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Flat knot 6.526

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,-1,0,2,3,4,0,1,1,1,1,1,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.526']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 10K12 - 6K1K2 - 2K1K3 - 3K1 - 2K22 + 4K2 + K3 + K4 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.526']
Outer characteristic polynomial of the knot is: t^7+55t^5+92t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.526']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 448K1*4K2 - 960K14 + 512K1*3K2K3 - 224K1*3K3 - 448K12K2*4 + 2016K1*2K2*3 - 320K1*2K2*2K3*2 + 192K1*2K2*2K4 - 7264K12K2*2 + 128K1*2K2K32 + 32K1*2K2K3K5 - 576K12K2K4 + 7888K1*2K2 - 416K12K3*2 - 32K1*2K4*2 - 5992K1*2 + 1728K1K23K3 + 288K1K2*2K3K4 - 1632K1K22K3 + 160K1K2*2K4K5 - 288K1K22K5 + 96K1K2K33 - 672K1K2K3K4 - 64K1K2K3K6 + 7184K1K2K3 - 128K1K2K4K5 - 32K1K2K4K7 + 1400K1K3K4 + 272K1K4K5 + 24K1K5K6 - 64K2*6 - 128K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 2008K24 + 96K2*3K3K5 + 32K2*3K4K6 + 128K2*2K3*2K4 - 1408K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 376K2*2K4*2 - 32K2*2K4K8 + 1784K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 3406K2*2 - 128K2K32K4 - 32K2K3K4K5 + 912K2K3K5 + 232K2K4K6 + 16K2K5K7 + 8K2K6K8 - 32K34 - 32K3*2K4*2 + 40K3*2K6 - 2180K32 + 16K3K4K7 - 8K44 + 8K4*2K8 - 840K42 - 204K5*2 - 26K6*2 - 2K8**2 + 4512
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.526']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4758', 'vk6.5085', 'vk6.6308', 'vk6.6747', 'vk6.8269', 'vk6.8718', 'vk6.9647', 'vk6.9962', 'vk6.20399', 'vk6.21748', 'vk6.27735', 'vk6.29273', 'vk6.39173', 'vk6.41401', 'vk6.45899', 'vk6.47540', 'vk6.48790', 'vk6.49001', 'vk6.49610', 'vk6.49813', 'vk6.50810', 'vk6.51025', 'vk6.51289', 'vk6.51484', 'vk6.57260', 'vk6.58477', 'vk6.61902', 'vk6.63011', 'vk6.66873', 'vk6.67743', 'vk6.69495', 'vk6.70217']
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,-1,0,0,2,2,-1,0,2,3,4,0,1,1,1,1,1,1,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.526']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 - 2*K2**2 + 4*K2 + K3 + K4 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.526']

Outer characteristic polynomial of the knot is: t^7+55t^5+92t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.526']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 960*K1**4 + 512*K1**3*K2*K3 - 224*K1**3*K3 - 448*K1**2*K2**4 + 2016*K1**2*K2**3 - 320*K1**2*K2**2*K3**2 + 192*K1**2*K2**2*K4 - 7264*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 576*K1**2*K2*K4 + 7888*K1**2*K2 - 416*K1**2*K3**2 - 32*K1**2*K4**2 - 5992*K1**2 + 1728*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 1632*K1*K2**2*K3 + 160*K1*K2**2*K4*K5 - 288*K1*K2**2*K5 + 96*K1*K2*K3**3 - 672*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7184*K1*K2*K3 - 128*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1400*K1*K3*K4 + 272*K1*K4*K5 + 24*K1*K5*K6 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 2008*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 + 128*K2**2*K3**2*K4 - 1408*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 376*K2**2*K4**2 - 32*K2**2*K4*K8 + 1784*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3406*K2**2 - 128*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 912*K2*K3*K5 + 232*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 32*K3**4 - 32*K3**2*K4**2 + 40*K3**2*K6 - 2180*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 840*K4**2 - 204*K5**2 - 26*K6**2 - 2*K8**2 + 4512

