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Flat knot 6.1144

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,2,3,3,1,1,1,2,0,0,0,-1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1144']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 4K12 - 6K1K2 - 2K1K3 - 2K2*2 + K2 + 2K3 + K4 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.959', '6.1144']
Outer characteristic polynomial of the knot is: t^7+50t^5+69t^3+15t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1144']
2-strand cable arrow polynomial of the knot is: -384K14K2*2 + 192K1*4K2 - 304K14 + 128K1*3K2*3K3 + 672K13K2K3 - 96K1*3K3 - 704K12K2*4 - 384K1*2K2*3K4 + 1728K12K2*3 - 192K1*2K2*2K3*2 - 256K1*2K2*2K4*2 + 288K1*2K2*2K4 - 6560K12K2*2 + 64K1*2K2K42 - 1632K1*2K2K4 + 4888K1*2K2 - 448K12K3*2 - 208K1*2K4*2 - 3804K1*2 - 128K1K24K3 - 128K1K2*3K3K4 + 1760K1K23K3 + 1088K1K2*2K3K4 - 1344K1K22K3 + 224K1K2*2K4K5 - 256K1K22K5 + 96K1K2K33 + 160K1K2K3K42 - 544K1K2K3K4 - 64K1K2K3K6 + 6872K1K2K3 - 64K1K2K4K5 - 32K1K2K4K7 + 2056K1K3K4 + 392K1K4K5 + 8K1K5K6 - 32K2*6 - 32K2*4K4*2 + 480K2*4K4 - 1808K24 + 64K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 1104K22K3*2 + 32K2*2K4*3 - 824K2*2K4*2 + 2256K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 2352K2*2 - 160K2K32K4 - 96K2K3K4K5 + 496K2K3K5 - 32K2K42K6 + 264K2K4K6 + 24K2K5K7 + 8K2K6K8 - 16K3*4 - 32K3*2K4*2 + 16K3*2K6 - 2068K32 + 24K3K4K7 - 8K44 + 8K4*2K8 - 1202K42 - 180K5*2 - 32K6*2 - 4K7*2 - 2K8**2 + 3282
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1144']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4720', 'vk6.5037', 'vk6.6243', 'vk6.6695', 'vk6.8221', 'vk6.8661', 'vk6.9601', 'vk6.9928', 'vk6.20296', 'vk6.21631', 'vk6.27592', 'vk6.29146', 'vk6.39010', 'vk6.41260', 'vk6.45778', 'vk6.47457', 'vk6.48760', 'vk6.48963', 'vk6.49559', 'vk6.49773', 'vk6.50774', 'vk6.50982', 'vk6.51253', 'vk6.51458', 'vk6.57147', 'vk6.58333', 'vk6.61773', 'vk6.62894', 'vk6.66768', 'vk6.67646', 'vk6.69416', 'vk6.70140']
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,-1,0,1,1,2,1,1,2,3,3,1,1,1,2,0,0,0,-1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.1144']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.959', '6.1144']

Outer characteristic polynomial of the knot is: t^7+50t^5+69t^3+15t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1144']

2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 192*K1**4*K2 - 304*K1**4 + 128*K1**3*K2**3*K3 + 672*K1**3*K2*K3 - 96*K1**3*K3 - 704*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 1728*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 256*K1**2*K2**2*K4**2 + 288*K1**2*K2**2*K4 - 6560*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 1632*K1**2*K2*K4 + 4888*K1**2*K2 - 448*K1**2*K3**2 - 208*K1**2*K4**2 - 3804*K1**2 - 128*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 1760*K1*K2**3*K3 + 1088*K1*K2**2*K3*K4 - 1344*K1*K2**2*K3 + 224*K1*K2**2*K4*K5 - 256*K1*K2**2*K5 + 96*K1*K2*K3**3 + 160*K1*K2*K3*K4**2 - 544*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6872*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2056*K1*K3*K4 + 392*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 32*K2**4*K4**2 + 480*K2**4*K4 - 1808*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1104*K2**2*K3**2 + 32*K2**2*K4**3 - 824*K2**2*K4**2 + 2256*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 2352*K2**2 - 160*K2*K3**2*K4 - 96*K2*K3*K4*K5 + 496*K2*K3*K5 - 32*K2*K4**2*K6 + 264*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**4 - 32*K3**2*K4**2 + 16*K3**2*K6 - 2068*K3**2 + 24*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1202*K4**2 - 180*K5**2 - 32*K6**2 - 4*K7**2 - 2*K8**2 + 3282

