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Flat knot 6.272

Min(phi) over symmetries of the knot is: [-4,-1,0,0,2,3,1,0,3,3,4,0,1,1,2,1,0,0,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.272']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 12K12 - 6K1K2 - 2K1K3 - 3K1 + 5*K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.272', '6.502']
Outer characteristic polynomial of the knot is: t^7+95t^5+60t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.272']
2-strand cable arrow polynomial of the knot is: 192K14K2 - 2112K14 + 32K1*3K2K3 - 320K1*3K3 - 128K12K2*4 + 576K1*2K2*3 - 5440K1*2K2*2 + 32K1*2K2K32 - 480K1*2K2K4 + 9328K1*2K2 - 128K12K3*2 - 16K1*2K4*2 - 6928K1*2 + 1024K1K23K3 + 128K1K2*2K3K4 - 1600K1K22K3 + 32K1K2*2K4K5 - 384K1K22K5 - 224K1K2K3K4 + 8400K1K2K3 + 1336K1K3K4 + 192K1K4K5 + 8K1K5K6 - 64K26 - 128K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 1824K24 + 96K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 - 1184K22K3*2 - 296K2*2K4*2 + 2528K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 5358K2*2 - 64K2K32K4 - 32K2K3K4K5 + 936K2K3K5 + 176K2K4K6 + 32K2K5K7 + 16K3*2K6 - 2884K32 - 1030K4*2 - 244K5*2 - 34K6**2 + 5780
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.272']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73353', 'vk6.73386', 'vk6.73514', 'vk6.73565', 'vk6.73719', 'vk6.73836', 'vk6.74252', 'vk6.74879', 'vk6.75324', 'vk6.75521', 'vk6.75838', 'vk6.76425', 'vk6.78243', 'vk6.78304', 'vk6.78490', 'vk6.78635', 'vk6.78828', 'vk6.79300', 'vk6.80064', 'vk6.80088', 'vk6.80212', 'vk6.80265', 'vk6.80397', 'vk6.80761', 'vk6.81944', 'vk6.82675', 'vk6.84739', 'vk6.85039', 'vk6.85139', 'vk6.86528', 'vk6.87336', 'vk6.89436']
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,-1,0,0,2,3,1,0,3,3,4,0,1,1,2,1,0,0,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.272']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.272', '6.502']

Outer characteristic polynomial of the knot is: t^7+95t^5+60t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.272']

2-strand cable arrow polynomial of the knot is: 192*K1**4*K2 - 2112*K1**4 + 32*K1**3*K2*K3 - 320*K1**3*K3 - 128*K1**2*K2**4 + 576*K1**2*K2**3 - 5440*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 480*K1**2*K2*K4 + 9328*K1**2*K2 - 128*K1**2*K3**2 - 16*K1**2*K4**2 - 6928*K1**2 + 1024*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 1600*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 8400*K1*K2*K3 + 1336*K1*K3*K4 + 192*K1*K4*K5 + 8*K1*K5*K6 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1824*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1184*K2**2*K3**2 - 296*K2**2*K4**2 + 2528*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 5358*K2**2 - 64*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 936*K2*K3*K5 + 176*K2*K4*K6 + 32*K2*K5*K7 + 16*K3**2*K6 - 2884*K3**2 - 1030*K4**2 - 244*K5**2 - 34*K6**2 + 5780

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.272']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73353', 'vk6.73386', 'vk6.73514', 'vk6.73565', 'vk6.73719', 'vk6.73836', 'vk6.74252', 'vk6.74879', 'vk6.75324', 'vk6.75521', 'vk6.75838', 'vk6.76425', 'vk6.78243', 'vk6.78304', 'vk6.78490', 'vk6.78635', 'vk6.78828', 'vk6.79300', 'vk6.80064', 'vk6.80088', 'vk6.80212', 'vk6.80265', 'vk6.80397', 'vk6.80761', 'vk6.81944', 'vk6.82675', 'vk6.84739', 'vk6.85039', 'vk6.85139', 'vk6.86528', 'vk6.87336', 'vk6.89436']

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	O1O2O3O4O5U1U4O6U3U5U2U6
R3 orbit	{'O1O2O3O4O5U1U4O6U3U5U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U4U1U3O6U2U5
Gauss code of K*	O1O2O3O4U5U3U1U6U2O5O6U4
Gauss code of -K*	O1O2O3O4U1O5O6U3U5U4U2U6
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 -1 0 2 3],[ 4 0 4 2 1 3 3],[ 0 -4 0 -1 -1 2 3],[ 1 -2 1 0 0 2 2],[ 0 -1 1 0 0 1 1],[-2 -3 -2 -2 -1 0 1],[-3 -3 -3 -2 -1 -1 0]]
Primitive based matrix	[[ 0 3 2 0 0 -1 -4],[-3 0 -1 -1 -3 -2 -3],[-2 1 0 -1 -2 -2 -3],[ 0 1 1 0 1 0 -1],[ 0 3 2 -1 0 -1 -4],[ 1 2 2 0 1 0 -2],[ 4 3 3 1 4 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,0,0,1,4,1,1,3,2,3,1,2,2,3,-1,0,1,1,4,2]
Phi over symmetry	[-4,-1,0,0,2,3,1,0,3,3,4,0,1,1,2,1,0,0,1,2,0]
Phi of -K	[-4,-1,0,0,2,3,1,0,3,3,4,0,1,1,2,1,0,0,1,2,0]
Phi of K*	[-3,-2,0,0,1,4,0,0,2,2,4,0,1,1,3,-1,0,0,1,3,1]
Phi of -K*	[-4,-1,0,0,2,3,2,1,4,3,3,0,1,2,2,1,1,1,2,3,1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial	4z^2+23z+31
Enhanced Jones-Krushkal polynomial	4w^3z^2+23w^2z+31w
Inner characteristic polynomial	t^6+65t^4+17t^2+1
Outer characteristic polynomial	t^7+95t^5+60t^3+6t
Flat arrow polynomial	8K13 + 4K1*2K2 - 12K12 - 6K1K2 - 2K1K3 - 3K1 + 5*K2 + K3 + 6
2-strand cable arrow polynomial	192K14K2 - 2112K14 + 32K1*3K2K3 - 320K1*3K3 - 128K12K2*4 + 576K1*2K2*3 - 5440K1*2K2*2 + 32K1*2K2K32 - 480K1*2K2K4 + 9328K1*2K2 - 128K12K3*2 - 16K1*2K4*2 - 6928K1*2 + 1024K1K23K3 + 128K1K2*2K3K4 - 1600K1K22K3 + 32K1K2*2K4K5 - 384K1K22K5 - 224K1K2K3K4 + 8400K1K2K3 + 1336K1K3K4 + 192K1K4K5 + 8K1K5K6 - 64K26 - 128K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 1824K24 + 96K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 - 1184K22K3*2 - 296K2*2K4*2 + 2528K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 5358K2*2 - 64K2K32K4 - 32K2K3K4K5 + 936K2K3K5 + 176K2K4K6 + 32K2K5K7 + 16K3*2K6 - 2884K32 - 1030K4*2 - 244K5*2 - 34K6**2 + 5780
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}], [{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}], [{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}, {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}, {3, 5}, {4}, {2}, {1}], [{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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