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Flat knot 6.502

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,2,2,4,3,1,1,1,1,1,3,2,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.502']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 12K12 - 6K1K2 - 2K1K3 - 3K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.272', '6.502']
Outer characteristic polynomial of the knot is: t^7+92t^5+168t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.502']
2-strand cable arrow polynomial of the knot is: -128K16 + 256K1*4K2*3 - 1280K1*4K2*2 + 1920K1*4K2 - 3088K14 + 448K1*3K2K3 - 160K1*3K3 - 1280K12K2*4 + 4224K1*2K2*3 - 128K1*2K2*2K3*2 + 64K1*2K2*2K4 - 11184K12K2*2 + 192K1*2K2K32 + 32K1*2K2K3K5 - 672K12K2K4 + 9392K1*2K2 - 624K12K3*2 - 64K1*2K3K5 - 4124K1*2 + 2016K1K23K3 + 480K1K2*2K3K4 - 2656K1K22K3 - 320K1K2*2K5 - 608K1K2K3K4 - 64K1K2K3K6 + 8760K1K2K3 - 32K1K2K4K5 + 1312K1K3K4 + 160K1K4K5 + 8K1K5K6 - 64K26 - 64K2*4K3*2 - 32K2*4K4*2 + 288K2*4K4 - 3760K24 + 160K2*3K3K5 + 32K2*3K4K6 - 1712K2*2K3*2 - 408K2*2K4*2 + 2824K2*2K4 - 80K22K5*2 - 8K2*2K6*2 - 2322K2*2 - 96K2K32K4 + 1080K2K3K5 + 144K2K4K6 + 8K2K5K7 - 1952K3*2 - 702K4*2 - 196K5*2 - 14K6**2 + 4020
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.502']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19937', 'vk6.19976', 'vk6.21177', 'vk6.21243', 'vk6.26892', 'vk6.26991', 'vk6.28648', 'vk6.28715', 'vk6.38317', 'vk6.38403', 'vk6.40456', 'vk6.40582', 'vk6.45195', 'vk6.45295', 'vk6.47021', 'vk6.47075', 'vk6.56722', 'vk6.56790', 'vk6.57816', 'vk6.57922', 'vk6.61139', 'vk6.61286', 'vk6.62388', 'vk6.62475', 'vk6.66419', 'vk6.66494', 'vk6.67189', 'vk6.67285', 'vk6.69071', 'vk6.69148', 'vk6.69857', 'vk6.69905']
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,-1,1,2,3,0,2,2,4,3,1,1,1,1,1,3,2,1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.502']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 5*K2 + K3 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.272', '6.502']

Outer characteristic polynomial of the knot is: t^7+92t^5+168t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.502']

2-strand cable arrow polynomial of the knot is: -128*K1**6 + 256*K1**4*K2**3 - 1280*K1**4*K2**2 + 1920*K1**4*K2 - 3088*K1**4 + 448*K1**3*K2*K3 - 160*K1**3*K3 - 1280*K1**2*K2**4 + 4224*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 11184*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 672*K1**2*K2*K4 + 9392*K1**2*K2 - 624*K1**2*K3**2 - 64*K1**2*K3*K5 - 4124*K1**2 + 2016*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 2656*K1*K2**2*K3 - 320*K1*K2**2*K5 - 608*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 8760*K1*K2*K3 - 32*K1*K2*K4*K5 + 1312*K1*K3*K4 + 160*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 3760*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1712*K2**2*K3**2 - 408*K2**2*K4**2 + 2824*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 2322*K2**2 - 96*K2*K3**2*K4 + 1080*K2*K3*K5 + 144*K2*K4*K6 + 8*K2*K5*K7 - 1952*K3**2 - 702*K4**2 - 196*K5**2 - 14*K6**2 + 4020

