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Flat knot 6.493

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,0,3,4,2,4,2,1,1,2,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.493']
Arrow polynomial of the knot is: 12K13 + 4K1*2K2 - 6K12 - 8K1K2 - 2K1K3 - 5K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.493']
Outer characteristic polynomial of the knot is: t^7+67t^5+120t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.493']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 320K1*4K2 - 736K14 + 160K1*3K2K3 - 64K1*3K3 - 768K12K2*4 + 2016K1*2K2*3 + 128K1*2K2*2K4 - 5984K12K2*2 - 256K1*2K2K4 + 5544K1*2K2 - 64K12K3*2 - 3628K1*2 + 1632K1K23K3 + 160K1K2*2K3K4 - 1440K1K22K3 + 64K1K2*2K4K5 - 288K1K22K5 - 256K1K2K3K4 + 5592K1K2K3 - 32K1K2K4K5 + 384K1K3K4 + 64K1K4K5 + 16K1K5K6 - 736K26 - 320K2*4K3*2 - 32K2*4K4*2 + 1184K2*4K4 - 3992K24 + 320K2*3K3K5 + 32K2*3K4K6 - 352K2*3K6 + 64K22K3*2K4 - 1728K22K3*2 - 64K2*2K3K7 - 416K2*2K4*2 + 3184K2*2K4 - 112K22K5*2 - 8K2*2K6*2 - 1578K2*2 + 1040K2K3K5 + 168K2K4K6 + 16K2K5K7 - 1420K32 - 616K4*2 - 168K5*2 - 22K6**2 + 3182
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.493']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10136', 'vk6.10193', 'vk6.10338', 'vk6.10425', 'vk6.17661', 'vk6.17708', 'vk6.24228', 'vk6.24275', 'vk6.29919', 'vk6.29966', 'vk6.30031', 'vk6.30080', 'vk6.36492', 'vk6.36586', 'vk6.43589', 'vk6.43699', 'vk6.51618', 'vk6.51653', 'vk6.51700', 'vk6.51723', 'vk6.55699', 'vk6.55756', 'vk6.60269', 'vk6.60331', 'vk6.63333', 'vk6.63360', 'vk6.63383', 'vk6.63402', 'vk6.65407', 'vk6.65448', 'vk6.68549', 'vk6.68580']
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,-1,0,1,2,2,0,3,4,2,4,2,1,1,2,1,1,1,1,1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.493']

Arrow polynomial of the knot is: 12*K1**3 + 4*K1**2*K2 - 6*K1**2 - 8*K1*K2 - 2*K1*K3 - 5*K1 + 2*K2 + K3 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.493']

Outer characteristic polynomial of the knot is: t^7+67t^5+120t^3+16t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.493']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 320*K1**4*K2 - 736*K1**4 + 160*K1**3*K2*K3 - 64*K1**3*K3 - 768*K1**2*K2**4 + 2016*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 5984*K1**2*K2**2 - 256*K1**2*K2*K4 + 5544*K1**2*K2 - 64*K1**2*K3**2 - 3628*K1**2 + 1632*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1440*K1*K2**2*K3 + 64*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 - 256*K1*K2*K3*K4 + 5592*K1*K2*K3 - 32*K1*K2*K4*K5 + 384*K1*K3*K4 + 64*K1*K4*K5 + 16*K1*K5*K6 - 736*K2**6 - 320*K2**4*K3**2 - 32*K2**4*K4**2 + 1184*K2**4*K4 - 3992*K2**4 + 320*K2**3*K3*K5 + 32*K2**3*K4*K6 - 352*K2**3*K6 + 64*K2**2*K3**2*K4 - 1728*K2**2*K3**2 - 64*K2**2*K3*K7 - 416*K2**2*K4**2 + 3184*K2**2*K4 - 112*K2**2*K5**2 - 8*K2**2*K6**2 - 1578*K2**2 + 1040*K2*K3*K5 + 168*K2*K4*K6 + 16*K2*K5*K7 - 1420*K3**2 - 616*K4**2 - 168*K5**2 - 22*K6**2 + 3182

