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Flat knot 6.170

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,1,1,3,4,2,1,2,2,2,-1,-1,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.170']
Arrow polynomial of the knot is: 4K13 - 4K1K2 - 2K1*K3 - K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.71', '6.148', '6.170', '6.253', '6.259', '6.298', '6.439', '6.453', '6.499', '6.503']
Outer characteristic polynomial of the knot is: t^7+72t^5+80t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.170']
2-strand cable arrow polynomial of the knot is: 384K14K2 - 1904K14 + 800K1*3K2K3 + 32K1*3K3K4 - 800K1*3K3 + 480K12K2*3 + 96K1*2K2*2K4 - 5024K12K2*2 - 864K1*2K2K4 + 7728K1*2K2 - 912K12K3*2 - 64K1*2K4*2 - 5480K1*2 + 864K1K23K3 - 1344K1K2*2K3 - 352K1K2*2K5 + 192K1K2K33 - 416K1K2K3K4 - 32K1K2K3K6 + 7944K1K2K3 - 64K1K3*2K5 + 1464K1K3K4 + 176K1K4K5 + 16K1K5K6 - 32K2*6 + 64K2*4K4 - 1264K24 + 32K2*3K3K5 - 64K2*3K6 - 1184K22K3*2 - 32K2*2K3K7 - 16K2*2K4*2 + 2016K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 4602K2*2 + 1104K2K3K5 + 80K2K4K6 + 16K2K5K7 + 8K2K6K8 - 128K3*4 + 104K3*2K6 - 2556K32 + 8K3K4K7 - 830K42 - 264K5*2 - 54K6*2 - 4K7*2 - 2K8**2 + 4702
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.170']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11489', 'vk6.11795', 'vk6.12811', 'vk6.13144', 'vk6.17079', 'vk6.17321', 'vk6.20896', 'vk6.21067', 'vk6.22306', 'vk6.22495', 'vk6.23800', 'vk6.28368', 'vk6.31250', 'vk6.31601', 'vk6.32821', 'vk6.35589', 'vk6.36045', 'vk6.40014', 'vk6.40312', 'vk6.42070', 'vk6.43281', 'vk6.46548', 'vk6.46769', 'vk6.48026', 'vk6.52252', 'vk6.53407', 'vk6.57708', 'vk6.57721', 'vk6.58892', 'vk6.59956', 'vk6.64421', 'vk6.69748']
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,-1,1,1,1,2,1,1,3,4,2,1,2,2,2,-1,-1,0,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.170']

Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.71', '6.148', '6.170', '6.253', '6.259', '6.298', '6.439', '6.453', '6.499', '6.503']

Outer characteristic polynomial of the knot is: t^7+72t^5+80t^3+11t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.170']

2-strand cable arrow polynomial of the knot is: 384*K1**4*K2 - 1904*K1**4 + 800*K1**3*K2*K3 + 32*K1**3*K3*K4 - 800*K1**3*K3 + 480*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 5024*K1**2*K2**2 - 864*K1**2*K2*K4 + 7728*K1**2*K2 - 912*K1**2*K3**2 - 64*K1**2*K4**2 - 5480*K1**2 + 864*K1*K2**3*K3 - 1344*K1*K2**2*K3 - 352*K1*K2**2*K5 + 192*K1*K2*K3**3 - 416*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7944*K1*K2*K3 - 64*K1*K3**2*K5 + 1464*K1*K3*K4 + 176*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 1264*K2**4 + 32*K2**3*K3*K5 - 64*K2**3*K6 - 1184*K2**2*K3**2 - 32*K2**2*K3*K7 - 16*K2**2*K4**2 + 2016*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 4602*K2**2 + 1104*K2*K3*K5 + 80*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 128*K3**4 + 104*K3**2*K6 - 2556*K3**2 + 8*K3*K4*K7 - 830*K4**2 - 264*K5**2 - 54*K6**2 - 4*K7**2 - 2*K8**2 + 4702

