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Flat knot 6.354

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,1,4,2,5,0,1,0,2,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.354']
Arrow polynomial of the knot is: -8K14 + 4K1*3 + 4K1*2K2 - 6K1K2 + 2K2 + 2K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.354']
Outer characteristic polynomial of the knot is: t^7+91t^5+142t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.354']
2-strand cable arrow polynomial of the knot is: -64K14K2*2 + 64K1*4K2 - 160K14 + 128K1*3K2K3 - 192K1*3K3 + 512K12K2*5 - 2432K1*2K2*4 + 4096K1*2K2*3 - 64K1*2K2*2K3*2 + 64K1*2K2*2K4 - 6848K12K2*2 - 256K1*2K2K4 + 4536K1*2K2 - 176K12K3*2 - 32K1*2K3K5 - 2952K1*2 + 256K1K25K3 - 768K1K2*4K3 + 3680K1K2*3K3 + 96K1K2*2K3K4 - 2016K1K22K3 + 32K1K2*2K4K5 - 288K1K22K5 + 64K1K2K33 - 192K1K2K3K4 - 32K1K2K3K6 + 5120K1K2K3 + 328K1K3K4 + 32K1K4K5 - 128K28 + 128K2*6K4 - 1504K26 - 384K2*4K3*2 - 32K2*4K4*2 + 1184K2*4K4 - 3552K24 + 128K2*3K3K5 - 1536K2*2K3*2 - 200K2*2K4*2 + 2248K2*2K4 - 32K22K5*2 - 148K2*2 + 568K2K3K5 + 24K2K4K6 + 8K2K5K7 - 16K34 + 8K3*2K6 - 1200K32 - 236K4*2 - 80K5*2 - 4K6**2 + 2218
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.354']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70465', 'vk6.70482', 'vk6.70520', 'vk6.70593', 'vk6.70639', 'vk6.70664', 'vk6.70745', 'vk6.70835', 'vk6.70918', 'vk6.70949', 'vk6.71001', 'vk6.71104', 'vk6.71155', 'vk6.71172', 'vk6.71234', 'vk6.71295', 'vk6.71322', 'vk6.71339', 'vk6.73556', 'vk6.74352', 'vk6.74996', 'vk6.75315', 'vk6.76566', 'vk6.76639', 'vk6.76983', 'vk6.78293', 'vk6.79392', 'vk6.79937', 'vk6.81508', 'vk6.86872', 'vk6.88072', 'vk6.89226']
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,-2,0,1,2,3,0,1,4,2,5,0,1,0,2,1,1,1,1,1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.354']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.354']

Outer characteristic polynomial of the knot is: t^7+91t^5+142t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.354']

2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 64*K1**4*K2 - 160*K1**4 + 128*K1**3*K2*K3 - 192*K1**3*K3 + 512*K1**2*K2**5 - 2432*K1**2*K2**4 + 4096*K1**2*K2**3 - 64*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 6848*K1**2*K2**2 - 256*K1**2*K2*K4 + 4536*K1**2*K2 - 176*K1**2*K3**2 - 32*K1**2*K3*K5 - 2952*K1**2 + 256*K1*K2**5*K3 - 768*K1*K2**4*K3 + 3680*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 + 64*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 5120*K1*K2*K3 + 328*K1*K3*K4 + 32*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 1504*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 1184*K2**4*K4 - 3552*K2**4 + 128*K2**3*K3*K5 - 1536*K2**2*K3**2 - 200*K2**2*K4**2 + 2248*K2**2*K4 - 32*K2**2*K5**2 - 148*K2**2 + 568*K2*K3*K5 + 24*K2*K4*K6 + 8*K2*K5*K7 - 16*K3**4 + 8*K3**2*K6 - 1200*K3**2 - 236*K4**2 - 80*K5**2 - 4*K6**2 + 2218

