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Flat knot 6.119

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,-1,2,1,2,3,2,1,2,2,1,1,1,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.119']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 6K12 - 2K1K2 - 4K1K3 - 2K1 + 3*K2 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.119', '6.852']
Outer characteristic polynomial of the knot is: t^7+69t^5+82t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.119']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 448K1*4K2 - 1216K14 + 1024K1*3K2K3 - 256K1*3K3 - 256K12K2*4 + 576K1*2K2*3 - 576K1*2K2*2K3*2 + 64K1*2K2*2K4 - 5744K12K2*2 + 160K1*2K2K32 + 32K1*2K2K3K5 - 544K12K2K4 + 5240K1*2K2 - 672K12K3*2 - 3940K1*2 + 1952K1K23K3 + 512K1K2*2K3K4 - 768K1K22K3 - 128K1K2*2K5 + 96K1K2K33 - 384K1K2K3K4 - 128K1K2K3K6 + 7568K1K2K3 + 1008K1K3K4 + 32K1K4K5 + 40K1K5K6 - 32K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 96K2*4K4 - 1576K24 + 128K2*3K3K5 + 64K2*3K4K6 - 32K2*3K6 + 128K22K3*2K6 - 2544K22K3*2 - 328K2*2K4*2 + 1248K2*2K4 - 112K22K5*2 - 176K2*2K6*2 - 2836K2*2 - 192K2K32K4 + 1408K2K3K5 + 504K2K4K6 + 32K2K5K7 + 48K2K6K8 - 32K34 + 128K3*2K6 - 2528K32 - 634K4*2 - 236K5*2 - 196K6*2 - 2K8**2 + 3882
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.119']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4750', 'vk6.5077', 'vk6.6292', 'vk6.6731', 'vk6.8253', 'vk6.8702', 'vk6.9639', 'vk6.9954', 'vk6.20407', 'vk6.21764', 'vk6.27751', 'vk6.29281', 'vk6.39181', 'vk6.41417', 'vk6.45915', 'vk6.47548', 'vk6.48782', 'vk6.48993', 'vk6.49594', 'vk6.49797', 'vk6.50794', 'vk6.51009', 'vk6.51281', 'vk6.51476', 'vk6.57268', 'vk6.58493', 'vk6.61918', 'vk6.63019', 'vk6.66881', 'vk6.67759', 'vk6.69511', 'vk6.70225']
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,-2,0,1,2,2,-1,2,1,2,3,2,1,2,2,1,1,1,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.119']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.119', '6.852']

Outer characteristic polynomial of the knot is: t^7+69t^5+82t^3+13t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.119']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 1216*K1**4 + 1024*K1**3*K2*K3 - 256*K1**3*K3 - 256*K1**2*K2**4 + 576*K1**2*K2**3 - 576*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 5744*K1**2*K2**2 + 160*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 544*K1**2*K2*K4 + 5240*K1**2*K2 - 672*K1**2*K3**2 - 3940*K1**2 + 1952*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 - 128*K1*K2**2*K5 + 96*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 7568*K1*K2*K3 + 1008*K1*K3*K4 + 32*K1*K4*K5 + 40*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 1576*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 + 128*K2**2*K3**2*K6 - 2544*K2**2*K3**2 - 328*K2**2*K4**2 + 1248*K2**2*K4 - 112*K2**2*K5**2 - 176*K2**2*K6**2 - 2836*K2**2 - 192*K2*K3**2*K4 + 1408*K2*K3*K5 + 504*K2*K4*K6 + 32*K2*K5*K7 + 48*K2*K6*K8 - 32*K3**4 + 128*K3**2*K6 - 2528*K3**2 - 634*K4**2 - 236*K5**2 - 196*K6**2 - 2*K8**2 + 3882

