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Flat knot 6.312

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,2,2,5,3,1,1,2,1,1,2,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.312']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 10K12 - 10K1K2 - 2K1K3 - K1 + 4K2 + 3*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.312']
Outer characteristic polynomial of the knot is: t^7+97t^5+54t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.312']
2-strand cable arrow polynomial of the knot is: -448K14K2*2 + 1088K1*4K2 - 1984K14 + 896K1*3K2K3 - 544K1*3K3 - 1024K12K2*4 + 3328K1*2K2*3 - 320K1*2K2*2K3*2 + 384K1*2K2*2K4 - 12768K12K2*2 + 64K1*2K2K32 - 1440K1*2K2K4 + 11712K1*2K2 - 480K12K3*2 - 32K1*2K4*2 - 7160K1*2 + 3712K1K23K3 + 480K1K2*2K3K4 - 3072K1K22K3 + 32K1K2*2K4K5 - 896K1K22K5 - 288K1K2K3K4 + 11632K1K2K3 - 64K1K2K4K5 + 1208K1K3K4 + 96K1K4K5 + 8K1K5K6 - 192K26 - 256K2*4K3*2 - 32K2*4K4*2 + 448K2*4K4 - 4952K24 + 224K2*3K3K5 + 32K2*3K4K6 - 96K2*3K6 - 2080K22K3*2 - 440K2*2K4*2 + 3912K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 3646K2*2 - 32K2K32K4 + 888K2K3K5 + 184K2K4K6 + 16K2K5K7 - 2664K3*2 - 748K4*2 - 120K5*2 - 26K6**2 + 5562
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.312']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72574', 'vk6.72575', 'vk6.72681', 'vk6.72682', 'vk6.72996', 'vk6.72997', 'vk6.73150', 'vk6.73151', 'vk6.74210', 'vk6.74212', 'vk6.74840', 'vk6.74842', 'vk6.76401', 'vk6.76403', 'vk6.76890', 'vk6.77859', 'vk6.77891', 'vk6.77892', 'vk6.78004', 'vk6.79250', 'vk6.79252', 'vk6.79734', 'vk6.80745', 'vk6.80747', 'vk6.81148', 'vk6.81157', 'vk6.82305', 'vk6.83987', 'vk6.86357', 'vk6.87285', 'vk6.88237', 'vk6.88246']
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,-2,0,1,2,3,0,2,2,5,3,1,1,2,1,1,2,2,0,1,2]

Flat knots (up to 7 crossings) with same phi are :['6.312']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 10*K1*K2 - 2*K1*K3 - K1 + 4*K2 + 3*K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.312']

Outer characteristic polynomial of the knot is: t^7+97t^5+54t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.312']

2-strand cable arrow polynomial of the knot is: -448*K1**4*K2**2 + 1088*K1**4*K2 - 1984*K1**4 + 896*K1**3*K2*K3 - 544*K1**3*K3 - 1024*K1**2*K2**4 + 3328*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 + 384*K1**2*K2**2*K4 - 12768*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 1440*K1**2*K2*K4 + 11712*K1**2*K2 - 480*K1**2*K3**2 - 32*K1**2*K4**2 - 7160*K1**2 + 3712*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 3072*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 896*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 11632*K1*K2*K3 - 64*K1*K2*K4*K5 + 1208*K1*K3*K4 + 96*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 448*K2**4*K4 - 4952*K2**4 + 224*K2**3*K3*K5 + 32*K2**3*K4*K6 - 96*K2**3*K6 - 2080*K2**2*K3**2 - 440*K2**2*K4**2 + 3912*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3646*K2**2 - 32*K2*K3**2*K4 + 888*K2*K3*K5 + 184*K2*K4*K6 + 16*K2*K5*K7 - 2664*K3**2 - 748*K4**2 - 120*K5**2 - 26*K6**2 + 5562

