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Flat knot 6.86

Min(phi) over symmetries of the knot is: [-4,-3,0,1,3,3,0,3,2,3,5,2,1,2,3,0,2,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.86']
Arrow polynomial of the knot is: 12K13 + 4K1*2K2 - 6K12 - 6K1K2 - 2K1K3 - 6K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.86']
Outer characteristic polynomial of the knot is: t^7+116t^5+98t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.86']
2-strand cable arrow polynomial of the knot is: -400K14 - 32K1*3K3 - 128K12K2*4 + 384K1*2K2*3 - 2608K1*2K2*2 - 288K1*2K2K4 + 3424K1*2K2 - 16K12K3*2 - 16K1*2K4*2 - 2632K1*2 + 896K1K23K3 + 128K1K2*2K3K4 - 704K1K22K3 + 32K1K2*2K4K5 - 224K1K22K5 + 32K1K2K3K4*2 - 256K1K2K3K4 - 32K1K2K3K6 + 3456K1K2K3 + 456K1K3K4 + 40K1K4K5 - 96K26 - 128K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 1240K24 + 128K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 + 64K22K3*2K4 - 912K22K3*2 - 32K2*2K3K7 - 216K2*2K4*2 + 1352K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 1708K2*2 - 32K2K32K4 + 480K2K3K5 + 104K2K4K6 - 32K32K4*2 - 984K3*2 - 364K4*2 - 32K5*2 - 12K6**2 + 2058
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.86']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71620', 'vk6.71780', 'vk6.72199', 'vk6.72347', 'vk6.73373', 'vk6.73534', 'vk6.75278', 'vk6.75544', 'vk6.77238', 'vk6.77317', 'vk6.77565', 'vk6.77677', 'vk6.78261', 'vk6.78510', 'vk6.80077', 'vk6.80225', 'vk6.81122', 'vk6.81168', 'vk6.81189', 'vk6.81243', 'vk6.81324', 'vk6.81510', 'vk6.81605', 'vk6.81773', 'vk6.82294', 'vk6.82417', 'vk6.82445', 'vk6.82749', 'vk6.84202', 'vk6.84383', 'vk6.84397', 'vk6.85435', 'vk6.85996', 'vk6.86917', 'vk6.87125', 'vk6.88088', 'vk6.88182', 'vk6.88671', 'vk6.88785', 'vk6.89111']
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,-3,0,1,3,3,0,3,2,3,5,2,1,2,3,0,2,2,1,1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.86']

Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 2*K1*K3 - 6*K1 + 2*K2 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.86']

Outer characteristic polynomial of the knot is: t^7+116t^5+98t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.86']

2-strand cable arrow polynomial of the knot is: -400*K1**4 - 32*K1**3*K3 - 128*K1**2*K2**4 + 384*K1**2*K2**3 - 2608*K1**2*K2**2 - 288*K1**2*K2*K4 + 3424*K1**2*K2 - 16*K1**2*K3**2 - 16*K1**2*K4**2 - 2632*K1**2 + 896*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 704*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 224*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3456*K1*K2*K3 + 456*K1*K3*K4 + 40*K1*K4*K5 - 96*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1240*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 912*K2**2*K3**2 - 32*K2**2*K3*K7 - 216*K2**2*K4**2 + 1352*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 1708*K2**2 - 32*K2*K3**2*K4 + 480*K2*K3*K5 + 104*K2*K4*K6 - 32*K3**2*K4**2 - 984*K3**2 - 364*K4**2 - 32*K5**2 - 12*K6**2 + 2058

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.86']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71620', 'vk6.71780', 'vk6.72199', 'vk6.72347', 'vk6.73373', 'vk6.73534', 'vk6.75278', 'vk6.75544', 'vk6.77238', 'vk6.77317', 'vk6.77565', 'vk6.77677', 'vk6.78261', 'vk6.78510', 'vk6.80077', 'vk6.80225', 'vk6.81122', 'vk6.81168', 'vk6.81189', 'vk6.81243', 'vk6.81324', 'vk6.81510', 'vk6.81605', 'vk6.81773', 'vk6.82294', 'vk6.82417', 'vk6.82445', 'vk6.82749', 'vk6.84202', 'vk6.84383', 'vk6.84397', 'vk6.85435', 'vk6.85996', 'vk6.86917', 'vk6.87125', 'vk6.88088', 'vk6.88182', 'vk6.88671', 'vk6.88785', 'vk6.89111']

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	O1O2O3O4O5O6U2U5U1U6U3U4
R3 orbit	{'O1O2O3O4O5U1U6U3U5U2O6U4', 'O1O2O3O4O5O6U2U5U1U6U3U4'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5O6U3U4U1U6U2U5
Gauss code of K*	O1O2O3O4O5O6U3U1U5U6U2U4
Gauss code of -K*	O1O2O3O4O5O6U3U5U1U2U6U4
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 -4 1 3 0 3],[ 3 0 -1 3 4 1 3],[ 4 1 0 3 4 1 2],[-1 -3 -3 0 1 -1 1],[-3 -4 -4 -1 0 -1 1],[ 0 -1 -1 1 1 0 1],[-3 -3 -2 -1 -1 -1 0]]
Primitive based matrix	[[ 0 3 3 1 0 -3 -4],[-3 0 1 -1 -1 -4 -4],[-3 -1 0 -1 -1 -3 -2],[-1 1 1 0 -1 -3 -3],[ 0 1 1 1 0 -1 -1],[ 3 4 3 3 1 0 -1],[ 4 4 2 3 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,-1,0,3,4,-1,1,1,4,4,1,1,3,2,1,3,3,1,1,1]
Phi over symmetry	[-4,-3,0,1,3,3,0,3,2,3,5,2,1,2,3,0,2,2,1,1,-1]
Phi of -K	[-4,-3,0,1,3,3,0,3,2,3,5,2,1,2,3,0,2,2,1,1,-1]
Phi of K*	[-3,-3,-1,0,3,4,-1,1,2,3,5,1,2,2,3,0,1,2,2,3,0]
Phi of -K*	[-4,-3,0,1,3,3,1,1,3,2,4,1,3,3,4,1,1,1,1,1,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	4w^3z^2+17w^2z+19w
Inner characteristic polynomial	t^6+72t^4+32t^2+1
Outer characteristic polynomial	t^7+116t^5+98t^3+4t
Flat arrow polynomial	12K13 + 4K1*2K2 - 6K12 - 6K1K2 - 2K1K3 - 6K1 + 2*K2 + 3
2-strand cable arrow polynomial	-400K14 - 32K1*3K3 - 128K12K2*4 + 384K1*2K2*3 - 2608K1*2K2*2 - 288K1*2K2K4 + 3424K1*2K2 - 16K12K3*2 - 16K1*2K4*2 - 2632K1*2 + 896K1K23K3 + 128K1K2*2K3K4 - 704K1K22K3 + 32K1K2*2K4K5 - 224K1K22K5 + 32K1K2K3K4*2 - 256K1K2K3K4 - 32K1K2K3K6 + 3456K1K2K3 + 456K1K3K4 + 40K1K4K5 - 96K26 - 128K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 1240K24 + 128K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 + 64K22K3*2K4 - 912K22K3*2 - 32K2*2K3K7 - 216K2*2K4*2 + 1352K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 1708K2*2 - 32K2K32K4 + 480K2K3K5 + 104K2K4K6 - 32K32K4*2 - 984K3*2 - 364K4*2 - 32K5*2 - 12K6**2 + 2058
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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}], [{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}, {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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]]
If K is slice	False

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