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Flat knot 6.852

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,2,3,1,0,0,1,1,1,1,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.852']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 6K12 - 2K1K2 - 4K1K3 - 2K1 + 3*K2 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.119', '6.852']
Outer characteristic polynomial of the knot is: t^7+44t^5+71t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.852']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 448K1*4K2 - 1584K14 + 960K1*3K2K3 - 576K1*3K3 - 256K12K2*4 + 576K1*2K2*3 - 128K1*2K2*2K3*2 + 64K1*2K2*2K4 - 4160K12K2*2 + 64K1*2K2K32 + 32K1*2K2K3K5 - 320K12K2K4 + 5256K1*2K2 - 1392K12K3*2 - 4072K1*2 + 992K1K23K3 + 128K1K2*2K3K4 - 768K1K22K3 - 384K1K2*2K5 + 288K1K2K33 - 192K1K2K3K4 - 128K1K2K3K6 + 6936K1K2K3 - 128K1K2K4K5 + 1640K1K3K4 + 112K1K4K5 + 56K1K5K6 - 32K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 96K2*4K4 - 808K24 + 96K2*3K3K5 + 64K2*3K4K6 - 32K2*3K6 - 1296K22K3*2 - 32K2*2K3K7 - 168K2*2K4*2 - 32K2*2K4K8 + 952K2*2K4 - 16K22K5*2 - 16K2*2K6*2 - 3328K2*2 - 256K2K32K4 + 1048K2K3K5 + 240K2K4K6 + 8K2K5K7 + 16K2K6K8 - 192K34 + 232K3*2K6 - 2492K32 - 650K4*2 - 244K5*2 - 96K6*2 - 2K8**2 + 3826
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.852']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4419', 'vk6.4516', 'vk6.5805', 'vk6.5934', 'vk6.7864', 'vk6.7971', 'vk6.9284', 'vk6.9405', 'vk6.10175', 'vk6.10246', 'vk6.10389', 'vk6.17883', 'vk6.17946', 'vk6.18286', 'vk6.18621', 'vk6.24386', 'vk6.25172', 'vk6.30058', 'vk6.30119', 'vk6.36904', 'vk6.37362', 'vk6.43813', 'vk6.44121', 'vk6.44444', 'vk6.48613', 'vk6.50512', 'vk6.50592', 'vk6.51116', 'vk6.51674', 'vk6.55834', 'vk6.56084', 'vk6.65498']
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: [-3,-1,0,1,1,2,0,1,2,3,1,0,0,1,1,1,1,1,0,2,2]

Flat knots (up to 7 crossings) with same phi are :['6.852']

Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 3*K2 + K4 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.119', '6.852']

Outer characteristic polynomial of the knot is: t^7+44t^5+71t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.852']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 1584*K1**4 + 960*K1**3*K2*K3 - 576*K1**3*K3 - 256*K1**2*K2**4 + 576*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 4160*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 320*K1**2*K2*K4 + 5256*K1**2*K2 - 1392*K1**2*K3**2 - 4072*K1**2 + 992*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 - 384*K1*K2**2*K5 + 288*K1*K2*K3**3 - 192*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 6936*K1*K2*K3 - 128*K1*K2*K4*K5 + 1640*K1*K3*K4 + 112*K1*K4*K5 + 56*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 808*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 32*K2**3*K6 - 1296*K2**2*K3**2 - 32*K2**2*K3*K7 - 168*K2**2*K4**2 - 32*K2**2*K4*K8 + 952*K2**2*K4 - 16*K2**2*K5**2 - 16*K2**2*K6**2 - 3328*K2**2 - 256*K2*K3**2*K4 + 1048*K2*K3*K5 + 240*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 192*K3**4 + 232*K3**2*K6 - 2492*K3**2 - 650*K4**2 - 244*K5**2 - 96*K6**2 - 2*K8**2 + 3826

