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Flat knot 6.738

Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,2,1,3,3,1,0,2,2,1,1,2,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.738']
Arrow polynomial of the knot is: 8K13 - 2K1*2 - 6K1K2 - 3K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.677', '6.696', '6.738', '6.758']
Outer characteristic polynomial of the knot is: t^7+63t^5+60t^3+12t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.738']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 96K1*4K2 - 368K14 + 128K1*3K2*3K3 + 800K13K2K3 - 192K1*3K3 - 1088K12K2*4 + 1824K1*2K2*3 - 384K1*2K2*2K3*2 - 7472K1*2K2*2 + 128K1*2K2K32 - 608K1*2K2K4 + 5464K1*2K2 - 784K12K3*2 - 3404K1*2 - 384K1K24K3 + 2016K1K2*3K3 + 480K1K2*2K3K4 - 1216K1K22K3 - 64K1K2*2K5 + 32K1K2K33 - 192K1K2K3K4 + 6864K1K2K3 - 96K1K2K4K5 + 1128K1K3K4 + 8K1K4K5 + 24K1K5K6 - 192K2*6 + 384K2*4K4 - 1944K24 - 32K2*3K6 - 992K22K3*2 - 248K2*2K4*2 + 1312K2*2K4 - 1606K22 + 272K2K3K5 + 88K2K4K6 - 1744K3*2 - 454K4*2 - 20K5*2 - 18K6**2 + 2708
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.738']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16904', 'vk6.17148', 'vk6.20516', 'vk6.21902', 'vk6.23296', 'vk6.23597', 'vk6.27959', 'vk6.29436', 'vk6.35310', 'vk6.35748', 'vk6.39369', 'vk6.41551', 'vk6.42815', 'vk6.43099', 'vk6.45938', 'vk6.47623', 'vk6.55059', 'vk6.55306', 'vk6.57385', 'vk6.58553', 'vk6.59455', 'vk6.59746', 'vk6.62042', 'vk6.63038', 'vk6.64900', 'vk6.65115', 'vk6.66934', 'vk6.67788', 'vk6.68209', 'vk6.68355', 'vk6.69541', 'vk6.70244']
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,-2,0,1,2,2,0,2,1,3,3,1,0,2,2,1,1,2,-1,-1,1]

Flat knots (up to 7 crossings) with same phi are :['6.738']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.677', '6.696', '6.738', '6.758']

Outer characteristic polynomial of the knot is: t^7+63t^5+60t^3+12t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.738']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 368*K1**4 + 128*K1**3*K2**3*K3 + 800*K1**3*K2*K3 - 192*K1**3*K3 - 1088*K1**2*K2**4 + 1824*K1**2*K2**3 - 384*K1**2*K2**2*K3**2 - 7472*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 608*K1**2*K2*K4 + 5464*K1**2*K2 - 784*K1**2*K3**2 - 3404*K1**2 - 384*K1*K2**4*K3 + 2016*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1216*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 - 192*K1*K2*K3*K4 + 6864*K1*K2*K3 - 96*K1*K2*K4*K5 + 1128*K1*K3*K4 + 8*K1*K4*K5 + 24*K1*K5*K6 - 192*K2**6 + 384*K2**4*K4 - 1944*K2**4 - 32*K2**3*K6 - 992*K2**2*K3**2 - 248*K2**2*K4**2 + 1312*K2**2*K4 - 1606*K2**2 + 272*K2*K3*K5 + 88*K2*K4*K6 - 1744*K3**2 - 454*K4**2 - 20*K5**2 - 18*K6**2 + 2708

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.738']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16904', 'vk6.17148', 'vk6.20516', 'vk6.21902', 'vk6.23296', 'vk6.23597', 'vk6.27959', 'vk6.29436', 'vk6.35310', 'vk6.35748', 'vk6.39369', 'vk6.41551', 'vk6.42815', 'vk6.43099', 'vk6.45938', 'vk6.47623', 'vk6.55059', 'vk6.55306', 'vk6.57385', 'vk6.58553', 'vk6.59455', 'vk6.59746', 'vk6.62042', 'vk6.63038', 'vk6.64900', 'vk6.65115', 'vk6.66934', 'vk6.67788', 'vk6.68209', 'vk6.68355', 'vk6.69541', 'vk6.70244']

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	O1O2O3O4U2O5U1U3U4O6U5U6
R3 orbit	{'O1O2O3O4U2O5U1U3U4O6U5U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U6O5U1U2U4O6U3
Gauss code of K*	O1O2O3U4O5O4U1U6U2U3O6U5
Gauss code of -K*	O1O2O3U4O5U1U2U5U3O6O4U6
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 -2 0 2 2 1],[ 3 0 0 2 3 3 1],[ 2 0 0 1 2 2 0],[ 0 -2 -1 0 1 2 1],[-2 -3 -2 -1 0 1 1],[-2 -3 -2 -2 -1 0 1],[-1 -1 0 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 2 1 0 -2 -3],[-2 0 1 1 -1 -2 -3],[-2 -1 0 1 -2 -2 -3],[-1 -1 -1 0 -1 0 -1],[ 0 1 2 1 0 -1 -2],[ 2 2 2 0 1 0 0],[ 3 3 3 1 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,2,3,-1,-1,1,2,3,-1,2,2,3,1,0,1,1,2,0]
Phi over symmetry	[-3,-2,0,1,2,2,0,2,1,3,3,1,0,2,2,1,1,2,-1,-1,1]
Phi of -K	[-3,-2,0,1,2,2,1,1,3,2,2,1,3,2,2,0,0,1,2,2,1]
Phi of K*	[-2,-2,-1,0,2,3,-1,2,0,2,2,2,1,2,2,0,3,3,1,1,1]
Phi of -K*	[-3,-2,0,1,2,2,0,2,1,3,3,1,0,2,2,1,1,2,-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+41t^4+17t^2+1
Outer characteristic polynomial	t^7+63t^5+60t^3+12t
Flat arrow polynomial	8K13 - 2K1*2 - 6K1K2 - 3K1 + K2 + K3 + 2
2-strand cable arrow polynomial	-256K14K2*2 + 96K1*4K2 - 368K14 + 128K1*3K2*3K3 + 800K13K2K3 - 192K1*3K3 - 1088K12K2*4 + 1824K1*2K2*3 - 384K1*2K2*2K3*2 - 7472K1*2K2*2 + 128K1*2K2K32 - 608K1*2K2K4 + 5464K1*2K2 - 784K12K3*2 - 3404K1*2 - 384K1K24K3 + 2016K1K2*3K3 + 480K1K2*2K3K4 - 1216K1K22K3 - 64K1K2*2K5 + 32K1K2K33 - 192K1K2K3K4 + 6864K1K2K3 - 96K1K2K4K5 + 1128K1K3K4 + 8K1K4K5 + 24K1K5K6 - 192K2*6 + 384K2*4K4 - 1944K24 - 32K2*3K6 - 992K22K3*2 - 248K2*2K4*2 + 1312K2*2K4 - 1606K22 + 272K2K3K5 + 88K2K4K6 - 1744K3*2 - 454K4*2 - 20K5*2 - 18K6**2 + 2708
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}], [{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}, {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}, {3}, {1, 2}]]
If K is slice	False

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