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Flat knot 6.222

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,3,2,2,5,1,1,1,2,0,1,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.222']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 8K12 - 6K1K2 - 2K2*2 + 2K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.168', '6.222', '6.288']
Outer characteristic polynomial of the knot is: t^7+91t^5+61t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.222']
2-strand cable arrow polynomial of the knot is: -144K14 + 32K1*3K2K3 - 352K1*3K3 - 128K12K2*4 + 608K1*2K2*3 + 128K1*2K2*2K4 - 3264K12K2*2 - 544K1*2K2K4 + 5640K1*2K2 - 304K12K3*2 - 96K1*2K3K5 - 32K1*2K4*2 - 5740K1*2 + 864K1K23K3 + 96K1K2*2K3K4 - 1504K1K22K3 - 160K1K2*2K5 - 640K1K2K3K4 - 32K1K2K3K6 + 6672K1K2K3 + 1472K1K3K4 + 368K1K4K5 + 48K1K5K6 - 32K26 - 128K2*4K3*2 - 32K2*4K4*2 + 224K2*4K4 - 1360K24 + 64K2*3K3K5 - 32K2*3K6 + 160K22K3*2K4 - 1072K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 328K2*2K4*2 + 2272K2*2K4 - 4304K22 - 32K2K32K4 - 32K2K3K4K5 + 864K2K3K5 + 224K2K4K6 - 64K32K4*2 + 8K3*2K6 - 2480K32 + 24K3K4K7 - 8K44 - 1088K4*2 - 268K5*2 - 72K6**2 + 4502
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.222']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71456', 'vk6.71507', 'vk6.71517', 'vk6.71978', 'vk6.71990', 'vk6.72033', 'vk6.72043', 'vk6.73222', 'vk6.73233', 'vk6.73253', 'vk6.73266', 'vk6.73656', 'vk6.73676', 'vk6.75145', 'vk6.75161', 'vk6.77085', 'vk6.77133', 'vk6.77136', 'vk6.77424', 'vk6.77426', 'vk6.78080', 'vk6.78089', 'vk6.78113', 'vk6.78125', 'vk6.81287', 'vk6.81532', 'vk6.81546', 'vk6.85467', 'vk6.85469', 'vk6.86881', 'vk6.87727', 'vk6.89512']
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,-2,0,1,2,3,0,3,2,2,5,1,1,1,2,0,1,2,1,1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.222']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.168', '6.222', '6.288']

Outer characteristic polynomial of the knot is: t^7+91t^5+61t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.222']

2-strand cable arrow polynomial of the knot is: -144*K1**4 + 32*K1**3*K2*K3 - 352*K1**3*K3 - 128*K1**2*K2**4 + 608*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 3264*K1**2*K2**2 - 544*K1**2*K2*K4 + 5640*K1**2*K2 - 304*K1**2*K3**2 - 96*K1**2*K3*K5 - 32*K1**2*K4**2 - 5740*K1**2 + 864*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 1504*K1*K2**2*K3 - 160*K1*K2**2*K5 - 640*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6672*K1*K2*K3 + 1472*K1*K3*K4 + 368*K1*K4*K5 + 48*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 1360*K2**4 + 64*K2**3*K3*K5 - 32*K2**3*K6 + 160*K2**2*K3**2*K4 - 1072*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 328*K2**2*K4**2 + 2272*K2**2*K4 - 4304*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 864*K2*K3*K5 + 224*K2*K4*K6 - 64*K3**2*K4**2 + 8*K3**2*K6 - 2480*K3**2 + 24*K3*K4*K7 - 8*K4**4 - 1088*K4**2 - 268*K5**2 - 72*K6**2 + 4502

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.222']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71456', 'vk6.71507', 'vk6.71517', 'vk6.71978', 'vk6.71990', 'vk6.72033', 'vk6.72043', 'vk6.73222', 'vk6.73233', 'vk6.73253', 'vk6.73266', 'vk6.73656', 'vk6.73676', 'vk6.75145', 'vk6.75161', 'vk6.77085', 'vk6.77133', 'vk6.77136', 'vk6.77424', 'vk6.77426', 'vk6.78080', 'vk6.78089', 'vk6.78113', 'vk6.78125', 'vk6.81287', 'vk6.81532', 'vk6.81546', 'vk6.85467', 'vk6.85469', 'vk6.86881', 'vk6.87727', 'vk6.89512']

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	O1O2O3O4O5U3O6U1U5U6U2U4
R3 orbit	{'O1O2O3O4O5U3O6U1U5U6U2U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U4U6U1U5O6U3
Gauss code of K*	O1O2O3O4O5U1U4U6U5U2O6U3
Gauss code of -K*	O1O2O3O4O5U3O6U4U1U6U2U5
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 0 -2 3 1 2],[ 4 0 3 0 5 2 2],[ 0 -3 0 -1 2 0 1],[ 2 0 1 0 2 1 1],[-3 -5 -2 -2 0 -1 1],[-1 -2 0 -1 1 0 1],[-2 -2 -1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 3 2 1 0 -2 -4],[-3 0 1 -1 -2 -2 -5],[-2 -1 0 -1 -1 -1 -2],[-1 1 1 0 0 -1 -2],[ 0 2 1 0 0 -1 -3],[ 2 2 1 1 1 0 0],[ 4 5 2 2 3 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,0,2,4,-1,1,2,2,5,1,1,1,2,0,1,2,1,3,0]
Phi over symmetry	[-4,-2,0,1,2,3,0,3,2,2,5,1,1,1,2,0,1,2,1,1,-1]
Phi of -K	[-4,-2,0,1,2,3,2,1,3,4,2,1,2,3,3,1,1,1,0,1,2]
Phi of K*	[-3,-2,-1,0,2,4,2,1,1,3,2,0,1,3,4,1,2,3,1,1,2]
Phi of -K*	[-4,-2,0,1,2,3,0,3,2,2,5,1,1,1,2,0,1,2,1,1,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	6z^2+25z+27
Enhanced Jones-Krushkal polynomial	6w^3z^2+25w^2z+27w
Inner characteristic polynomial	t^6+57t^4+15t^2
Outer characteristic polynomial	t^7+91t^5+61t^3+5t
Flat arrow polynomial	4K13 + 4K1*2K2 - 8K12 - 6K1K2 - 2K2*2 + 2K2 + 2*K3 + 5
2-strand cable arrow polynomial	-144K14 + 32K1*3K2K3 - 352K1*3K3 - 128K12K2*4 + 608K1*2K2*3 + 128K1*2K2*2K4 - 3264K12K2*2 - 544K1*2K2K4 + 5640K1*2K2 - 304K12K3*2 - 96K1*2K3K5 - 32K1*2K4*2 - 5740K1*2 + 864K1K23K3 + 96K1K2*2K3K4 - 1504K1K22K3 - 160K1K2*2K5 - 640K1K2K3K4 - 32K1K2K3K6 + 6672K1K2K3 + 1472K1K3K4 + 368K1K4K5 + 48K1K5K6 - 32K26 - 128K2*4K3*2 - 32K2*4K4*2 + 224K2*4K4 - 1360K24 + 64K2*3K3K5 - 32K2*3K6 + 160K22K3*2K4 - 1072K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 328K2*2K4*2 + 2272K2*2K4 - 4304K22 - 32K2K32K4 - 32K2K3K4K5 + 864K2K3K5 + 224K2K4K6 - 64K32K4*2 + 8K3*2K6 - 2480K32 + 24K3K4K7 - 8K44 - 1088K4*2 - 268K5*2 - 72K6**2 + 4502
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}, {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}, {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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