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.493']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10136', 'vk6.10193', 'vk6.10338', 'vk6.10425', 'vk6.17661', 'vk6.17708', 'vk6.24228', 'vk6.24275', 'vk6.29919', 'vk6.29966', 'vk6.30031', 'vk6.30080', 'vk6.36492', 'vk6.36586', 'vk6.43589', 'vk6.43699', 'vk6.51618', 'vk6.51653', 'vk6.51700', 'vk6.51723', 'vk6.55699', 'vk6.55756', 'vk6.60269', 'vk6.60331', 'vk6.63333', 'vk6.63360', 'vk6.63383', 'vk6.63402', 'vk6.65407', 'vk6.65448', 'vk6.68549', 'vk6.68580']

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	O1O2O3O4O5U1U4U5O6U2U6U3
R3 orbit	{'O1O2O3O4O5U1U4U5O6U2U6U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U6U4O6U1U2U5
Gauss code of K*	O1O2O3U4O5O4O6U1U5U6U2U3
Gauss code of -K*	O1O2O3U2O4O5O6U4U5U1U3U6
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 2 0 2 1],[ 4 0 3 4 1 2 1],[ 1 -3 0 2 -1 1 1],[-2 -4 -2 0 -1 1 0],[ 0 -1 1 1 0 1 0],[-2 -2 -1 -1 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix	[[ 0 2 2 1 0 -1 -4],[-2 0 1 0 -1 -2 -4],[-2 -1 0 0 -1 -1 -2],[-1 0 0 0 0 -1 -1],[ 0 1 1 0 0 1 -1],[ 1 2 1 1 -1 0 -3],[ 4 4 2 1 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,1,4,-1,0,1,2,4,0,1,1,2,0,1,1,-1,1,3]
Phi over symmetry	[-4,-1,0,1,2,2,0,3,4,2,4,2,1,1,2,1,1,1,1,1,-1]
Phi of -K	[-4,-1,0,1,2,2,0,3,4,2,4,2,1,1,2,1,1,1,1,1,-1]
Phi of K*	[-2,-2,-1,0,1,4,-1,1,1,2,4,1,1,1,2,1,1,4,2,3,0]
Phi of -K*	[-4,-1,0,1,2,2,3,1,1,2,4,-1,1,1,2,0,1,1,0,0,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^2
Normalized Jones-Krushkal polynomial	3z^2+16z+21
Enhanced Jones-Krushkal polynomial	-2w^4z^2+5w^3z^2-8w^3z+24w^2z+21w
Inner characteristic polynomial	t^6+41t^4+38t^2+1
Outer characteristic polynomial	t^7+67t^5+120t^3+16t
Flat arrow polynomial	12K13 + 4K1*2K2 - 6K12 - 8K1K2 - 2K1K3 - 5K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial	-192K14K2*2 + 320K1*4K2 - 736K14 + 160K1*3K2K3 - 64K1*3K3 - 768K12K2*4 + 2016K1*2K2*3 + 128K1*2K2*2K4 - 5984K12K2*2 - 256K1*2K2K4 + 5544K1*2K2 - 64K12K3*2 - 3628K1*2 + 1632K1K23K3 + 160K1K2*2K3K4 - 1440K1K22K3 + 64K1K2*2K4K5 - 288K1K22K5 - 256K1K2K3K4 + 5592K1K2K3 - 32K1K2K4K5 + 384K1K3K4 + 64K1K4K5 + 16K1K5K6 - 736K26 - 320K2*4K3*2 - 32K2*4K4*2 + 1184K2*4K4 - 3992K24 + 320K2*3K3K5 + 32K2*3K4K6 - 352K2*3K6 + 64K22K3*2K4 - 1728K22K3*2 - 64K2*2K3K7 - 416K2*2K4*2 + 3184K2*2K4 - 112K22K5*2 - 8K2*2K6*2 - 1578K2*2 + 1040K2K3K5 + 168K2K4K6 + 16K2K5K7 - 1420K32 - 616K4*2 - 168K5*2 - 22K6**2 + 3182
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}, {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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