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1144']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4720', 'vk6.5037', 'vk6.6243', 'vk6.6695', 'vk6.8221', 'vk6.8661', 'vk6.9601', 'vk6.9928', 'vk6.20296', 'vk6.21631', 'vk6.27592', 'vk6.29146', 'vk6.39010', 'vk6.41260', 'vk6.45778', 'vk6.47457', 'vk6.48760', 'vk6.48963', 'vk6.49559', 'vk6.49773', 'vk6.50774', 'vk6.50982', 'vk6.51253', 'vk6.51458', 'vk6.57147', 'vk6.58333', 'vk6.61773', 'vk6.62894', 'vk6.66768', 'vk6.67646', 'vk6.69416', 'vk6.70140']

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	O1O2O3O4U1U2U5O6O5U4U6U3
R3 orbit	{'O1O2O3O4U1U2U5O6O5U4U6U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U2U5U1O6O5U6U3U4
Gauss code of K*	O1O2O3U4U3O5O4O6U1U2U6U5
Gauss code of -K*	O1O2O3U4U2O4O5O6U3U1U5U6
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 2 1 1 0],[ 3 0 1 3 2 3 1],[ 1 -1 0 2 1 1 1],[-2 -3 -2 0 -1 -1 0],[-1 -2 -1 1 0 -1 0],[-1 -3 -1 1 1 0 0],[ 0 -1 -1 0 0 0 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 -1 -1 0 -2 -3],[-1 1 0 1 0 -1 -3],[-1 1 -1 0 0 -1 -2],[ 0 0 0 0 0 -1 -1],[ 1 2 1 1 1 0 -1],[ 3 3 3 2 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,1,1,0,2,3,-1,0,1,3,0,1,2,1,1,1]
Phi over symmetry	[-3,-1,0,1,1,2,1,1,2,3,3,1,1,1,2,0,0,0,-1,1,1]
Phi of -K	[-3,-1,0,1,1,2,1,2,1,2,2,0,1,1,1,1,1,2,-1,0,0]
Phi of K*	[-2,-1,-1,0,1,3,0,0,2,1,2,-1,1,1,2,1,1,1,0,2,1]
Phi of -K*	[-3,-1,0,1,1,2,1,1,2,3,3,1,1,1,2,0,0,0,-1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	4w^3z^2-8w^3z+25w^2z+19w
Inner characteristic polynomial	t^6+34t^4+36t^2+4
Outer characteristic polynomial	t^7+50t^5+69t^3+15t
Flat arrow polynomial	4K13 + 4K1*2K2 - 4K12 - 6K1K2 - 2K1K3 - 2K2*2 + K2 + 2K3 + K4 + 3
2-strand cable arrow polynomial	-384K14K2*2 + 192K1*4K2 - 304K14 + 128K1*3K2*3K3 + 672K13K2K3 - 96K1*3K3 - 704K12K2*4 - 384K1*2K2*3K4 + 1728K12K2*3 - 192K1*2K2*2K3*2 - 256K1*2K2*2K4*2 + 288K1*2K2*2K4 - 6560K12K2*2 + 64K1*2K2K42 - 1632K1*2K2K4 + 4888K1*2K2 - 448K12K3*2 - 208K1*2K4*2 - 3804K1*2 - 128K1K24K3 - 128K1K2*3K3K4 + 1760K1K23K3 + 1088K1K2*2K3K4 - 1344K1K22K3 + 224K1K2*2K4K5 - 256K1K22K5 + 96K1K2K33 + 160K1K2K3K42 - 544K1K2K3K4 - 64K1K2K3K6 + 6872K1K2K3 - 64K1K2K4K5 - 32K1K2K4K7 + 2056K1K3K4 + 392K1K4K5 + 8K1K5K6 - 32K2*6 - 32K2*4K4*2 + 480K2*4K4 - 1808K24 + 64K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 1104K22K3*2 + 32K2*2K4*3 - 824K2*2K4*2 + 2256K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 2352K2*2 - 160K2K32K4 - 96K2K3K4K5 + 496K2K3K5 - 32K2K42K6 + 264K2K4K6 + 24K2K5K7 + 8K2K6K8 - 16K3*4 - 32K3*2K4*2 + 16K3*2K6 - 2068K32 + 24K3K4K7 - 8K44 + 8K4*2K8 - 1202K42 - 180K5*2 - 32K6*2 - 4K7*2 - 2K8**2 + 3282
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}, {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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

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