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.526']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4758', 'vk6.5085', 'vk6.6308', 'vk6.6747', 'vk6.8269', 'vk6.8718', 'vk6.9647', 'vk6.9962', 'vk6.20399', 'vk6.21748', 'vk6.27735', 'vk6.29273', 'vk6.39173', 'vk6.41401', 'vk6.45899', 'vk6.47540', 'vk6.48790', 'vk6.49001', 'vk6.49610', 'vk6.49813', 'vk6.50810', 'vk6.51025', 'vk6.51289', 'vk6.51484', 'vk6.57260', 'vk6.58477', 'vk6.61902', 'vk6.63011', 'vk6.66873', 'vk6.67743', 'vk6.69495', 'vk6.70217']

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	O1O2O3O4O5U2U4U5O6U3U1U6
R3 orbit	{'O1O2O3O4O5U2U4U5O6U3U1U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U5U3O6U1U2U4
Gauss code of K*	O1O2O3U4O5O6O4U6U1U5U2U3
Gauss code of -K*	O1O2O3U1O4O5O6U4U5U3U6U2
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 -1 -3 0 0 2 2],[ 1 0 -3 1 0 2 2],[ 3 3 0 3 1 2 1],[ 0 -1 -3 0 -1 1 1],[ 0 0 -1 1 0 1 0],[-2 -2 -2 -1 -1 0 0],[-2 -2 -1 -1 0 0 0]]
Primitive based matrix	[[ 0 2 2 0 0 -1 -3],[-2 0 0 0 -1 -2 -1],[-2 0 0 -1 -1 -2 -2],[ 0 0 1 0 1 0 -1],[ 0 1 1 -1 0 -1 -3],[ 1 2 2 0 1 0 -3],[ 3 1 2 1 3 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,0,1,3,0,0,1,2,1,1,1,2,2,-1,0,1,1,3,3]
Phi over symmetry	[-3,-1,0,0,2,2,-1,0,2,3,4,0,1,1,1,1,1,1,1,2,0]
Phi of -K	[-3,-1,0,0,2,2,-1,0,2,3,4,0,1,1,1,1,1,1,1,2,0]
Phi of K*	[-2,-2,0,0,1,3,0,1,1,1,3,1,2,1,4,-1,0,0,1,2,-1]
Phi of -K*	[-3,-1,0,0,2,2,3,1,3,1,2,0,1,2,2,1,0,1,1,1,0]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^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+37t^4+34t^2
Outer characteristic polynomial	t^7+55t^5+92t^3+5t
Flat arrow polynomial	8K13 + 4K1*2K2 - 10K12 - 6K1K2 - 2K1K3 - 3K1 - 2K22 + 4K2 + K3 + K4 + 6
2-strand cable arrow polynomial	-256K14K2*2 + 448K1*4K2 - 960K14 + 512K1*3K2K3 - 224K1*3K3 - 448K12K2*4 + 2016K1*2K2*3 - 320K1*2K2*2K3*2 + 192K1*2K2*2K4 - 7264K12K2*2 + 128K1*2K2K32 + 32K1*2K2K3K5 - 576K12K2K4 + 7888K1*2K2 - 416K12K3*2 - 32K1*2K4*2 - 5992K1*2 + 1728K1K23K3 + 288K1K2*2K3K4 - 1632K1K22K3 + 160K1K2*2K4K5 - 288K1K22K5 + 96K1K2K33 - 672K1K2K3K4 - 64K1K2K3K6 + 7184K1K2K3 - 128K1K2K4K5 - 32K1K2K4K7 + 1400K1K3K4 + 272K1K4K5 + 24K1K5K6 - 64K2*6 - 128K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 2008K24 + 96K2*3K3K5 + 32K2*3K4K6 + 128K2*2K3*2K4 - 1408K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 376K2*2K4*2 - 32K2*2K4K8 + 1784K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 3406K2*2 - 128K2K32K4 - 32K2K3K4K5 + 912K2K3K5 + 232K2K4K6 + 16K2K5K7 + 8K2K6K8 - 32K34 - 32K3*2K4*2 + 40K3*2K6 - 2180K32 + 16K3K4K7 - 8K44 + 8K4*2K8 - 840K42 - 204K5*2 - 26K6*2 - 2K8**2 + 4512
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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}], [{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}, {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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