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Flat knot 6.487

Min(phi) over symmetries of the knot is: [-4,-1,0,0,2,3,1,1,3,4,3,0,1,1,1,0,0,1,1,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.487']
Arrow polynomial of the knot is: 4K13 - 8K1*2 - 6K1K2 - 2K1K3 + 5K2 + 2*K3 + K4 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.487']
Outer characteristic polynomial of the knot is: t^7+89t^5+44t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.487']
2-strand cable arrow polynomial of the knot is: 160K14K2 - 1632K14 + 160K1*3K2K3 + 32K1*3K3K4 - 1504K1*3K3 + 256K12K2*3 - 2864K1*2K2*2 + 256K1*2K2K32 - 448K1*2K2K4 + 8792K1*2K2 - 1584K12K3*2 - 192K1*2K3K5 - 128K1*2K4*2 - 128K1*2K4K6 - 64K1*2K6*2 - 8332K1*2 + 160K1K23K3 - 1024K1K2*2K3 - 128K1K2*2K5 + 32K1K2K33 - 352K1K2K3K4 + 8864K1K2K3 - 32K1K2K4K5 - 32K1K3*2K5 - 32K1K3K4K6 + 2496K1K3K4 + 592K1K4K5 + 264K1K5K6 + 80K1K6K7 - 32K26 + 96K2*4K4 - 576K24 - 32K2*3K6 - 480K22K3*2 - 88K2*2K4*2 + 1264K2*2K4 - 8K22K6*2 - 5756K2*2 + 656K2K3K5 + 248K2K4K6 + 8K2K6K8 - 48K34 + 64K3*2K6 - 3452K32 + 16K3K4K7 - 1190K42 - 400K5*2 - 212K6*2 - 24K7*2 - 2K8**2 + 6278
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.487']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73297', 'vk6.73307', 'vk6.73440', 'vk6.73450', 'vk6.74074', 'vk6.74087', 'vk6.74645', 'vk6.74656', 'vk6.75440', 'vk6.75454', 'vk6.76112', 'vk6.76125', 'vk6.78170', 'vk6.78188', 'vk6.78402', 'vk6.78420', 'vk6.79084', 'vk6.79089', 'vk6.79991', 'vk6.80013', 'vk6.80144', 'vk6.80166', 'vk6.80592', 'vk6.80597', 'vk6.83805', 'vk6.83824', 'vk6.85108', 'vk6.85129', 'vk6.86598', 'vk6.86603', 'vk6.87379', 'vk6.87389']
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,-1,0,0,2,3,1,1,3,4,3,0,1,1,1,0,0,1,1,2,2]

Flat knots (up to 7 crossings) with same phi are :['6.487']

Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 6*K1*K2 - 2*K1*K3 + 5*K2 + 2*K3 + K4 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.487']

Outer characteristic polynomial of the knot is: t^7+89t^5+44t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.487']

2-strand cable arrow polynomial of the knot is: 160*K1**4*K2 - 1632*K1**4 + 160*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1504*K1**3*K3 + 256*K1**2*K2**3 - 2864*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 448*K1**2*K2*K4 + 8792*K1**2*K2 - 1584*K1**2*K3**2 - 192*K1**2*K3*K5 - 128*K1**2*K4**2 - 128*K1**2*K4*K6 - 64*K1**2*K6**2 - 8332*K1**2 + 160*K1*K2**3*K3 - 1024*K1*K2**2*K3 - 128*K1*K2**2*K5 + 32*K1*K2*K3**3 - 352*K1*K2*K3*K4 + 8864*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2496*K1*K3*K4 + 592*K1*K4*K5 + 264*K1*K5*K6 + 80*K1*K6*K7 - 32*K2**6 + 96*K2**4*K4 - 576*K2**4 - 32*K2**3*K6 - 480*K2**2*K3**2 - 88*K2**2*K4**2 + 1264*K2**2*K4 - 8*K2**2*K6**2 - 5756*K2**2 + 656*K2*K3*K5 + 248*K2*K4*K6 + 8*K2*K6*K8 - 48*K3**4 + 64*K3**2*K6 - 3452*K3**2 + 16*K3*K4*K7 - 1190*K4**2 - 400*K5**2 - 212*K6**2 - 24*K7**2 - 2*K8**2 + 6278

