,1,1,2,-1,1,3]
Phi over symmetry	[-4,-1,0,1,2,2,0,3,3,2,4,2,1,1,1,0,1,1,1,0,-1]
Phi of -K	[-4,-1,0,1,2,2,0,3,3,2,4,2,1,1,1,0,1,1,1,0,-1]
Phi of K*	[-2,-2,-1,0,1,4,-1,0,1,1,4,1,1,1,2,0,1,3,2,3,0]
Phi of -K*	[-4,-1,0,1,2,2,3,1,2,2,4,-1,1,2,2,1,1,1,1,0,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^2
Normalized Jones-Krushkal polynomial	6z^2+25z+27
Enhanced Jones-Krushkal polynomial	6w^3z^2-2w^3z+27w^2z+27w
Inner characteristic polynomial	t^6+49t^4+25t^2+1
Outer characteristic polynomial	t^7+75t^5+103t^3+10t
Flat arrow polynomial	4K13 + 4K1*2K2 - 8K12 - 8K1K2 + K1 - 2K2*2 + 2K2 + 3*K3 + 5
2-strand cable arrow polynomial	32K14K2 - 320K14 - 352K1*3K3 - 128K12K2*4 + 608K1*2K2*3 + 160K1*2K2*2K4 - 3376K12K2*2 - 768K1*2K2K4 + 6280K1*2K2 - 224K12K3*2 - 64K1*2K4*2 - 6420K1*2 + 960K1K23K3 + 96K1K2*2K3K4 - 1888K1K22K3 - 320K1K2*2K5 - 544K1K2K3K4 - 32K1K2K3K6 + 7824K1K2K3 - 32K1K2K4K5 + 1896K1K3K4 + 208K1K4K5 + 16K1K5K6 - 32K26 - 128K2*4K3*2 - 32K2*4K4*2 + 224K2*4K4 - 1328K24 + 64K2*3K3K5 - 32K2*3K6 + 128K22K3*2K4 - 1440K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 368K2*2K4*2 + 2616K2*2K4 - 5166K22 - 96K2K32K4 - 32K2K3K4K5 + 1152K2K3K5 + 280K2K4K6 - 32K32K4*2 + 40K3*2K6 - 3028K32 + 16K3K4K7 - 8K44 - 1288K4*2 - 224K5*2 - 66K6**2 + 5166
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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