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Flat knot 6.302

Min(phi) over symmetries of the knot is: [-4,-1,-1,0,3,3,0,2,2,3,4,1,1,1,1,1,2,3,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.302']
Arrow polynomial of the knot is: -4K1K2 + 2K1 - 2K2*2 + 2K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.87', '6.88', '6.184', '6.302', '6.459', '6.467', '6.506']
Outer characteristic polynomial of the knot is: t^7+92t^5+92t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.302']
2-strand cable arrow polynomial of the knot is: -560K14 + 384K1*3K2K3 + 32K1*3K3K4 - 288K1*3K3 - 1088K12K2*2 - 288K1*2K2K4 + 2056K1*2K2 - 784K12K3*2 - 96K1*2K3K5 - 96K1*2K4*2 - 1644K1*2 + 64K1K23K3 - 288K1K2*2K3 - 64K1K2*2K5 + 128K1K2K33 + 64K1K2K3K42 - 128K1K2K3K4 - 64K1K2K3K6 + 2360K1K2K3 - 32K1K3*2K5 + 1136K1K3K4 + 216K1K4K5 - 32K24 - 160K2*2K3*2 - 80K2*2K4*2 + 376K2*2K4 - 1300K22 - 32K2K32K4 - 32K2K3K4K5 + 248K2K3K5 + 32K2K4K6 + 8K2K5K7 - 128K3*4 - 80K3*2K4*2 + 96K3*2K6 - 872K32 + 40K3K4K7 - 8K44 + 8K4*2K8 - 396K42 - 104K5*2 - 4K6*2 - 4K7*2 - 2K8**2 + 1404
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.302']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16986', 'vk6.17228', 'vk6.19925', 'vk6.20218', 'vk6.21143', 'vk6.21512', 'vk6.23387', 'vk6.26841', 'vk6.26864', 'vk6.27416', 'vk6.28614', 'vk6.29028', 'vk6.35447', 'vk6.38277', 'vk6.38300', 'vk6.38825', 'vk6.40408', 'vk6.41016', 'vk6.42880', 'vk6.45152', 'vk6.45163', 'vk6.45590', 'vk6.46997', 'vk6.47355', 'vk6.55149', 'vk6.56705', 'vk6.56707', 'vk6.57793', 'vk6.58169', 'vk6.59522', 'vk6.61115', 'vk6.61564', 'vk6.62366', 'vk6.62738', 'vk6.64965', 'vk6.66398', 'vk6.67160', 'vk6.68254', 'vk6.69053', 'vk6.69841']
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,-1,-1,0,3,3,0,2,2,3,4,1,1,1,1,1,2,3,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.302']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.87', '6.88', '6.184', '6.302', '6.459', '6.467', '6.506']

Outer characteristic polynomial of the knot is: t^7+92t^5+92t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.302']

2-strand cable arrow polynomial of the knot is: -560*K1**4 + 384*K1**3*K2*K3 + 32*K1**3*K3*K4 - 288*K1**3*K3 - 1088*K1**2*K2**2 - 288*K1**2*K2*K4 + 2056*K1**2*K2 - 784*K1**2*K3**2 - 96*K1**2*K3*K5 - 96*K1**2*K4**2 - 1644*K1**2 + 64*K1*K2**3*K3 - 288*K1*K2**2*K3 - 64*K1*K2**2*K5 + 128*K1*K2*K3**3 + 64*K1*K2*K3*K4**2 - 128*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 2360*K1*K2*K3 - 32*K1*K3**2*K5 + 1136*K1*K3*K4 + 216*K1*K4*K5 - 32*K2**4 - 160*K2**2*K3**2 - 80*K2**2*K4**2 + 376*K2**2*K4 - 1300*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 248*K2*K3*K5 + 32*K2*K4*K6 + 8*K2*K5*K7 - 128*K3**4 - 80*K3**2*K4**2 + 96*K3**2*K6 - 872*K3**2 + 40*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 396*K4**2 - 104*K5**2 - 4*K6**2 - 4*K7**2 - 2*K8**2 + 1404

