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Flat knot 6.452

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,1,2,3,1,0,1,1,1,2,3,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.452']
Arrow polynomial of the knot is: 4K12K2 - 2K12 - 2K1K2 - 2K1K3 + K1 - 2K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']
Outer characteristic polynomial of the knot is: t^7+64t^5+104t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.452']
2-strand cable arrow polynomial of the knot is: 384K14K2 - 2496K14 + 736K1*3K2K3 + 64K1*3K3K4 - 1024K1*3K3 + 352K12K2*2K4 - 3600K12K2*2 + 64K1*2K2K32 + 128K1*2K2K42 - 1248K1*2K2K4 + 7952K1*2K2 - 864K12K3*2 - 624K1*2K4*2 - 5796K1*2 + 448K1K23K3 - 1120K1K2*2K3 - 480K1K2*2K5 + 32K1K2K3K4*2 - 768K1K2K3K4 + 7304K1K2K3 - 192K1K2K4K5 - 32K1K2K4K7 + 2688K1K3K4 + 824K1K4K5 + 32K1K5K6 - 32K2*4K4*2 + 224K2*4K4 - 664K24 + 32K2*3K4K6 - 160K2*3K6 - 432K22K3*2 + 32K2*2K4*3 - 488K2*2K4*2 - 32K2*2K4K8 + 2424K2*2K4 - 8K22K6*2 - 5346K2*2 - 64K2K32K4 + 784K2K3K5 + 568K2K4K6 + 8K2K5K7 + 8K2K6K8 - 16K32K4*2 + 16K3*2K6 - 2728K32 + 16K3K4K7 - 8K44 + 8K4*2K8 - 1644K42 - 344K5*2 - 118K6*2 - 4K7*2 - 2K8**2 + 5204
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.452']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11647', 'vk6.11650', 'vk6.11998', 'vk6.12003', 'vk6.12993', 'vk6.12996', 'vk6.13266', 'vk6.20436', 'vk6.20450', 'vk6.21797', 'vk6.27808', 'vk6.27834', 'vk6.29318', 'vk6.29344', 'vk6.31449', 'vk6.32628', 'vk6.32639', 'vk6.32972', 'vk6.32983', 'vk6.39236', 'vk6.39258', 'vk6.47565', 'vk6.52364', 'vk6.53250', 'vk6.53269', 'vk6.57305', 'vk6.57311', 'vk6.61994', 'vk6.64323', 'vk6.64334', 'vk6.64503', 'vk6.66899']
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,-1,1,2,2,0,1,1,2,3,1,0,1,1,1,2,3,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.452']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']

Outer characteristic polynomial of the knot is: t^7+64t^5+104t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.452']

2-strand cable arrow polynomial of the knot is: 384*K1**4*K2 - 2496*K1**4 + 736*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1024*K1**3*K3 + 352*K1**2*K2**2*K4 - 3600*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 1248*K1**2*K2*K4 + 7952*K1**2*K2 - 864*K1**2*K3**2 - 624*K1**2*K4**2 - 5796*K1**2 + 448*K1*K2**3*K3 - 1120*K1*K2**2*K3 - 480*K1*K2**2*K5 + 32*K1*K2*K3*K4**2 - 768*K1*K2*K3*K4 + 7304*K1*K2*K3 - 192*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 2688*K1*K3*K4 + 824*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**4*K4**2 + 224*K2**4*K4 - 664*K2**4 + 32*K2**3*K4*K6 - 160*K2**3*K6 - 432*K2**2*K3**2 + 32*K2**2*K4**3 - 488*K2**2*K4**2 - 32*K2**2*K4*K8 + 2424*K2**2*K4 - 8*K2**2*K6**2 - 5346*K2**2 - 64*K2*K3**2*K4 + 784*K2*K3*K5 + 568*K2*K4*K6 + 8*K2*K5*K7 + 8*K2*K6*K8 - 16*K3**2*K4**2 + 16*K3**2*K6 - 2728*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1644*K4**2 - 344*K5**2 - 118*K6**2 - 4*K7**2 - 2*K8**2 + 5204

