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Flat knot 6.433

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,1,2,1,4,3,1,1,3,2,1,2,1,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.433']
Arrow polynomial of the knot is: -8K14 + 4K1*3 + 4K1*2K2 - 2K1K2 - 2K1 + 2K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.152', '6.164', '6.256', '6.433']
Outer characteristic polynomial of the knot is: t^7+103t^5+302t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.433']
2-strand cable arrow polynomial of the knot is: -736K14 - 96K1*3K3 + 1152K12K2*5 - 4608K1*2K2*4 - 128K1*2K2*3K4 + 3424K12K2*3 - 4400K1*2K2*2 - 160K1*2K2K4 + 3864K1*2K2 - 64K12K3*2 - 2520K1*2 + 768K1K25K3 - 640K1K2*4K3 - 128K1K2*4K5 + 3296K1K2*3K3 + 32K1K2*2K3K4 - 544K1K22K3 - 64K1K2*2K5 - 64K1K2K3K4 + 3120K1K2K3 + 344K1K3K4 + 56K1K4K5 - 128K2*8 + 256K2*6K4 - 1952K26 - 640K2*4K3*2 - 288K2*4K4*2 + 1344K2*4K4 - 1184K24 + 192K2*3K3K5 + 96K2*3K4K6 - 720K2*2K3*2 - 264K2*2K4*2 + 944K2*2K4 - 552K22 + 184K2K3K5 + 32K2K4K6 - 868K3*2 - 304K4*2 - 52K5**2 + 1998
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.433']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16900', 'vk6.17144', 'vk6.20200', 'vk6.21478', 'vk6.23288', 'vk6.23589', 'vk6.27364', 'vk6.29004', 'vk6.35294', 'vk6.35734', 'vk6.38799', 'vk6.40968', 'vk6.42805', 'vk6.43089', 'vk6.45550', 'vk6.47339', 'vk6.55049', 'vk6.55294', 'vk6.57043', 'vk6.58139', 'vk6.59441', 'vk6.59730', 'vk6.61534', 'vk6.62718', 'vk6.64892', 'vk6.65107', 'vk6.66659', 'vk6.67484', 'vk6.68201', 'vk6.68347', 'vk6.69304', 'vk6.70068']
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,-2,0,1,2,3,1,2,1,4,3,1,1,3,2,1,2,1,-1,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.433']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.152', '6.164', '6.256', '6.433']

Outer characteristic polynomial of the knot is: t^7+103t^5+302t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.433']

2-strand cable arrow polynomial of the knot is: -736*K1**4 - 96*K1**3*K3 + 1152*K1**2*K2**5 - 4608*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3424*K1**2*K2**3 - 4400*K1**2*K2**2 - 160*K1**2*K2*K4 + 3864*K1**2*K2 - 64*K1**2*K3**2 - 2520*K1**2 + 768*K1*K2**5*K3 - 640*K1*K2**4*K3 - 128*K1*K2**4*K5 + 3296*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 544*K1*K2**2*K3 - 64*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3120*K1*K2*K3 + 344*K1*K3*K4 + 56*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1952*K2**6 - 640*K2**4*K3**2 - 288*K2**4*K4**2 + 1344*K2**4*K4 - 1184*K2**4 + 192*K2**3*K3*K5 + 96*K2**3*K4*K6 - 720*K2**2*K3**2 - 264*K2**2*K4**2 + 944*K2**2*K4 - 552*K2**2 + 184*K2*K3*K5 + 32*K2*K4*K6 - 868*K3**2 - 304*K4**2 - 52*K5**2 + 1998

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.433']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16900', 'vk6.17144', 'vk6.20200', 'vk6.21478', 'vk6.23288', 'vk6.23589', 'vk6.27364', 'vk6.29004', 'vk6.35294', 'vk6.35734', 'vk6.38799', 'vk6.40968', 'vk6.42805', 'vk6.43089', 'vk6.45550', 'vk6.47339', 'vk6.55049', 'vk6.55294', 'vk6.57043', 'vk6.58139', 'vk6.59441', 'vk6.59730', 'vk6.61534', 'vk6.62718', 'vk6.64892', 'vk6.65107', 'vk6.66659', 'vk6.67484', 'vk6.68201', 'vk6.68347', 'vk6.69304', 'vk6.70068']

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	O1O2O3O4O5U6U2O6U1U3U4U5
R3 orbit	{'O1O2O3O4O5U6U2O6U1U3U4U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U2U3U5O6U4U6
Gauss code of K*	O1O2O3O4U1U5U2U3U4O6O5U6
Gauss code of -K*	O1O2O3O4U5O6O5U1U2U3U6U4
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 0 2 4 -1],[ 3 0 1 2 3 4 2],[ 2 -1 0 0 1 2 2],[ 0 -2 0 0 1 2 0],[-2 -3 -1 -1 0 1 -2],[-4 -4 -2 -2 -1 0 -4],[ 1 -2 -2 0 2 4 0]]
Primitive based matrix	[[ 0 4 2 0 -1 -2 -3],[-4 0 -1 -2 -4 -2 -4],[-2 1 0 -1 -2 -1 -3],[ 0 2 1 0 0 0 -2],[ 1 4 2 0 0 -2 -2],[ 2 2 1 0 2 0 -1],[ 3 4 3 2 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,0,1,2,3,1,2,4,2,4,1,2,1,3,0,0,2,2,2,1]
Phi over symmetry	[-4,-2,0,1,2,3,1,2,1,4,3,1,1,3,2,1,2,1,-1,0,0]
Phi of -K	[-3,-2,-1,0,2,4,0,0,1,2,3,-1,2,3,4,1,1,1,1,2,1]
Phi of K*	[-4,-2,0,1,2,3,1,2,1,4,3,1,1,3,2,1,2,1,-1,0,0]
Phi of -K*	[-3,-2,-1,0,2,4,1,2,2,3,4,2,0,1,2,0,2,4,1,2,1]
Symmetry type of based matrix	c
u-polynomial	-t^4+t^3+t
Normalized Jones-Krushkal polynomial	5z+11
Enhanced Jones-Krushkal polynomial	-6w^4z^2+6w^3z^2-10w^3z+15w^2z+11w
Inner characteristic polynomial	t^6+69t^4+170t^2
Outer characteristic polynomial	t^7+103t^5+302t^3+6t
Flat arrow polynomial	-8K14 + 4K1*3 + 4K1*2K2 - 2K1K2 - 2K1 + 2K2 + 3
2-strand cable arrow polynomial	-736K14 - 96K1*3K3 + 1152K12K2*5 - 4608K1*2K2*4 - 128K1*2K2*3K4 + 3424K12K2*3 - 4400K1*2K2*2 - 160K1*2K2K4 + 3864K1*2K2 - 64K12K3*2 - 2520K1*2 + 768K1K25K3 - 640K1K2*4K3 - 128K1K2*4K5 + 3296K1K2*3K3 + 32K1K2*2K3K4 - 544K1K22K3 - 64K1K2*2K5 - 64K1K2K3K4 + 3120K1K2K3 + 344K1K3K4 + 56K1K4K5 - 128K2*8 + 256K2*6K4 - 1952K26 - 640K2*4K3*2 - 288K2*4K4*2 + 1344K2*4K4 - 1184K24 + 192K2*3K3K5 + 96K2*3K4K6 - 720K2*2K3*2 - 264K2*2K4*2 + 944K2*2K4 - 552K22 + 184K2K3K5 + 32K2K4K6 - 868K3*2 - 304K4*2 - 52K5**2 + 1998
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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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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