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.312']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72574', 'vk6.72575', 'vk6.72681', 'vk6.72682', 'vk6.72996', 'vk6.72997', 'vk6.73150', 'vk6.73151', 'vk6.74210', 'vk6.74212', 'vk6.74840', 'vk6.74842', 'vk6.76401', 'vk6.76403', 'vk6.76890', 'vk6.77859', 'vk6.77891', 'vk6.77892', 'vk6.78004', 'vk6.79250', 'vk6.79252', 'vk6.79734', 'vk6.80745', 'vk6.80747', 'vk6.81148', 'vk6.81157', 'vk6.82305', 'vk6.83987', 'vk6.86357', 'vk6.87285', 'vk6.88237', 'vk6.88246']

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	O1O2O3O4O5U2U3O6U4U1U6U5
R3 orbit	{'O1O2O3O4O5U2U3O6U4U1U6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U6U5U2O6U3U4
Gauss code of K*	O1O2O3O4U2U5U6U1U4O5O6U3
Gauss code of -K*	O1O2O3O4U2O5O6U1U4U5U6U3
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 -3 -1 0 4 2],[ 2 0 -2 0 2 5 2],[ 3 2 0 1 2 3 1],[ 1 0 -1 0 1 2 1],[ 0 -2 -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 0 -1 -2 -3],[-4 0 0 -2 -2 -5 -3],[-2 0 0 -1 -1 -2 -1],[ 0 2 1 0 -1 -2 -2],[ 1 2 1 1 0 0 -1],[ 2 5 2 2 0 0 -2],[ 3 3 1 2 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,0,1,2,3,0,2,2,5,3,1,1,2,1,1,2,2,0,1,2]
Phi over symmetry	[-4,-2,0,1,2,3,0,2,2,5,3,1,1,2,1,1,2,2,0,1,2]
Phi of -K	[-3,-2,-1,0,2,4,-1,1,1,4,4,1,0,2,1,0,2,3,1,2,2]
Phi of K*	[-4,-2,0,1,2,3,2,2,3,1,4,1,2,2,4,0,0,1,1,1,-1]
Phi of -K*	[-3,-2,-1,0,2,4,2,1,2,1,3,0,2,2,5,1,1,2,1,2,0]
Symmetry type of based matrix	c
u-polynomial	-t^4+t^3+t
Normalized Jones-Krushkal polynomial	8z^2+29z+27
Enhanced Jones-Krushkal polynomial	8w^3z^2+29w^2z+27w
Inner characteristic polynomial	t^6+63t^4+8t^2
Outer characteristic polynomial	t^7+97t^5+54t^3+4t
Flat arrow polynomial	8K13 + 4K1*2K2 - 10K12 - 10K1K2 - 2K1K3 - K1 + 4K2 + 3*K3 + 5
2-strand cable arrow polynomial	-448K14K2*2 + 1088K1*4K2 - 1984K14 + 896K1*3K2K3 - 544K1*3K3 - 1024K12K2*4 + 3328K1*2K2*3 - 320K1*2K2*2K3*2 + 384K1*2K2*2K4 - 12768K12K2*2 + 64K1*2K2K32 - 1440K1*2K2K4 + 11712K1*2K2 - 480K12K3*2 - 32K1*2K4*2 - 7160K1*2 + 3712K1K23K3 + 480K1K2*2K3K4 - 3072K1K22K3 + 32K1K2*2K4K5 - 896K1K22K5 - 288K1K2K3K4 + 11632K1K2K3 - 64K1K2K4K5 + 1208K1K3K4 + 96K1K4K5 + 8K1K5K6 - 192K26 - 256K2*4K3*2 - 32K2*4K4*2 + 448K2*4K4 - 4952K24 + 224K2*3K3K5 + 32K2*3K4K6 - 96K2*3K6 - 2080K22K3*2 - 440K2*2K4*2 + 3912K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 3646K2*2 - 32K2K32K4 + 888K2K3K5 + 184K2K4K6 + 16K2K5K7 - 2664K3*2 - 748K4*2 - 120K5*2 - 26K6**2 + 5562
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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{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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