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Flat knot 6.427

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,3,1,2,4,1,1,1,2,1,1,3,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.427']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 10K12 - 2K1K2 - 2K1K3 - 2K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.290', '6.427']
Outer characteristic polynomial of the knot is: t^7+84t^5+70t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.427']
2-strand cable arrow polynomial of the knot is: -128K16 - 256K1*4K2*2 + 1536K1*4K2 - 3824K14 - 512K1*3K2*2K3 + 1728K13K2K3 - 1088K1*3K3 - 256K12K2*4 + 2496K1*2K2*3 - 640K1*2K2*2K3*2 + 320K1*2K2*2K4 - 10064K12K2*2 + 160K1*2K2K32 - 1344K1*2K2K4 + 10800K1*2K2 - 528K12K3*2 - 96K1*2K4*2 - 5104K1*2 - 256K1K23K3K4 + 2368K1K23K3 + 672K1K2*2K3K4 - 2016K1K22K3 + 96K1K2*2K4K5 - 320K1K22K5 + 32K1K2K33 - 512K1K2K3K4 + 8280K1K2K3 + 1072K1K3K4 + 184K1K4K5 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 256K2*4K4 - 2184K24 + 160K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 1312K22K3*2 - 360K2*2K4*2 + 1840K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 3324K2*2 - 32K2K32K4 + 440K2K3K5 + 72K2K4K6 - 1880K32 - 536K4*2 - 64K5*2 - 4K6**2 + 4310
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.427']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19933', 'vk6.19975', 'vk6.21170', 'vk6.21244', 'vk6.26886', 'vk6.26990', 'vk6.28644', 'vk6.28716', 'vk6.38313', 'vk6.38402', 'vk6.40448', 'vk6.40583', 'vk6.45187', 'vk6.45294', 'vk6.47017', 'vk6.47076', 'vk6.56726', 'vk6.56789', 'vk6.57824', 'vk6.57923', 'vk6.61147', 'vk6.61285', 'vk6.62392', 'vk6.62476', 'vk6.66423', 'vk6.66493', 'vk6.67195', 'vk6.67286', 'vk6.69078', 'vk6.69147', 'vk6.69861', 'vk6.69906']
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,-1,0,1,1,3,1,3,1,2,4,1,1,1,2,1,1,3,-1,-1,1]

Flat knots (up to 7 crossings) with same phi are :['6.427']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.290', '6.427']

Outer characteristic polynomial of the knot is: t^7+84t^5+70t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.427']

2-strand cable arrow polynomial of the knot is: -128*K1**6 - 256*K1**4*K2**2 + 1536*K1**4*K2 - 3824*K1**4 - 512*K1**3*K2**2*K3 + 1728*K1**3*K2*K3 - 1088*K1**3*K3 - 256*K1**2*K2**4 + 2496*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 + 320*K1**2*K2**2*K4 - 10064*K1**2*K2**2 + 160*K1**2*K2*K3**2 - 1344*K1**2*K2*K4 + 10800*K1**2*K2 - 528*K1**2*K3**2 - 96*K1**2*K4**2 - 5104*K1**2 - 256*K1*K2**3*K3*K4 + 2368*K1*K2**3*K3 + 672*K1*K2**2*K3*K4 - 2016*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 320*K1*K2**2*K5 + 32*K1*K2*K3**3 - 512*K1*K2*K3*K4 + 8280*K1*K2*K3 + 1072*K1*K3*K4 + 184*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 256*K2**4*K4 - 2184*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1312*K2**2*K3**2 - 360*K2**2*K4**2 + 1840*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 3324*K2**2 - 32*K2*K3**2*K4 + 440*K2*K3*K5 + 72*K2*K4*K6 - 1880*K3**2 - 536*K4**2 - 64*K5**2 - 4*K6**2 + 4310