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.354']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.70465', 'vk6.70482', 'vk6.70520', 'vk6.70593', 'vk6.70639', 'vk6.70664', 'vk6.70745', 'vk6.70835', 'vk6.70918', 'vk6.70949', 'vk6.71001', 'vk6.71104', 'vk6.71155', 'vk6.71172', 'vk6.71234', 'vk6.71295', 'vk6.71322', 'vk6.71339', 'vk6.73556', 'vk6.74352', 'vk6.74996', 'vk6.75315', 'vk6.76566', 'vk6.76639', 'vk6.76983', 'vk6.78293', 'vk6.79392', 'vk6.79937', 'vk6.81508', 'vk6.86872', 'vk6.88072', 'vk6.89226']

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	O1O2O3O4O5U3U4O6U1U2U6U5
R3 orbit	{'O1O2O3O4O5U3U4O6U1U2U6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U6U4U5O6U2U3
Gauss code of K*	O1O2O3O4U1U2U5U6U4O5O6U3
Gauss code of -K*	O1O2O3O4U2O5O6U1U5U6U3U4
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 -3 -1 -2 0 4 2],[ 3 0 1 -1 1 5 2],[ 1 -1 0 -1 1 4 1],[ 2 1 1 0 1 2 0],[ 0 -1 -1 -1 0 1 0],[-4 -5 -4 -2 -1 0 0],[-2 -2 -1 0 0 0 0]]
Primitive based matrix	[[ 0 4 2 0 -1 -2 -3],[-4 0 0 -1 -4 -2 -5],[-2 0 0 0 -1 0 -2],[ 0 1 0 0 -1 -1 -1],[ 1 4 1 1 0 -1 -1],[ 2 2 0 1 1 0 1],[ 3 5 2 1 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,0,1,2,3,0,1,4,2,5,0,1,0,2,1,1,1,1,1,-1]
Phi over symmetry	[-4,-2,0,1,2,3,0,1,4,2,5,0,1,0,2,1,1,1,1,1,-1]
Phi of -K	[-3,-2,-1,0,2,4,2,1,2,3,2,0,1,4,4,0,2,1,2,3,2]
Phi of K*	[-4,-2,0,1,2,3,2,3,1,4,2,2,2,4,3,0,1,2,0,1,2]
Phi of -K*	[-3,-2,-1,0,2,4,-1,1,1,2,5,1,1,0,2,1,1,4,0,1,0]
Symmetry type of based matrix	c
u-polynomial	-t^4+t^3+t
Normalized Jones-Krushkal polynomial	5z^2+18z+17
Enhanced Jones-Krushkal polynomial	-4w^4z^2+9w^3z^2-4w^3z+22w^2z+17w
Inner characteristic polynomial	t^6+57t^4+52t^2
Outer characteristic polynomial	t^7+91t^5+142t^3+7t
Flat arrow polynomial	-8K14 + 4K1*3 + 4K1*2K2 - 6K1K2 + 2K2 + 2K3 + 3
2-strand cable arrow polynomial	-64K14K2*2 + 64K1*4K2 - 160K14 + 128K1*3K2K3 - 192K1*3K3 + 512K12K2*5 - 2432K1*2K2*4 + 4096K1*2K2*3 - 64K1*2K2*2K3*2 + 64K1*2K2*2K4 - 6848K12K2*2 - 256K1*2K2K4 + 4536K1*2K2 - 176K12K3*2 - 32K1*2K3K5 - 2952K1*2 + 256K1K25K3 - 768K1K2*4K3 + 3680K1K2*3K3 + 96K1K2*2K3K4 - 2016K1K22K3 + 32K1K2*2K4K5 - 288K1K22K5 + 64K1K2K33 - 192K1K2K3K4 - 32K1K2K3K6 + 5120K1K2K3 + 328K1K3K4 + 32K1K4K5 - 128K28 + 128K2*6K4 - 1504K26 - 384K2*4K3*2 - 32K2*4K4*2 + 1184K2*4K4 - 3552K24 + 128K2*3K3K5 - 1536K2*2K3*2 - 200K2*2K4*2 + 2248K2*2K4 - 32K22K5*2 - 148K2*2 + 568K2K3K5 + 24K2K4K6 + 8K2K5K7 - 16K34 + 8K3*2K6 - 1200K32 - 236K4*2 - 80K5*2 - 4K6**2 + 2218
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}, {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}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]]
If K is slice	False

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