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.170']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11489', 'vk6.11795', 'vk6.12811', 'vk6.13144', 'vk6.17079', 'vk6.17321', 'vk6.20896', 'vk6.21067', 'vk6.22306', 'vk6.22495', 'vk6.23800', 'vk6.28368', 'vk6.31250', 'vk6.31601', 'vk6.32821', 'vk6.35589', 'vk6.36045', 'vk6.40014', 'vk6.40312', 'vk6.42070', 'vk6.43281', 'vk6.46548', 'vk6.46769', 'vk6.48026', 'vk6.52252', 'vk6.53407', 'vk6.57708', 'vk6.57721', 'vk6.58892', 'vk6.59956', 'vk6.64421', 'vk6.69748']

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	O1O2O3O4O5U1O6U4U6U5U3U2
R3 orbit	{'O1O2O3O4O5U1O6U4U6U5U3U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U3U1U6U2O6U5
Gauss code of K*	O1O2O3O4O5U6U5U4U1U3O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U3U5U2U1U6
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 1 1 -1 2 1],[ 4 0 4 3 1 2 1],[-1 -4 0 0 -2 1 1],[-1 -3 0 0 -2 1 1],[ 1 -1 2 2 0 2 1],[-2 -2 -1 -1 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 2 1 1 1 -1 -4],[-2 0 0 -1 -1 -2 -2],[-1 0 0 -1 -1 -1 -1],[-1 1 1 0 0 -2 -3],[-1 1 1 0 0 -2 -4],[ 1 2 1 2 2 0 -1],[ 4 2 1 3 4 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,-1,1,4,0,1,1,2,2,1,1,1,1,0,2,3,2,4,1]
Phi over symmetry	[-4,-1,1,1,1,2,1,1,3,4,2,1,2,2,2,-1,-1,0,0,1,1]
Phi of -K	[-4,-1,1,1,1,2,2,1,2,4,4,0,0,1,1,0,-1,0,-1,0,1]
Phi of K*	[-2,-1,-1,-1,1,4,0,0,1,1,4,0,1,0,1,1,0,2,1,4,2]
Phi of -K*	[-4,-1,1,1,1,2,1,1,3,4,2,1,2,2,2,-1,-1,0,0,1,1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^2-2t
Normalized Jones-Krushkal polynomial	7z^2+28z+29
Enhanced Jones-Krushkal polynomial	7w^3z^2+28w^2z+29w
Inner characteristic polynomial	t^6+48t^4+21t^2+1
Outer characteristic polynomial	t^7+72t^5+80t^3+11t
Flat arrow polynomial	4K13 - 4K1K2 - 2K1*K3 - K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial	384K14K2 - 1904K14 + 800K1*3K2K3 + 32K1*3K3K4 - 800K1*3K3 + 480K12K2*3 + 96K1*2K2*2K4 - 5024K12K2*2 - 864K1*2K2K4 + 7728K1*2K2 - 912K12K3*2 - 64K1*2K4*2 - 5480K1*2 + 864K1K23K3 - 1344K1K2*2K3 - 352K1K2*2K5 + 192K1K2K33 - 416K1K2K3K4 - 32K1K2K3K6 + 7944K1K2K3 - 64K1K3*2K5 + 1464K1K3K4 + 176K1K4K5 + 16K1K5K6 - 32K2*6 + 64K2*4K4 - 1264K24 + 32K2*3K3K5 - 64K2*3K6 - 1184K22K3*2 - 32K2*2K3K7 - 16K2*2K4*2 + 2016K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 4602K2*2 + 1104K2K3K5 + 80K2K4K6 + 16K2K5K7 + 8K2K6K8 - 128K3*4 + 104K3*2K6 - 2556K32 + 8K3K4K7 - 830K42 - 264K5*2 - 54K6*2 - 4K7*2 - 2K8**2 + 4702
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}, {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}]]
If K is slice	False

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