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Flat knot 6.869

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,1,2,2,1,1,2,2,-1,-1,-1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.869']
Arrow polynomial of the knot is: -2K1K2 - 4K1K3 + K1 + 2K2 + K3 + 2K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.126', '6.195', '6.367', '6.438', '6.869', '6.872', '6.896', '6.1147']
Outer characteristic polynomial of the knot is: t^7+72t^5+67t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.869']
2-strand cable arrow polynomial of the knot is: -144K14 + 384K1*3K2K3 - 544K1*3K3 - 256K12K2*2K3*2 - 1024K1*2K2*2 + 384K1*2K2K32 + 32K1*2K2K3K5 - 128K12K2K4 + 2864K1*2K2 - 1872K12K3*2 - 64K1*2K3K5 - 4044K1*2 + 416K1K23K3 + 160K1K2*2K3K4 - 1216K1K22K3 - 32K1K2*2K5 + 256K1K2K33 - 192K1K2K3K4 - 192K1K2K3K6 + 6840K1K2K3 - 32K1K2K5K6 + 1952K1K3K4 + 72K1K4K5 + 48K1K5K6 + 8K1K6K7 - 192K24 - 2192K2*2K3*2 - 40K2*2K4*2 + 504K2*2K4 - 16K22K5*2 - 48K2*2K6*2 - 3270K2*2 - 96K2K32K4 + 1632K2K3K5 + 152K2K4K6 + 24K2K5K7 + 32K2K6K8 - 64K34 + 88K3*2K6 - 2716K32 - 516K4*2 - 332K5*2 - 74K6*2 - 4K7*2 - 4K8**2 + 3454
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.869']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16927', 'vk6.17168', 'vk6.20223', 'vk6.21518', 'vk6.23321', 'vk6.23614', 'vk6.27425', 'vk6.29036', 'vk6.35359', 'vk6.35781', 'vk6.38841', 'vk6.41034', 'vk6.42840', 'vk6.43118', 'vk6.45602', 'vk6.47362', 'vk6.55087', 'vk6.55337', 'vk6.57056', 'vk6.58181', 'vk6.59486', 'vk6.59774', 'vk6.61573', 'vk6.62746', 'vk6.64933', 'vk6.65139', 'vk6.66677', 'vk6.67516', 'vk6.68226', 'vk6.68367', 'vk6.69330', 'vk6.70082']
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: [-3,-2,1,1,1,2,0,1,1,2,2,1,1,2,2,-1,-1,-1,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.869']

Arrow polynomial of the knot is: -2*K1*K2 - 4*K1*K3 + K1 + 2*K2 + K3 + 2*K4 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.126', '6.195', '6.367', '6.438', '6.869', '6.872', '6.896', '6.1147']

Outer characteristic polynomial of the knot is: t^7+72t^5+67t^3+10t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.869']

2-strand cable arrow polynomial of the knot is: -144*K1**4 + 384*K1**3*K2*K3 - 544*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 1024*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 128*K1**2*K2*K4 + 2864*K1**2*K2 - 1872*K1**2*K3**2 - 64*K1**2*K3*K5 - 4044*K1**2 + 416*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1216*K1*K2**2*K3 - 32*K1*K2**2*K5 + 256*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 192*K1*K2*K3*K6 + 6840*K1*K2*K3 - 32*K1*K2*K5*K6 + 1952*K1*K3*K4 + 72*K1*K4*K5 + 48*K1*K5*K6 + 8*K1*K6*K7 - 192*K2**4 - 2192*K2**2*K3**2 - 40*K2**2*K4**2 + 504*K2**2*K4 - 16*K2**2*K5**2 - 48*K2**2*K6**2 - 3270*K2**2 - 96*K2*K3**2*K4 + 1632*K2*K3*K5 + 152*K2*K4*K6 + 24*K2*K5*K7 + 32*K2*K6*K8 - 64*K3**4 + 88*K3**2*K6 - 2716*K3**2 - 516*K4**2 - 332*K5**2 - 74*K6**2 - 4*K7**2 - 4*K8**2 + 3454

