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Flat knot 6.51

Min(phi) over symmetries of the knot is: [-5,0,1,1,1,2,1,3,4,5,2,1,1,1,1,0,0,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.51']
Arrow polynomial of the knot is: -2K12 + 4K1K22 - 6K1K2 - 2K1K4 + K1 + K2 + 3K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.16', '6.51', '6.55']
Outer characteristic polynomial of the knot is: t^7+94t^5+94t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.51']
2-strand cable arrow polynomial of the knot is: -288K14 - 848K1*2K2*2 - 128K1*2K2K4 + 1688K1*2K2 - 192K12K3*2 - 128K1*2K3K5 - 128K1*2K4*2 - 32K1*2K4K6 - 2028K1*2 + 480K1K22K3K4 - 384K1K22K3 + 384K1K2*2K4K5 - 320K1K22K5 - 768K1K2K3K4 - 64K1K2K3K6 + 2272K1K2K3 - 96K1K2K4K5 - 32K1K2K4K7 - 64K1K32K5 + 1336K1K3K4 + 672K1K4K5 + 72K1K5K6 - 72K2*4 + 128K2*3K3K5 - 416K2*2K3*2 - 32K2*2K4*4 + 64K2*2K4*3 + 32K2*2K4*2K8 - 760K22K4*2 - 32K2*2K4K8 + 1448K2*2K4 - 352K22K5*2 - 8K2*2K8*2 - 2066K2*2 - 96K2K32K4 - 64K2K3K4K5 + 1120K2K3K5 - 32K2K42K6 + 360K2K4K6 + 96K2K5K7 + 16K2K6K8 + 48K3*2K6 - 1396K32 - 16K4*4 + 16K4*2K8 - 1070K42 - 472K5*2 - 62K6*2 - 4K8**2 + 2216
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.51']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20127', 'vk6.20131', 'vk6.21415', 'vk6.21423', 'vk6.27223', 'vk6.27231', 'vk6.28893', 'vk6.28897', 'vk6.38634', 'vk6.38642', 'vk6.40840', 'vk6.40854', 'vk6.45525', 'vk6.45541', 'vk6.47239', 'vk6.47246', 'vk6.56948', 'vk6.56952', 'vk6.58100', 'vk6.58108', 'vk6.61514', 'vk6.61522', 'vk6.62641', 'vk6.62645', 'vk6.66650', 'vk6.66652', 'vk6.67443', 'vk6.67447', 'vk6.69292', 'vk6.69296', 'vk6.70021', 'vk6.70023']
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: [-5,0,1,1,1,2,1,3,4,5,2,1,1,1,1,0,0,1,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.51']

Arrow polynomial of the knot is: -2*K1**2 + 4*K1*K2**2 - 6*K1*K2 - 2*K1*K4 + K1 + K2 + 3*K3 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.16', '6.51', '6.55']

Outer characteristic polynomial of the knot is: t^7+94t^5+94t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.51']

2-strand cable arrow polynomial of the knot is: -288*K1**4 - 848*K1**2*K2**2 - 128*K1**2*K2*K4 + 1688*K1**2*K2 - 192*K1**2*K3**2 - 128*K1**2*K3*K5 - 128*K1**2*K4**2 - 32*K1**2*K4*K6 - 2028*K1**2 + 480*K1*K2**2*K3*K4 - 384*K1*K2**2*K3 + 384*K1*K2**2*K4*K5 - 320*K1*K2**2*K5 - 768*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 2272*K1*K2*K3 - 96*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 64*K1*K3**2*K5 + 1336*K1*K3*K4 + 672*K1*K4*K5 + 72*K1*K5*K6 - 72*K2**4 + 128*K2**3*K3*K5 - 416*K2**2*K3**2 - 32*K2**2*K4**4 + 64*K2**2*K4**3 + 32*K2**2*K4**2*K8 - 760*K2**2*K4**2 - 32*K2**2*K4*K8 + 1448*K2**2*K4 - 352*K2**2*K5**2 - 8*K2**2*K8**2 - 2066*K2**2 - 96*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1120*K2*K3*K5 - 32*K2*K4**2*K6 + 360*K2*K4*K6 + 96*K2*K5*K7 + 16*K2*K6*K8 + 48*K3**2*K6 - 1396*K3**2 - 16*K4**4 + 16*K4**2*K8 - 1070*K4**2 - 472*K5**2 - 62*K6**2 - 4*K8**2 + 2216