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.302']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16986', 'vk6.17228', 'vk6.19925', 'vk6.20218', 'vk6.21143', 'vk6.21512', 'vk6.23387', 'vk6.26841', 'vk6.26864', 'vk6.27416', 'vk6.28614', 'vk6.29028', 'vk6.35447', 'vk6.38277', 'vk6.38300', 'vk6.38825', 'vk6.40408', 'vk6.41016', 'vk6.42880', 'vk6.45152', 'vk6.45163', 'vk6.45590', 'vk6.46997', 'vk6.47355', 'vk6.55149', 'vk6.56705', 'vk6.56707', 'vk6.57793', 'vk6.58169', 'vk6.59522', 'vk6.61115', 'vk6.61564', 'vk6.62366', 'vk6.62738', 'vk6.64965', 'vk6.66398', 'vk6.67160', 'vk6.68254', 'vk6.69053', 'vk6.69841']

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	O1O2O3O4O5U2U1O6U4U6U3U5
R3 orbit	{'O1O2O3O4O5U2U1O6U4U6U3U5', 'O1O2O3O4O5U2U1U3O6U4U6U5'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U1U3U6U2O6U5U4
Gauss code of K*	O1O2O3O4U5U6U3U1U4O6O5U2
Gauss code of -K*	O1O2O3O4U3O5O6U1U4U2U6U5
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 -3 1 0 4 1],[ 3 0 0 3 2 4 1],[ 3 0 0 2 1 3 1],[-1 -3 -2 0 -1 2 1],[ 0 -2 -1 1 0 2 1],[-4 -4 -3 -2 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 4 1 1 0 -3 -3],[-4 0 0 -2 -2 -3 -4],[-1 0 0 -1 -1 -1 -1],[-1 2 1 0 -1 -2 -3],[ 0 2 1 1 0 -1 -2],[ 3 3 1 2 1 0 0],[ 3 4 1 3 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-1,-1,0,3,3,0,2,2,3,4,1,1,1,1,1,2,3,1,2,0]
Phi over symmetry	[-4,-1,-1,0,3,3,0,2,2,3,4,1,1,1,1,1,2,3,1,2,0]
Phi of -K	[-3,-3,0,1,1,4,0,1,1,3,3,2,2,3,4,0,0,2,-1,1,3]
Phi of K*	[-4,-1,-1,0,3,3,1,3,2,3,4,1,0,1,2,0,3,3,1,2,0]
Phi of -K*	[-3,-3,0,1,1,4,0,1,1,2,3,2,1,3,4,1,1,2,-1,0,2]
Symmetry type of based matrix	c
u-polynomial	-t^4+2t^3-2t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	4w^3z^2+17w^2z+19w
Inner characteristic polynomial	t^6+56t^4+24t^2
Outer characteristic polynomial	t^7+92t^5+92t^3+3t
Flat arrow polynomial	-4K1K2 + 2K1 - 2K2*2 + 2K3 + K4 + 2
2-strand cable arrow polynomial	-560K14 + 384K1*3K2K3 + 32K1*3K3K4 - 288K1*3K3 - 1088K12K2*2 - 288K1*2K2K4 + 2056K1*2K2 - 784K12K3*2 - 96K1*2K3K5 - 96K1*2K4*2 - 1644K1*2 + 64K1K23K3 - 288K1K2*2K3 - 64K1K2*2K5 + 128K1K2K33 + 64K1K2K3K42 - 128K1K2K3K4 - 64K1K2K3K6 + 2360K1K2K3 - 32K1K3*2K5 + 1136K1K3K4 + 216K1K4K5 - 32K24 - 160K2*2K3*2 - 80K2*2K4*2 + 376K2*2K4 - 1300K22 - 32K2K32K4 - 32K2K3K4K5 + 248K2K3K5 + 32K2K4K6 + 8K2K5K7 - 128K3*4 - 80K3*2K4*2 + 96K3*2K6 - 872K32 + 40K3K4K7 - 8K44 + 8K4*2K8 - 396K42 - 104K5*2 - 4K6*2 - 4K7*2 - 2K8**2 + 1404
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}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {3}, {1, 2}]]
If K is slice	False

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