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Flat knot 6.457

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,1,3,1,4,2,1,0,1,1,1,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.457']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 8K12 - 4K1K2 - 2K1K3 - K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.96', '6.149', '6.269', '6.441', '6.457']
Outer characteristic polynomial of the knot is: t^7+71t^5+136t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.457']
2-strand cable arrow polynomial of the knot is: 256K14K2*3 - 640K1*4K2*2 + 704K1*4K2 - 832K14 + 96K1*3K2K3 + 32K1*3K3K4 - 384K1*2K2*4 + 640K1*2K2*3 - 1536K1*2K2*2 + 1472K1*2K2 - 96K12K3*2 - 32K1*2K4*2 - 768K1*2 + 288K1K23K3 + 1056K1K2K3 + 200K1K3K4 + 24K1K4K5 - 32K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 560K24 + 96K2*3K3K5 + 32K2*3K4K6 - 240K2*2K3*2 - 144K2*2K4*2 + 296K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 466K2*2 + 168K2K3K5 + 48K2K4K6 + 8K2K5K7 - 348K32 - 186K4*2 - 52K5*2 - 6K6**2 + 896
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.457']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11456', 'vk6.11756', 'vk6.12773', 'vk6.13112', 'vk6.13378', 'vk6.13455', 'vk6.13646', 'vk6.13760', 'vk6.14155', 'vk6.14386', 'vk6.15615', 'vk6.16079', 'vk6.16474', 'vk6.16489', 'vk6.17630', 'vk6.22106', 'vk6.22885', 'vk6.22916', 'vk6.28163', 'vk6.29588', 'vk6.33133', 'vk6.33180', 'vk6.33244', 'vk6.33293', 'vk6.34862', 'vk6.34893', 'vk6.39607', 'vk6.41848', 'vk6.42444', 'vk6.42459', 'vk6.46223', 'vk6.47830', 'vk6.52218', 'vk6.52498', 'vk6.53057', 'vk6.53374', 'vk6.53554', 'vk6.53595', 'vk6.53628', 'vk6.53688']
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: [-4,0,0,1,1,2,1,3,1,4,2,1,0,1,1,1,1,1,-1,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.457']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.96', '6.149', '6.269', '6.441', '6.457']

Outer characteristic polynomial of the knot is: t^7+71t^5+136t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.457']

2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 640*K1**4*K2**2 + 704*K1**4*K2 - 832*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 - 384*K1**2*K2**4 + 640*K1**2*K2**3 - 1536*K1**2*K2**2 + 1472*K1**2*K2 - 96*K1**2*K3**2 - 32*K1**2*K4**2 - 768*K1**2 + 288*K1*K2**3*K3 + 1056*K1*K2*K3 + 200*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 560*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 240*K2**2*K3**2 - 144*K2**2*K4**2 + 296*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 466*K2**2 + 168*K2*K3*K5 + 48*K2*K4*K6 + 8*K2*K5*K7 - 348*K3**2 - 186*K4**2 - 52*K5**2 - 6*K6**2 + 896

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.457']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11456', 'vk6.11756', 'vk6.12773', 'vk6.13112', 'vk6.13378', 'vk6.13455', 'vk6.13646', 'vk6.13760', 'vk6.14155', 'vk6.14386', 'vk6.15615', 'vk6.16079', 'vk6.16474', 'vk6.16489', 'vk6.17630', 'vk6.22106', 'vk6.22885', 'vk6.22916', 'vk6.28163', 'vk6.29588', 'vk6.33133', 'vk6.33180', 'vk6.33244', 'vk6.33293', 'vk6.34862', 'vk6.34893', 'vk6.39607', 'vk6.41848', 'vk6.42444', 'vk6.42459', 'vk6.46223', 'vk6.47830', 'vk6.52218', 'vk6.52498', 'vk6.53057', 'vk6.53374', 'vk6.53554', 'vk6.53595', 'vk6.53628', 'vk6.53688']

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	O1O2O3O4O5U6U3O6U4U1U2U5
R3 orbit	{'O1O2O3O4O5U6U3O6U4U1U2U5', 'O1O2O3O4O5U4U6U3O6U1U2U5'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U1U4U5U2O6U3U6
Gauss code of K*	O1O2O3O4U2U3U5U1U4O6O5U6
Gauss code of -K*	O1O2O3O4U5O6O5U1U4U6U2U3
Diagrammatic symmetry type	c
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -2 0 -1 0 4 -1],[ 2 0 1 0 1 4 1],[ 0 -1 0 0 1 3 -1],[ 1 0 0 0 0 1 1],[ 0 -1 -1 0 0 1 0],[-4 -4 -3 -1 -1 0 -4],[ 1 -1 1 -1 0 4 0]]
Primitive based matrix	[[ 0 4 0 0 -1 -1 -2],[-4 0 -1 -3 -1 -4 -4],[ 0 1 0 -1 0 0 -1],[ 0 3 1 0 0 -1 -1],[ 1 1 0 0 0 1 0],[ 1 4 0 1 -1 0 -1],[ 2 4 1 1 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,0,0,1,1,2,1,3,1,4,4,1,0,0,1,0,1,1,-1,0,1]
Phi over symmetry	[-4,0,0,1,1,2,1,3,1,4,2,1,0,1,1,1,1,1,-1,0,1]
Phi of -K	[-2,-1,-1,0,0,4,0,1,1,1,2,1,0,1,1,1,1,4,-1,1,3]
Phi of K*	[-4,0,0,1,1,2,1,3,1,4,2,1,0,1,1,1,1,1,-1,0,1]
Phi of -K*	[-2,-1,-1,0,0,4,0,1,1,1,4,1,0,0,1,0,1,4,-1,1,3]
Symmetry type of based matrix	c
u-polynomial	-t^4+t^2+2t
Normalized Jones-Krushkal polynomial	7z+15
Enhanced Jones-Krushkal polynomial	-4w^3z+11w^2z+15w
Inner characteristic polynomial	t^6+49t^4+83t^2
Outer characteristic polynomial	t^7+71t^5+136t^3
Flat arrow polynomial	4K13 + 4K1*2K2 - 8K12 - 4K1K2 - 2K1K3 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	256K14K2*3 - 640K1*4K2*2 + 704K1*4K2 - 832K14 + 96K1*3K2K3 + 32K1*3K3K4 - 384K1*2K2*4 + 640K1*2K2*3 - 1536K1*2K2*2 + 1472K1*2K2 - 96K12K3*2 - 32K1*2K4*2 - 768K1*2 + 288K1K23K3 + 1056K1K2K3 + 200K1K3K4 + 24K1K4K5 - 32K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 560K24 + 96K2*3K3K5 + 32K2*3K4K6 - 240K2*2K3*2 - 144K2*2K4*2 + 296K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 466K2*2 + 168K2K3K5 + 48K2K4K6 + 8K2K5K7 - 348K32 - 186K4*2 - 52K5*2 - 6K6**2 + 896
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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {2, 5}, {4}, {3}, {1}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}], [{6}, {5}, {4}, {3}, {2}, {1}]]
If K is slice	False

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