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Flat knot 6.784

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,1,3,3,1,2,3,2,0,-1,0,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.784']
Arrow polynomial of the knot is: 4K13 - 4K1*2 - 6K1K2 + 2K2 + 2*K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.552', '6.652', '6.764', '6.776', '6.784', '6.839', '6.903', '6.1010', '6.1166']
Outer characteristic polynomial of the knot is: t^7+60t^5+97t^3+18t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.784']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 448K1*4K2 - 2032K14 + 832K1*3K2K3 - 640K1*3K3 - 320K12K2*4 + 800K1*2K2*3 - 128K1*2K2*2K3*2 + 64K1*2K2*2K4 - 5488K12K2*2 + 64K1*2K2K32 + 32K1*2K2K3K5 - 608K12K2K4 + 7496K1*2K2 - 1040K12K3*2 - 4784K1*2 + 992K1K23K3 + 32K1K2*2K3K4 - 768K1K22K3 - 96K1K2*2K5 + 192K1K2K33 - 256K1K2K3K4 - 32K1K2K3K6 + 7296K1K2K3 - 64K1K3*2K5 + 824K1K3K4 - 32K2*6 + 64K2*4K4 - 896K24 - 1168K2*2K3*2 - 56K2*2K4*2 + 808K2*2K4 - 3548K22 + 736K2K3K5 + 32K2K4K6 - 96K3*4 + 88K3*2K6 - 2016K32 - 220K4*2 - 104K5*2 - 28K6**2 + 3874
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.784']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16529', 'vk6.16620', 'vk6.17527', 'vk6.17583', 'vk6.18861', 'vk6.18939', 'vk6.19214', 'vk6.19508', 'vk6.23052', 'vk6.24127', 'vk6.25487', 'vk6.25563', 'vk6.26026', 'vk6.26410', 'vk6.34925', 'vk6.35037', 'vk6.36308', 'vk6.36378', 'vk6.37584', 'vk6.37675', 'vk6.42495', 'vk6.42606', 'vk6.43485', 'vk6.44607', 'vk6.54773', 'vk6.54863', 'vk6.56428', 'vk6.56561', 'vk6.59291', 'vk6.60199', 'vk6.66086', 'vk6.66131']
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,-2,1,1,1,2,0,1,1,3,3,1,2,3,2,0,-1,0,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.784']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.552', '6.652', '6.764', '6.776', '6.784', '6.839', '6.903', '6.1010', '6.1166']

Outer characteristic polynomial of the knot is: t^7+60t^5+97t^3+18t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.784']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 2032*K1**4 + 832*K1**3*K2*K3 - 640*K1**3*K3 - 320*K1**2*K2**4 + 800*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 5488*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 608*K1**2*K2*K4 + 7496*K1**2*K2 - 1040*K1**2*K3**2 - 4784*K1**2 + 992*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 - 96*K1*K2**2*K5 + 192*K1*K2*K3**3 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 7296*K1*K2*K3 - 64*K1*K3**2*K5 + 824*K1*K3*K4 - 32*K2**6 + 64*K2**4*K4 - 896*K2**4 - 1168*K2**2*K3**2 - 56*K2**2*K4**2 + 808*K2**2*K4 - 3548*K2**2 + 736*K2*K3*K5 + 32*K2*K4*K6 - 96*K3**4 + 88*K3**2*K6 - 2016*K3**2 - 220*K4**2 - 104*K5**2 - 28*K6**2 + 3874

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.784']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16529', 'vk6.16620', 'vk6.17527', 'vk6.17583', 'vk6.18861', 'vk6.18939', 'vk6.19214', 'vk6.19508', 'vk6.23052', 'vk6.24127', 'vk6.25487', 'vk6.25563', 'vk6.26026', 'vk6.26410', 'vk6.34925', 'vk6.35037', 'vk6.36308', 'vk6.36378', 'vk6.37584', 'vk6.37675', 'vk6.42495', 'vk6.42606', 'vk6.43485', 'vk6.44607', 'vk6.54773', 'vk6.54863', 'vk6.56428', 'vk6.56561', 'vk6.59291', 'vk6.60199', 'vk6.66086', 'vk6.66131']

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	O1O2O3O4U3O5U6U2U4O6U1U5
R3 orbit	{'O1O2O3O4U3O5U6U2U4O6U1U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U4O6U1U3U6O5U2
Gauss code of K*	O1O2O3U1O4O5U4U2U6U3O6U5
Gauss code of -K*	O1O2O3U4O5U1U5U2U6O4O6U3
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 -1 -1 2 3 -2],[ 1 0 0 -1 3 3 -1],[ 1 0 0 0 2 1 0],[ 1 1 0 0 1 1 0],[-2 -3 -2 -1 0 0 -2],[-3 -3 -1 -1 0 0 -3],[ 2 1 0 0 2 3 0]]
Primitive based matrix	[[ 0 3 2 -1 -1 -1 -2],[-3 0 0 -1 -1 -3 -3],[-2 0 0 -1 -2 -3 -2],[ 1 1 1 0 0 1 0],[ 1 1 2 0 0 0 0],[ 1 3 3 -1 0 0 -1],[ 2 3 2 0 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,1,1,1,2,0,1,1,3,3,1,2,3,2,0,-1,0,0,0,1]
Phi over symmetry	[-3,-2,1,1,1,2,0,1,1,3,3,1,2,3,2,0,-1,0,0,0,1]
Phi of -K	[-2,-1,-1,-1,2,3,0,1,1,2,2,0,1,0,1,0,1,3,2,3,1]
Phi of K*	[-3,-2,1,1,1,2,1,1,3,3,2,0,1,2,2,0,-1,0,0,1,1]
Phi of -K*	[-2,-1,-1,-1,2,3,0,0,1,2,3,0,0,2,1,1,1,1,3,3,0]
Symmetry type of based matrix	c
u-polynomial	-t^3+3t
Normalized Jones-Krushkal polynomial	3z^2+22z+33
Enhanced Jones-Krushkal polynomial	3w^3z^2-2w^3z+24w^2z+33w
Inner characteristic polynomial	t^6+40t^4+51t^2+9
Outer characteristic polynomial	t^7+60t^5+97t^3+18t
Flat arrow polynomial	4K13 - 4K1*2 - 6K1K2 + 2K2 + 2*K3 + 3
2-strand cable arrow polynomial	-256K14K2*2 + 448K1*4K2 - 2032K14 + 832K1*3K2K3 - 640K1*3K3 - 320K12K2*4 + 800K1*2K2*3 - 128K1*2K2*2K3*2 + 64K1*2K2*2K4 - 5488K12K2*2 + 64K1*2K2K32 + 32K1*2K2K3K5 - 608K12K2K4 + 7496K1*2K2 - 1040K12K3*2 - 4784K1*2 + 992K1K23K3 + 32K1K2*2K3K4 - 768K1K22K3 - 96K1K2*2K5 + 192K1K2K33 - 256K1K2K3K4 - 32K1K2K3K6 + 7296K1K2K3 - 64K1K3*2K5 + 824K1K3K4 - 32K2*6 + 64K2*4K4 - 896K24 - 1168K2*2K3*2 - 56K2*2K4*2 + 808K2*2K4 - 3548K22 + 736K2K3K5 + 32K2K4K6 - 96K3*4 + 88K3*2K6 - 2016K32 - 220K4*2 - 104K5*2 - 28K6**2 + 3874
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}, {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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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