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.502']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19937', 'vk6.19976', 'vk6.21177', 'vk6.21243', 'vk6.26892', 'vk6.26991', 'vk6.28648', 'vk6.28715', 'vk6.38317', 'vk6.38403', 'vk6.40456', 'vk6.40582', 'vk6.45195', 'vk6.45295', 'vk6.47021', 'vk6.47075', 'vk6.56722', 'vk6.56790', 'vk6.57816', 'vk6.57922', 'vk6.61139', 'vk6.61286', 'vk6.62388', 'vk6.62475', 'vk6.66419', 'vk6.66494', 'vk6.67189', 'vk6.67285', 'vk6.69071', 'vk6.69148', 'vk6.69857', 'vk6.69905']

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	O1O2O3O4O5U1U6U2O6U5U3U4
R3 orbit	{'O1O2O3O4O5U1U6U2O6U5U3U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U3U1O6U4U6U5
Gauss code of K*	O1O2O3U2O4O5O6U1U3U5U6U4
Gauss code of -K*	O1O2O3U4O5O4O6U3U1U2U5U6
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 1 3 2 -1],[ 4 0 1 3 4 2 3],[ 1 -1 0 1 2 0 1],[-1 -3 -1 0 1 0 -1],[-3 -4 -2 -1 0 0 -3],[-2 -2 0 0 0 0 -2],[ 1 -3 -1 1 3 2 0]]
Primitive based matrix	[[ 0 3 2 1 -1 -1 -4],[-3 0 0 -1 -2 -3 -4],[-2 0 0 0 0 -2 -2],[-1 1 0 0 -1 -1 -3],[ 1 2 0 1 0 1 -1],[ 1 3 2 1 -1 0 -3],[ 4 4 2 3 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,1,1,4,0,1,2,3,4,0,0,2,2,1,1,3,-1,1,3]
Phi over symmetry	[-4,-1,-1,1,2,3,0,2,2,4,3,1,1,1,1,1,3,2,1,1,1]
Phi of -K	[-4,-1,-1,1,2,3,0,2,2,4,3,1,1,1,1,1,3,2,1,1,1]
Phi of K*	[-3,-2,-1,1,1,4,1,1,1,2,3,1,1,3,4,1,1,2,-1,0,2]
Phi of -K*	[-4,-1,-1,1,2,3,1,3,3,2,4,1,1,0,2,1,2,3,0,1,0]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial	6z^2+27z+31
Enhanced Jones-Krushkal polynomial	6w^3z^2+27w^2z+31w
Inner characteristic polynomial	t^6+60t^4+91t^2+4
Outer characteristic polynomial	t^7+92t^5+168t^3+9t
Flat arrow polynomial	8K13 + 4K1*2K2 - 12K12 - 6K1K2 - 2K1K3 - 3K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial	-128K16 + 256K1*4K2*3 - 1280K1*4K2*2 + 1920K1*4K2 - 3088K14 + 448K1*3K2K3 - 160K1*3K3 - 1280K12K2*4 + 4224K1*2K2*3 - 128K1*2K2*2K3*2 + 64K1*2K2*2K4 - 11184K12K2*2 + 192K1*2K2K32 + 32K1*2K2K3K5 - 672K12K2K4 + 9392K1*2K2 - 624K12K3*2 - 64K1*2K3K5 - 4124K1*2 + 2016K1K23K3 + 480K1K2*2K3K4 - 2656K1K22K3 - 320K1K2*2K5 - 608K1K2K3K4 - 64K1K2K3K6 + 8760K1K2K3 - 32K1K2K4K5 + 1312K1K3K4 + 160K1K4K5 + 8K1K5K6 - 64K26 - 64K2*4K3*2 - 32K2*4K4*2 + 288K2*4K4 - 3760K24 + 160K2*3K3K5 + 32K2*3K4K6 - 1712K2*2K3*2 - 408K2*2K4*2 + 2824K2*2K4 - 80K22K5*2 - 8K2*2K6*2 - 2322K2*2 - 96K2K32K4 + 1080K2K3K5 + 144K2K4K6 + 8K2K5K7 - 1952K3*2 - 702K4*2 - 196K5*2 - 14K6**2 + 4020
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}], [{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}], [{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}, {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}]]
If K is slice	False

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