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.852']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4419', 'vk6.4516', 'vk6.5805', 'vk6.5934', 'vk6.7864', 'vk6.7971', 'vk6.9284', 'vk6.9405', 'vk6.10175', 'vk6.10246', 'vk6.10389', 'vk6.17883', 'vk6.17946', 'vk6.18286', 'vk6.18621', 'vk6.24386', 'vk6.25172', 'vk6.30058', 'vk6.30119', 'vk6.36904', 'vk6.37362', 'vk6.43813', 'vk6.44121', 'vk6.44444', 'vk6.48613', 'vk6.50512', 'vk6.50592', 'vk6.51116', 'vk6.51674', 'vk6.55834', 'vk6.56084', 'vk6.65498']

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	O1O2O3O4U1U3O5O6U5U4U6U2
R3 orbit	{'O1O2O3O4U1U3O5O6U5U4U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U5U1U6O5O6U2U4
Gauss code of K*	O1O2O3O4U5U4U6U2O5O6U1U3
Gauss code of -K*	O1O2O3O4U2U4O5O6U3U5U1U6
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 0 1 -1 2],[ 3 0 3 1 2 0 1],[-1 -3 0 -1 0 -1 2],[ 0 -1 1 0 1 0 1],[-1 -2 0 -1 0 0 2],[ 1 0 1 0 0 0 1],[-2 -1 -2 -1 -2 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 -2 -2 -1 -1 -1],[-1 2 0 0 -1 0 -2],[-1 2 0 0 -1 -1 -3],[ 0 1 1 1 0 0 -1],[ 1 1 0 1 0 0 0],[ 3 1 2 3 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,2,2,1,1,1,0,1,0,2,1,1,3,0,1,0]
Phi over symmetry	[-3,-1,0,1,1,2,0,1,2,3,1,0,0,1,1,1,1,1,0,2,2]
Phi of -K	[-3,-1,0,1,1,2,2,2,1,2,4,1,1,2,2,0,0,1,0,-1,-1]
Phi of K*	[-2,-1,-1,0,1,3,-1,-1,1,2,4,0,0,1,1,0,2,2,1,2,2]
Phi of -K*	[-3,-1,0,1,1,2,0,1,2,3,1,0,0,1,1,1,1,1,0,2,2]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	2z^2+21z+35
Enhanced Jones-Krushkal polynomial	2w^3z^2+21w^2z+35w
Inner characteristic polynomial	t^6+28t^4+26t^2
Outer characteristic polynomial	t^7+44t^5+71t^3+5t
Flat arrow polynomial	4K13 + 4K1*2K2 - 6K12 - 2K1K2 - 4K1K3 - 2K1 + 3*K2 + K4 + 3
2-strand cable arrow polynomial	-256K14K2*2 + 448K1*4K2 - 1584K14 + 960K1*3K2K3 - 576K1*3K3 - 256K12K2*4 + 576K1*2K2*3 - 128K1*2K2*2K3*2 + 64K1*2K2*2K4 - 4160K12K2*2 + 64K1*2K2K32 + 32K1*2K2K3K5 - 320K12K2K4 + 5256K1*2K2 - 1392K12K3*2 - 4072K1*2 + 992K1K23K3 + 128K1K2*2K3K4 - 768K1K22K3 - 384K1K2*2K5 + 288K1K2K33 - 192K1K2K3K4 - 128K1K2K3K6 + 6936K1K2K3 - 128K1K2K4K5 + 1640K1K3K4 + 112K1K4K5 + 56K1K5K6 - 32K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 96K2*4K4 - 808K24 + 96K2*3K3K5 + 64K2*3K4K6 - 32K2*3K6 - 1296K22K3*2 - 32K2*2K3K7 - 168K2*2K4*2 - 32K2*2K4K8 + 952K2*2K4 - 16K22K5*2 - 16K2*2K6*2 - 3328K2*2 - 256K2K32K4 + 1048K2K3K5 + 240K2K4K6 + 8K2K5K7 + 16K2K6K8 - 192K34 + 232K3*2K6 - 2492K32 - 650K4*2 - 244K5*2 - 96K6*2 - 2K8**2 + 3826
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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}, {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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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