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.487']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73297', 'vk6.73307', 'vk6.73440', 'vk6.73450', 'vk6.74074', 'vk6.74087', 'vk6.74645', 'vk6.74656', 'vk6.75440', 'vk6.75454', 'vk6.76112', 'vk6.76125', 'vk6.78170', 'vk6.78188', 'vk6.78402', 'vk6.78420', 'vk6.79084', 'vk6.79089', 'vk6.79991', 'vk6.80013', 'vk6.80144', 'vk6.80166', 'vk6.80592', 'vk6.80597', 'vk6.83805', 'vk6.83824', 'vk6.85108', 'vk6.85129', 'vk6.86598', 'vk6.86603', 'vk6.87379', 'vk6.87389']

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	O1O2O3O4O5U1U4U2O6U3U5U6
R3 orbit	{'O1O2O3O4O5U1U4U2O6U3U5U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U1U3O6U4U2U5
Gauss code of K*	O1O2O3U4O5O6O4U1U3U5U2U6
Gauss code of -K*	O1O2O3U1O4O5O6U2U5U3U4U6
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 0 0 3 2],[ 4 0 2 3 1 4 2],[ 1 -2 0 1 0 3 2],[ 0 -3 -1 0 0 2 2],[ 0 -1 0 0 0 1 1],[-3 -4 -3 -2 -1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix	[[ 0 3 2 0 0 -1 -4],[-3 0 1 -1 -2 -3 -4],[-2 -1 0 -1 -2 -2 -2],[ 0 1 1 0 0 0 -1],[ 0 2 2 0 0 -1 -3],[ 1 3 2 0 1 0 -2],[ 4 4 2 1 3 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,0,0,1,4,-1,1,2,3,4,1,2,2,2,0,0,1,1,3,2]
Phi over symmetry	[-4,-1,0,0,2,3,1,1,3,4,3,0,1,1,1,0,0,1,1,2,2]
Phi of -K	[-4,-1,0,0,2,3,1,1,3,4,3,0,1,1,1,0,0,1,1,2,2]
Phi of K*	[-3,-2,0,0,1,4,2,1,2,1,3,0,1,1,4,0,0,1,1,3,1]
Phi of -K*	[-4,-1,0,0,2,3,2,1,3,2,4,0,1,2,3,0,1,1,2,2,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial	z^2+20z+37
Enhanced Jones-Krushkal polynomial	w^3z^2+20w^2z+37w
Inner characteristic polynomial	t^6+59t^4+11t^2
Outer characteristic polynomial	t^7+89t^5+44t^3+5t
Flat arrow polynomial	4K13 - 8K1*2 - 6K1K2 - 2K1K3 + 5K2 + 2*K3 + K4 + 5
2-strand cable arrow polynomial	160K14K2 - 1632K14 + 160K1*3K2K3 + 32K1*3K3K4 - 1504K1*3K3 + 256K12K2*3 - 2864K1*2K2*2 + 256K1*2K2K32 - 448K1*2K2K4 + 8792K1*2K2 - 1584K12K3*2 - 192K1*2K3K5 - 128K1*2K4*2 - 128K1*2K4K6 - 64K1*2K6*2 - 8332K1*2 + 160K1K23K3 - 1024K1K2*2K3 - 128K1K2*2K5 + 32K1K2K33 - 352K1K2K3K4 + 8864K1K2K3 - 32K1K2K4K5 - 32K1K3*2K5 - 32K1K3K4K6 + 2496K1K3K4 + 592K1K4K5 + 264K1K5K6 + 80K1K6K7 - 32K26 + 96K2*4K4 - 576K24 - 32K2*3K6 - 480K22K3*2 - 88K2*2K4*2 + 1264K2*2K4 - 8K22K6*2 - 5756K2*2 + 656K2K3K5 + 248K2K4K6 + 8K2K6K8 - 48K34 + 64K3*2K6 - 3452K32 + 16K3K4K7 - 1190K42 - 400K5*2 - 212K6*2 - 24K7*2 - 2K8**2 + 6278
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}], [{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}, {4, 5}, {3}, {2}, {1}], [{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}, {4}, {2, 3}, {1}]]
If K is slice	False

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