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Flat knot 6.189

Min(phi) over symmetries of the knot is: [-4,-3,0,2,2,3,0,2,2,5,4,1,1,3,2,1,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.189']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 10K12 - 4K1K2 - 2K1K3 - K1 + 4K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.189']
Outer characteristic polynomial of the knot is: t^7+117t^5+126t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.189']
2-strand cable arrow polynomial of the knot is: 64K14K2 - 1392K14 - 96K1*3K3 - 128K12K2*4 + 384K1*2K2*3 - 3360K1*2K2*2 - 192K1*2K2K4 + 7320K1*2K2 - 48K12K3*2 - 6140K1*2 + 832K1K23K3 + 32K1K2*2K3K4 - 1696K1K22K3 - 128K1K2*2K5 - 672K1K2K3K4 - 32K1K2K3K6 + 6744K1K2K3 + 1432K1K3K4 + 184K1K4K5 + 8K1K5K6 - 32K26 - 128K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 1128K24 + 96K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 + 96K22K3*2K4 - 1168K22K3*2 - 64K2*2K3K7 - 176K2*2K4*2 + 2328K2*2K4 - 8K22K6*2 - 5250K2*2 - 32K2K32K4 + 960K2K3K5 + 224K2K4K6 - 32K32K4*2 + 24K3*2K6 - 2712K32 + 16K3K4K7 - 1156K42 - 172K5*2 - 70K6**2 + 5202
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.189']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73347', 'vk6.73395', 'vk6.73508', 'vk6.73574', 'vk6.73735', 'vk6.73852', 'vk6.74248', 'vk6.74876', 'vk6.75334', 'vk6.75515', 'vk6.75850', 'vk6.76421', 'vk6.78237', 'vk6.78318', 'vk6.78482', 'vk6.78651', 'vk6.78844', 'vk6.79296', 'vk6.80059', 'vk6.80100', 'vk6.80208', 'vk6.80277', 'vk6.80407', 'vk6.80757', 'vk6.81945', 'vk6.82676', 'vk6.84740', 'vk6.85040', 'vk6.85143', 'vk6.86527', 'vk6.87342', 'vk6.89444']
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,-3,0,2,2,3,0,2,2,5,4,1,1,3,2,1,2,2,-1,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.189']

Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 4*K1*K2 - 2*K1*K3 - K1 + 4*K2 + K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.189']

Outer characteristic polynomial of the knot is: t^7+117t^5+126t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.189']

2-strand cable arrow polynomial of the knot is: 64*K1**4*K2 - 1392*K1**4 - 96*K1**3*K3 - 128*K1**2*K2**4 + 384*K1**2*K2**3 - 3360*K1**2*K2**2 - 192*K1**2*K2*K4 + 7320*K1**2*K2 - 48*K1**2*K3**2 - 6140*K1**2 + 832*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1696*K1*K2**2*K3 - 128*K1*K2**2*K5 - 672*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 6744*K1*K2*K3 + 1432*K1*K3*K4 + 184*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 1128*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 + 96*K2**2*K3**2*K4 - 1168*K2**2*K3**2 - 64*K2**2*K3*K7 - 176*K2**2*K4**2 + 2328*K2**2*K4 - 8*K2**2*K6**2 - 5250*K2**2 - 32*K2*K3**2*K4 + 960*K2*K3*K5 + 224*K2*K4*K6 - 32*K3**2*K4**2 + 24*K3**2*K6 - 2712*K3**2 + 16*K3*K4*K7 - 1156*K4**2 - 172*K5**2 - 70*K6**2 + 5202

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.189']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73347', 'vk6.73395', 'vk6.73508', 'vk6.73574', 'vk6.73735', 'vk6.73852', 'vk6.74248', 'vk6.74876', 'vk6.75334', 'vk6.75515', 'vk6.75850', 'vk6.76421', 'vk6.78237', 'vk6.78318', 'vk6.78482', 'vk6.78651', 'vk6.78844', 'vk6.79296', 'vk6.80059', 'vk6.80100', 'vk6.80208', 'vk6.80277', 'vk6.80407', 'vk6.80757', 'vk6.81945', 'vk6.82676', 'vk6.84740', 'vk6.85040', 'vk6.85143', 'vk6.86527', 'vk6.87342', 'vk6.89444']

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	O1O2O3O4O5U2O6U1U4U6U5U3
R3 orbit	{'O1O2O3O4O5U2O6U1U4U6U5U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U1U6U2U5O6U4
Gauss code of K*	O1O2O3O4O5U1U6U5U2U4O6U3
Gauss code of -K*	O1O2O3O4O5U3O6U2U4U1U6U5
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 -3 2 0 3 2],[ 4 0 0 5 2 4 2],[ 3 0 0 3 1 2 1],[-2 -5 -3 0 -2 1 1],[ 0 -2 -1 2 0 2 1],[-3 -4 -2 -1 -2 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 3 2 2 0 -3 -4],[-3 0 0 -1 -2 -2 -4],[-2 0 0 -1 -1 -1 -2],[-2 1 1 0 -2 -3 -5],[ 0 2 1 2 0 -1 -2],[ 3 2 1 3 1 0 0],[ 4 4 2 5 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-2,0,3,4,0,1,2,2,4,1,1,1,2,2,3,5,1,2,0]
Phi over symmetry	[-4,-3,0,2,2,3,0,2,2,5,4,1,1,3,2,1,2,2,-1,0,1]
Phi of -K	[-4,-3,0,2,2,3,1,2,1,4,3,2,2,4,4,0,1,1,-1,0,1]
Phi of K*	[-3,-2,-2,0,3,4,0,1,1,4,3,1,0,2,1,1,4,4,2,2,1]
Phi of -K*	[-4,-3,0,2,2,3,0,2,2,5,4,1,1,3,2,1,2,2,-1,0,1]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^2
Normalized Jones-Krushkal polynomial	3z^2+22z+33
Enhanced Jones-Krushkal polynomial	3w^3z^2+22w^2z+33w
Inner characteristic polynomial	t^6+75t^4+14t^2
Outer characteristic polynomial	t^7+117t^5+126t^3+5t
Flat arrow polynomial	4K13 + 4K1*2K2 - 10K12 - 4K1K2 - 2K1K3 - K1 + 4K2 + K3 + 5
2-strand cable arrow polynomial	64K14K2 - 1392K14 - 96K1*3K3 - 128K12K2*4 + 384K1*2K2*3 - 3360K1*2K2*2 - 192K1*2K2K4 + 7320K1*2K2 - 48K12K3*2 - 6140K1*2 + 832K1K23K3 + 32K1K2*2K3K4 - 1696K1K22K3 - 128K1K2*2K5 - 672K1K2K3K4 - 32K1K2K3K6 + 6744K1K2K3 + 1432K1K3K4 + 184K1K4K5 + 8K1K5K6 - 32K26 - 128K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 1128K24 + 96K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 + 96K22K3*2K4 - 1168K22K3*2 - 64K2*2K3K7 - 176K2*2K4*2 + 2328K2*2K4 - 8K22K6*2 - 5250K2*2 - 32K2K32K4 + 960K2K3K5 + 224K2K4K6 - 32K32K4*2 + 24K3*2K6 - 2712K32 + 16K3K4K7 - 1156K42 - 172K5*2 - 70K6**2 + 5202
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}], [{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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {1, 5}, {4}, {3}, {2}], [{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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