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.427']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19933', 'vk6.19975', 'vk6.21170', 'vk6.21244', 'vk6.26886', 'vk6.26990', 'vk6.28644', 'vk6.28716', 'vk6.38313', 'vk6.38402', 'vk6.40448', 'vk6.40583', 'vk6.45187', 'vk6.45294', 'vk6.47017', 'vk6.47076', 'vk6.56726', 'vk6.56789', 'vk6.57824', 'vk6.57923', 'vk6.61147', 'vk6.61285', 'vk6.62392', 'vk6.62476', 'vk6.66423', 'vk6.66493', 'vk6.67195', 'vk6.67286', 'vk6.69078', 'vk6.69147', 'vk6.69861', 'vk6.69906']

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	O1O2O3O4O5U6U1O6U4U2U3U5
R3 orbit	{'O1O2O3O4O5U6U1O6U4U2U3U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U3U4U2O6U5U6
Gauss code of K*	O1O2O3O4U5U2U3U1U4O6O5U6
Gauss code of -K*	O1O2O3O4U5O6O5U1U4U2U3U6
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 -1 1 0 4 -1],[ 3 0 1 2 0 3 3],[ 1 -1 0 1 0 3 1],[-1 -2 -1 0 0 2 -1],[ 0 0 0 0 0 1 0],[-4 -3 -3 -2 -1 0 -4],[ 1 -3 -1 1 0 4 0]]
Primitive based matrix	[[ 0 4 1 0 -1 -1 -3],[-4 0 -2 -1 -3 -4 -3],[-1 2 0 0 -1 -1 -2],[ 0 1 0 0 0 0 0],[ 1 3 1 0 0 1 -1],[ 1 4 1 0 -1 0 -3],[ 3 3 2 0 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-1,0,1,1,3,2,1,3,4,3,0,1,1,2,0,0,0,-1,1,3]
Phi over symmetry	[-4,-1,0,1,1,3,1,3,1,2,4,1,1,1,2,1,1,3,-1,-1,1]
Phi of -K	[-3,-1,-1,0,1,4,-1,1,3,2,4,1,1,1,1,1,1,2,1,3,1]
Phi of K*	[-4,-1,0,1,1,3,1,3,1,2,4,1,1,1,2,1,1,3,-1,-1,1]
Phi of -K*	[-3,-1,-1,0,1,4,1,3,0,2,3,1,0,1,3,0,1,4,0,1,2]
Symmetry type of based matrix	c
u-polynomial	-t^4+t^3+t
Normalized Jones-Krushkal polynomial	6z^2+27z+31
Enhanced Jones-Krushkal polynomial	6w^3z^2+27w^2z+31w
Inner characteristic polynomial	t^6+56t^4+24t^2
Outer characteristic polynomial	t^7+84t^5+70t^3+7t
Flat arrow polynomial	4K13 + 4K1*2K2 - 10K12 - 2K1K2 - 2K1K3 - 2K1 + 4*K2 + 5
2-strand cable arrow polynomial	-128K16 - 256K1*4K2*2 + 1536K1*4K2 - 3824K14 - 512K1*3K2*2K3 + 1728K13K2K3 - 1088K1*3K3 - 256K12K2*4 + 2496K1*2K2*3 - 640K1*2K2*2K3*2 + 320K1*2K2*2K4 - 10064K12K2*2 + 160K1*2K2K32 - 1344K1*2K2K4 + 10800K1*2K2 - 528K12K3*2 - 96K1*2K4*2 - 5104K1*2 - 256K1K23K3K4 + 2368K1K23K3 + 672K1K2*2K3K4 - 2016K1K22K3 + 96K1K2*2K4K5 - 320K1K22K5 + 32K1K2K33 - 512K1K2K3K4 + 8280K1K2K3 + 1072K1K3K4 + 184K1K4K5 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 256K2*4K4 - 2184K24 + 160K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 1312K22K3*2 - 360K2*2K4*2 + 1840K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 3324K2*2 - 32K2K32K4 + 440K2K3K5 + 72K2K4K6 - 1880K32 - 536K4*2 - 64K5*2 - 4K6**2 + 4310
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}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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