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Flat knot 6.169

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,1,3,1,2,4,2,1,2,2,1,1,1,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.169']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 6K12 - 4K1K2 - 2K1K3 - 4K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.169', '6.359']
Outer characteristic polynomial of the knot is: t^7+75t^5+116t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.169']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 448K1*4K2 - 704K14 + 288K1*3K2K3 - 160K1*3K3 - 640K12K2*4 + 1440K1*2K2*3 - 128K1*2K2*2K3*2 + 128K1*2K2*2K4 - 5520K12K2*2 + 64K1*2K2K32 + 32K1*2K2K3K5 - 192K12K2K4 + 5016K1*2K2 - 160K12K3*2 - 3380K1*2 + 1856K1K23K3 + 128K1K2*2K3K4 - 896K1K22K3 - 160K1K2*2K5 - 352K1K2K3K4 + 5288K1K2K3 + 224K1K3K4 + 8K1K4K5 - 576K2*6 - 320K2*4K3*2 - 32K2*4K4*2 + 640K2*4K4 - 2968K24 + 192K2*3K3K5 + 32K2*3K4K6 - 160K2*3K6 + 64K22K3*2K4 - 1680K22K3*2 - 32K2*2K3K7 - 112K2*2K4*2 + 2048K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 1388K2*2 + 904K2K3K5 + 40K2K4K6 + 8K3*2K6 - 1376K32 - 312K4*2 - 116K5*2 - 12K6**2 + 2806
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.169']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11452', 'vk6.11749', 'vk6.12765', 'vk6.13107', 'vk6.20676', 'vk6.22116', 'vk6.28181', 'vk6.29606', 'vk6.31206', 'vk6.31545', 'vk6.32372', 'vk6.32785', 'vk6.39629', 'vk6.41870', 'vk6.46233', 'vk6.47840', 'vk6.52214', 'vk6.52487', 'vk6.53047', 'vk6.53367', 'vk6.57607', 'vk6.58768', 'vk6.62267', 'vk6.63210', 'vk6.63781', 'vk6.63894', 'vk6.64209', 'vk6.64393', 'vk6.67065', 'vk6.67932', 'vk6.69681', 'vk6.70364']
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,-1,0,1,2,2,1,3,1,2,4,2,1,2,2,1,1,1,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.169']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.169', '6.359']

Outer characteristic polynomial of the knot is: t^7+75t^5+116t^3+16t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.169']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 704*K1**4 + 288*K1**3*K2*K3 - 160*K1**3*K3 - 640*K1**2*K2**4 + 1440*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 5520*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 192*K1**2*K2*K4 + 5016*K1**2*K2 - 160*K1**2*K3**2 - 3380*K1**2 + 1856*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 160*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 5288*K1*K2*K3 + 224*K1*K3*K4 + 8*K1*K4*K5 - 576*K2**6 - 320*K2**4*K3**2 - 32*K2**4*K4**2 + 640*K2**4*K4 - 2968*K2**4 + 192*K2**3*K3*K5 + 32*K2**3*K4*K6 - 160*K2**3*K6 + 64*K2**2*K3**2*K4 - 1680*K2**2*K3**2 - 32*K2**2*K3*K7 - 112*K2**2*K4**2 + 2048*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 1388*K2**2 + 904*K2*K3*K5 + 40*K2*K4*K6 + 8*K3**2*K6 - 1376*K3**2 - 312*K4**2 - 116*K5**2 - 12*K6**2 + 2806

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.169']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11452', 'vk6.11749', 'vk6.12765', 'vk6.13107', 'vk6.20676', 'vk6.22116', 'vk6.28181', 'vk6.29606', 'vk6.31206', 'vk6.31545', 'vk6.32372', 'vk6.32785', 'vk6.39629', 'vk6.41870', 'vk6.46233', 'vk6.47840', 'vk6.52214', 'vk6.52487', 'vk6.53047', 'vk6.53367', 'vk6.57607', 'vk6.58768', 'vk6.62267', 'vk6.63210', 'vk6.63781', 'vk6.63894', 'vk6.64209', 'vk6.64393', 'vk6.67065', 'vk6.67932', 'vk6.69681', 'vk6.70364']

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	O1O2O3O4O5U1O6U4U6U5U2U3
R3 orbit	{'O1O2O3O4O5U1O6U4U6U5U2U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U4U1U6U2O6U5
Gauss code of K*	O1O2O3O4O5U6U4U5U1U3O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U3U5U1U2U6
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 0 2 -1 2 1],[ 4 0 3 4 1 2 1],[ 0 -3 0 1 -2 1 1],[-2 -4 -1 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 2 1 0 -1 -4],[-2 0 1 1 -1 -2 -4],[-2 -1 0 0 -1 -2 -2],[-1 -1 0 0 -1 -1 -1],[ 0 1 1 1 0 -2 -3],[ 1 2 2 1 2 0 -1],[ 4 4 2 1 3 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,1,4,-1,-1,1,2,4,0,1,2,2,1,1,1,2,3,1]
Phi over symmetry	[-4,-1,0,1,2,2,1,3,1,2,4,2,1,2,2,1,1,1,0,-1,-1]
Phi of -K	[-4,-1,0,1,2,2,2,1,4,2,4,-1,1,1,1,0,1,1,2,1,-1]
Phi of K*	[-2,-2,-1,0,1,4,-1,1,1,1,4,2,1,1,2,0,1,4,-1,1,2]
Phi of -K*	[-4,-1,0,1,2,2,1,3,1,2,4,2,1,2,2,1,1,1,0,-1,-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+49t^4+30t^2+1
Outer characteristic polynomial	t^7+75t^5+116t^3+16t
Flat arrow polynomial	8K13 + 4K1*2K2 - 6K12 - 4K1K2 - 2K1K3 - 4K1 + 2*K2 + 3
2-strand cable arrow polynomial	-256K14K2*2 + 448K1*4K2 - 704K14 + 288K1*3K2K3 - 160K1*3K3 - 640K12K2*4 + 1440K1*2K2*3 - 128K1*2K2*2K3*2 + 128K1*2K2*2K4 - 5520K12K2*2 + 64K1*2K2K32 + 32K1*2K2K3K5 - 192K12K2K4 + 5016K1*2K2 - 160K12K3*2 - 3380K1*2 + 1856K1K23K3 + 128K1K2*2K3K4 - 896K1K22K3 - 160K1K2*2K5 - 352K1K2K3K4 + 5288K1K2K3 + 224K1K3K4 + 8K1K4K5 - 576K2*6 - 320K2*4K3*2 - 32K2*4K4*2 + 640K2*4K4 - 2968K24 + 192K2*3K3K5 + 32K2*3K4K6 - 160K2*3K6 + 64K22K3*2K4 - 1680K22K3*2 - 32K2*2K3K7 - 112K2*2K4*2 + 2048K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 1388K2*2 + 904K2K3K5 + 40K2K4K6 + 8K3*2K6 - 1376K32 - 312K4*2 - 116K5*2 - 12K6**2 + 2806
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}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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