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.119']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4750', 'vk6.5077', 'vk6.6292', 'vk6.6731', 'vk6.8253', 'vk6.8702', 'vk6.9639', 'vk6.9954', 'vk6.20407', 'vk6.21764', 'vk6.27751', 'vk6.29281', 'vk6.39181', 'vk6.41417', 'vk6.45915', 'vk6.47548', 'vk6.48782', 'vk6.48993', 'vk6.49594', 'vk6.49797', 'vk6.50794', 'vk6.51009', 'vk6.51281', 'vk6.51476', 'vk6.57268', 'vk6.58493', 'vk6.61918', 'vk6.63019', 'vk6.66881', 'vk6.67759', 'vk6.69511', 'vk6.70225']

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	O1O2O3O4O5O6U3U5U6U1U4U2
R3 orbit	{'O1O2O3O4O5O6U3U5U6U1U4U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U5U3U6U1U2U4
Gauss code of K*	O1O2O3O4O5O6U4U6U1U5U2U3
Gauss code of -K*	O1O2O3O4O5O6U4U5U2U6U1U3
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 -2 1 -3 2 0 2],[ 2 0 2 -2 2 0 2],[-1 -2 0 -3 1 0 2],[ 3 2 3 0 3 1 2],[-2 -2 -1 -3 0 -1 1],[ 0 0 0 -1 1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix	[[ 0 2 2 1 0 -2 -3],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -2 -1 -2 -2],[-1 1 2 0 0 -2 -3],[ 0 1 1 0 0 0 -1],[ 2 2 2 2 0 0 -2],[ 3 3 2 3 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,2,3,-1,1,1,2,3,2,1,2,2,0,2,3,0,1,2]
Phi over symmetry	[-3,-2,0,1,2,2,-1,2,1,2,3,2,1,2,2,1,1,1,0,-1,-1]
Phi of -K	[-3,-2,0,1,2,2,-1,2,1,2,3,2,1,2,2,1,1,1,0,-1,-1]
Phi of K*	[-2,-2,-1,0,2,3,-1,-1,1,2,3,0,1,2,2,1,1,1,2,2,-1]
Phi of -K*	[-3,-2,0,1,2,2,2,1,3,2,3,0,2,2,2,0,1,1,2,1,-1]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	5z^2+26z+33
Enhanced Jones-Krushkal polynomial	5w^3z^2+26w^2z+33w
Inner characteristic polynomial	t^6+47t^4+27t^2
Outer characteristic polynomial	t^7+69t^5+82t^3+13t
Flat arrow polynomial	4K13 + 4K1*2K2 - 6K12 - 2K1K2 - 4K1K3 - 2K1 + 3*K2 + K4 + 3
2-strand cable arrow polynomial	-256K14K2*2 + 448K1*4K2 - 1216K14 + 1024K1*3K2K3 - 256K1*3K3 - 256K12K2*4 + 576K1*2K2*3 - 576K1*2K2*2K3*2 + 64K1*2K2*2K4 - 5744K12K2*2 + 160K1*2K2K32 + 32K1*2K2K3K5 - 544K12K2K4 + 5240K1*2K2 - 672K12K3*2 - 3940K1*2 + 1952K1K23K3 + 512K1K2*2K3K4 - 768K1K22K3 - 128K1K2*2K5 + 96K1K2K33 - 384K1K2K3K4 - 128K1K2K3K6 + 7568K1K2K3 + 1008K1K3K4 + 32K1K4K5 + 40K1K5K6 - 32K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 96K2*4K4 - 1576K24 + 128K2*3K3K5 + 64K2*3K4K6 - 32K2*3K6 + 128K22K3*2K6 - 2544K22K3*2 - 328K2*2K4*2 + 1248K2*2K4 - 112K22K5*2 - 176K2*2K6*2 - 2836K2*2 - 192K2K32K4 + 1408K2K3K5 + 504K2K4K6 + 32K2K5K7 + 48K2K6K8 - 32K34 + 128K3*2K6 - 2528K32 - 634K4*2 - 236K5*2 - 196K6*2 - 2K8**2 + 3882
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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {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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