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.51']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20127', 'vk6.20131', 'vk6.21415', 'vk6.21423', 'vk6.27223', 'vk6.27231', 'vk6.28893', 'vk6.28897', 'vk6.38634', 'vk6.38642', 'vk6.40840', 'vk6.40854', 'vk6.45525', 'vk6.45541', 'vk6.47239', 'vk6.47246', 'vk6.56948', 'vk6.56952', 'vk6.58100', 'vk6.58108', 'vk6.61514', 'vk6.61522', 'vk6.62641', 'vk6.62645', 'vk6.66650', 'vk6.66652', 'vk6.67443', 'vk6.67447', 'vk6.69292', 'vk6.69296', 'vk6.70021', 'vk6.70023']

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	O1O2O3O4O5O6U1U5U6U4U3U2
R3 orbit	{'O1O2O3O4O5O6U1U5U6U4U3U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U5U4U3U1U2U6
Gauss code of K*	O1O2O3O4O5O6U1U6U5U4U2U3
Gauss code of -K*	O1O2O3O4O5O6U4U5U3U2U1U6
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 -5 1 1 1 0 2],[ 5 0 5 4 3 1 2],[-1 -5 0 0 0 -1 1],[-1 -4 0 0 0 -1 1],[-1 -3 0 0 0 -1 1],[ 0 -1 1 1 1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 1 0 -5],[-2 0 -1 -1 -1 -1 -2],[-1 1 0 0 0 -1 -3],[-1 1 0 0 0 -1 -4],[-1 1 0 0 0 -1 -5],[ 0 1 1 1 1 0 -1],[ 5 2 3 4 5 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,-1,0,5,1,1,1,1,2,0,0,1,3,0,1,4,1,5,1]
Phi over symmetry	[-5,0,1,1,1,2,1,3,4,5,2,1,1,1,1,0,0,1,0,1,1]
Phi of -K	[-5,0,1,1,1,2,4,1,2,3,5,0,0,0,1,0,0,0,0,0,0]
Phi of K*	[-2,-1,-1,-1,0,5,0,0,0,1,5,0,0,0,1,0,0,2,0,3,4]
Phi of -K*	[-5,0,1,1,1,2,1,3,4,5,2,1,1,1,1,0,0,1,0,1,1]
Symmetry type of based matrix	c
u-polynomial	t^5-t^2-3t
Normalized Jones-Krushkal polynomial	z^2+6z+9
Enhanced Jones-Krushkal polynomial	-4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w
Inner characteristic polynomial	t^6+62t^4+17t^2
Outer characteristic polynomial	t^7+94t^5+94t^3+6t
Flat arrow polynomial	-2K12 + 4K1K22 - 6K1K2 - 2K1K4 + K1 + K2 + 3K3 + 2
2-strand cable arrow polynomial	-288K14 - 848K1*2K2*2 - 128K1*2K2K4 + 1688K1*2K2 - 192K12K3*2 - 128K1*2K3K5 - 128K1*2K4*2 - 32K1*2K4K6 - 2028K1*2 + 480K1K22K3K4 - 384K1K22K3 + 384K1K2*2K4K5 - 320K1K22K5 - 768K1K2K3K4 - 64K1K2K3K6 + 2272K1K2K3 - 96K1K2K4K5 - 32K1K2K4K7 - 64K1K32K5 + 1336K1K3K4 + 672K1K4K5 + 72K1K5K6 - 72K2*4 + 128K2*3K3K5 - 416K2*2K3*2 - 32K2*2K4*4 + 64K2*2K4*3 + 32K2*2K4*2K8 - 760K22K4*2 - 32K2*2K4K8 + 1448K2*2K4 - 352K22K5*2 - 8K2*2K8*2 - 2066K2*2 - 96K2K32K4 - 64K2K3K4K5 + 1120K2K3K5 - 32K2K42K6 + 360K2K4K6 + 96K2K5K7 + 16K2K6K8 + 48K3*2K6 - 1396K32 - 16K4*4 + 16K4*2K8 - 1070K42 - 472K5*2 - 62K6*2 - 4K8**2 + 2216
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}], [{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}], [{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}, {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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