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.452']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11647', 'vk6.11650', 'vk6.11998', 'vk6.12003', 'vk6.12993', 'vk6.12996', 'vk6.13266', 'vk6.20436', 'vk6.20450', 'vk6.21797', 'vk6.27808', 'vk6.27834', 'vk6.29318', 'vk6.29344', 'vk6.31449', 'vk6.32628', 'vk6.32639', 'vk6.32972', 'vk6.32983', 'vk6.39236', 'vk6.39258', 'vk6.47565', 'vk6.52364', 'vk6.53250', 'vk6.53269', 'vk6.57305', 'vk6.57311', 'vk6.61994', 'vk6.64323', 'vk6.64334', 'vk6.64503', 'vk6.66899']

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	O1O2O3O4O5U6U3O6U1U5U4U2
R3 orbit	{'O1O2O3O4O5U6U3O6U1U5U4U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U2U1U5O6U3U6
Gauss code of K*	O1O2O3O4U1U4U5U3U2O6O5U6
Gauss code of -K*	O1O2O3O4U5O6O5U3U2U6U1U4
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 -1 2 2 -1],[ 3 0 3 1 3 2 2],[-1 -3 0 -1 1 1 -2],[ 1 -1 1 0 1 0 1],[-2 -3 -1 -1 0 0 -2],[-2 -2 -1 0 0 0 -2],[ 1 -2 2 -1 2 2 0]]
Primitive based matrix	[[ 0 2 2 1 -1 -1 -3],[-2 0 0 -1 0 -2 -2],[-2 0 0 -1 -1 -2 -3],[-1 1 1 0 -1 -2 -3],[ 1 0 1 1 0 1 -1],[ 1 2 2 2 -1 0 -2],[ 3 2 3 3 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,1,1,3,0,1,0,2,2,1,1,2,3,1,2,3,-1,1,2]
Phi over symmetry	[-3,-1,-1,1,2,2,0,1,1,2,3,1,0,1,1,1,2,3,0,0,0]
Phi of -K	[-3,-1,-1,1,2,2,0,1,1,2,3,1,0,1,1,1,2,3,0,0,0]
Phi of K*	[-2,-2,-1,1,1,3,0,0,1,2,2,0,1,3,3,0,1,1,-1,0,1]
Phi of -K*	[-3,-1,-1,1,2,2,1,2,3,2,3,1,1,0,1,2,2,2,1,1,0]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	7z^2+28z+29
Enhanced Jones-Krushkal polynomial	7w^3z^2+28w^2z+29w
Inner characteristic polynomial	t^6+44t^4+64t^2+1
Outer characteristic polynomial	t^7+64t^5+104t^3+8t
Flat arrow polynomial	4K12K2 - 2K12 - 2K1K2 - 2K1K3 + K1 - 2K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial	384K14K2 - 2496K14 + 736K1*3K2K3 + 64K1*3K3K4 - 1024K1*3K3 + 352K12K2*2K4 - 3600K12K2*2 + 64K1*2K2K32 + 128K1*2K2K42 - 1248K1*2K2K4 + 7952K1*2K2 - 864K12K3*2 - 624K1*2K4*2 - 5796K1*2 + 448K1K23K3 - 1120K1K2*2K3 - 480K1K2*2K5 + 32K1K2K3K4*2 - 768K1K2K3K4 + 7304K1K2K3 - 192K1K2K4K5 - 32K1K2K4K7 + 2688K1K3K4 + 824K1K4K5 + 32K1K5K6 - 32K2*4K4*2 + 224K2*4K4 - 664K24 + 32K2*3K4K6 - 160K2*3K6 - 432K22K3*2 + 32K2*2K4*3 - 488K2*2K4*2 - 32K2*2K4K8 + 2424K2*2K4 - 8K22K6*2 - 5346K2*2 - 64K2K32K4 + 784K2K3K5 + 568K2K4K6 + 8K2K5K7 + 8K2K6K8 - 16K32K4*2 + 16K3*2K6 - 2728K32 + 16K3K4K7 - 8K44 + 8K4*2K8 - 1644K42 - 344K5*2 - 118K6*2 - 4K7*2 - 2K8**2 + 5204
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}], [{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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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