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.869']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16927', 'vk6.17168', 'vk6.20223', 'vk6.21518', 'vk6.23321', 'vk6.23614', 'vk6.27425', 'vk6.29036', 'vk6.35359', 'vk6.35781', 'vk6.38841', 'vk6.41034', 'vk6.42840', 'vk6.43118', 'vk6.45602', 'vk6.47362', 'vk6.55087', 'vk6.55337', 'vk6.57056', 'vk6.58181', 'vk6.59486', 'vk6.59774', 'vk6.61573', 'vk6.62746', 'vk6.64933', 'vk6.65139', 'vk6.66677', 'vk6.67516', 'vk6.68226', 'vk6.68367', 'vk6.69330', 'vk6.70082']

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	O1O2O3O4U1U5O6O5U2U4U3U6
R3 orbit	{'O1O2O3O4U1U5O6O5U2U4U3U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U2U1U3O6O5U6U4
Gauss code of K*	O1O2O3O4U5U1U3U2O5O6U4U6
Gauss code of -K*	O1O2O3O4U5U1O5O6U3U2U4U6
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 -2 1 1 1 2],[ 3 0 1 3 2 3 3],[ 2 -1 0 2 1 2 2],[-1 -3 -2 0 0 -1 1],[-1 -2 -1 0 0 -1 0],[-1 -3 -2 1 1 0 2],[-2 -3 -2 -1 0 -2 0]]
Primitive based matrix	[[ 0 2 1 1 1 -2 -3],[-2 0 0 -1 -2 -2 -3],[-1 0 0 0 -1 -1 -2],[-1 1 0 0 -1 -2 -3],[-1 2 1 1 0 -2 -3],[ 2 2 1 2 2 0 -1],[ 3 3 2 3 3 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,-1,2,3,0,1,2,2,3,0,1,1,2,1,2,3,2,3,1]
Phi over symmetry	[-3,-2,1,1,1,2,0,1,1,2,2,1,1,2,2,-1,-1,-1,0,0,1]
Phi of -K	[-3,-2,1,1,1,2,0,1,1,2,2,1,1,2,2,-1,-1,-1,0,0,1]
Phi of K*	[-2,-1,-1,-1,2,3,-1,0,1,2,2,1,1,1,1,0,1,1,2,2,0]
Phi of -K*	[-3,-2,1,1,1,2,1,2,3,3,3,1,2,2,2,-1,0,0,1,2,1]
Symmetry type of based matrix	c
u-polynomial	t^3-3t
Normalized Jones-Krushkal polynomial	6z^2+25z+27
Enhanced Jones-Krushkal polynomial	-2w^4z^2+8w^3z^2+25w^2z+27w
Inner characteristic polynomial	t^6+52t^4+21t^2+1
Outer characteristic polynomial	t^7+72t^5+67t^3+10t
Flat arrow polynomial	-2K1K2 - 4K1K3 + K1 + 2K2 + K3 + 2K4 + 1
2-strand cable arrow polynomial	-144K14 + 384K1*3K2K3 - 544K1*3K3 - 256K12K2*2K3*2 - 1024K1*2K2*2 + 384K1*2K2K32 + 32K1*2K2K3K5 - 128K12K2K4 + 2864K1*2K2 - 1872K12K3*2 - 64K1*2K3K5 - 4044K1*2 + 416K1K23K3 + 160K1K2*2K3K4 - 1216K1K22K3 - 32K1K2*2K5 + 256K1K2K33 - 192K1K2K3K4 - 192K1K2K3K6 + 6840K1K2K3 - 32K1K2K5K6 + 1952K1K3K4 + 72K1K4K5 + 48K1K5K6 + 8K1K6K7 - 192K24 - 2192K2*2K3*2 - 40K2*2K4*2 + 504K2*2K4 - 16K22K5*2 - 48K2*2K6*2 - 3270K2*2 - 96K2K32K4 + 1632K2K3K5 + 152K2K4K6 + 24K2K5K7 + 32K2K6K8 - 64K34 + 88K3*2K6 - 2716K32 - 516K4*2 - 332K5*2 - 74K6*2 - 4K7*2 - 4K8**2 + 3454
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}], [{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}, {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}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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