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Flat knot 6.620

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,3,1,3,2,1,0,2,2,1,0,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.620']
Arrow polynomial of the knot is: -6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']
Outer characteristic polynomial of the knot is: t^7+52t^5+163t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.620']
2-strand cable arrow polynomial of the knot is: -64K16 + 480K1*4K2 - 2736K14 + 544K1*3K2K3 - 512K1*3K3 - 2880K12K2*2 + 32K1*2K2K32 - 160K1*2K2K4 + 7624K1*2K2 - 720K12K3*2 - 64K1*2K4*2 - 5548K1*2 + 160K1K23K3 - 960K1K2*2K3 - 64K1K2*2K5 - 352K1K2K3K4 + 6128K1K2K3 + 1424K1K3K4 + 200K1K4K5 - 184K2*4 - 352K2*2K3*2 - 8K2*2K4*2 + 824K2*2K4 - 4710K22 + 488K2K3K5 + 8K2K4K6 - 2508K3*2 - 678K4*2 - 160K5*2 - 2K6**2 + 4748
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.620']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20012', 'vk6.20089', 'vk6.21284', 'vk6.21369', 'vk6.27059', 'vk6.27154', 'vk6.28764', 'vk6.28841', 'vk6.38452', 'vk6.38547', 'vk6.40641', 'vk6.40742', 'vk6.45332', 'vk6.45447', 'vk6.47101', 'vk6.47187', 'vk6.56811', 'vk6.56894', 'vk6.57945', 'vk6.58030', 'vk6.61325', 'vk6.61424', 'vk6.62501', 'vk6.62579', 'vk6.66523', 'vk6.66594', 'vk6.67312', 'vk6.67383', 'vk6.69165', 'vk6.69246', 'vk6.69916', 'vk6.69985']
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,-1,0,1,1,2,0,3,1,3,2,1,0,2,2,1,0,1,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.620']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']

Outer characteristic polynomial of the knot is: t^7+52t^5+163t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.620']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 480*K1**4*K2 - 2736*K1**4 + 544*K1**3*K2*K3 - 512*K1**3*K3 - 2880*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 7624*K1**2*K2 - 720*K1**2*K3**2 - 64*K1**2*K4**2 - 5548*K1**2 + 160*K1*K2**3*K3 - 960*K1*K2**2*K3 - 64*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 6128*K1*K2*K3 + 1424*K1*K3*K4 + 200*K1*K4*K5 - 184*K2**4 - 352*K2**2*K3**2 - 8*K2**2*K4**2 + 824*K2**2*K4 - 4710*K2**2 + 488*K2*K3*K5 + 8*K2*K4*K6 - 2508*K3**2 - 678*K4**2 - 160*K5**2 - 2*K6**2 + 4748

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.620']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20012', 'vk6.20089', 'vk6.21284', 'vk6.21369', 'vk6.27059', 'vk6.27154', 'vk6.28764', 'vk6.28841', 'vk6.38452', 'vk6.38547', 'vk6.40641', 'vk6.40742', 'vk6.45332', 'vk6.45447', 'vk6.47101', 'vk6.47187', 'vk6.56811', 'vk6.56894', 'vk6.57945', 'vk6.58030', 'vk6.61325', 'vk6.61424', 'vk6.62501', 'vk6.62579', 'vk6.66523', 'vk6.66594', 'vk6.67312', 'vk6.67383', 'vk6.69165', 'vk6.69246', 'vk6.69916', 'vk6.69985']

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	O1O2O3O4U5O6U1O5U3U6U4U2
R3 orbit	{'O1O2O3O4U5O6U1O5U3U6U4U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U1U5U2O6U4O5U6
Gauss code of K*	O1O2O3O4U5U4U1U3O6U2O5U6
Gauss code of -K*	O1O2O3O4U5O6U3O5U2U4U1U6
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 0 1],[ 3 0 3 0 2 3 1],[-1 -3 0 -2 1 0 0],[ 1 0 2 0 2 1 0],[-2 -2 -1 -2 0 -1 -1],[ 0 -3 0 -1 1 0 1],[-1 -1 0 0 1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 0 0 -2 -3],[-1 1 0 0 -1 0 -1],[ 0 1 0 1 0 -1 -3],[ 1 2 2 0 1 0 0],[ 3 2 3 1 3 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,1,1,1,2,2,0,0,2,3,1,0,1,1,3,0]
Phi over symmetry	[-3,-1,0,1,1,2,0,3,1,3,2,1,0,2,2,1,0,1,0,1,1]
Phi of -K	[-3,-1,0,1,1,2,2,0,1,3,3,0,0,2,1,1,0,1,0,0,0]
Phi of K*	[-2,-1,-1,0,1,3,0,0,1,1,3,0,0,2,3,1,0,1,0,0,2]
Phi of -K*	[-3,-1,0,1,1,2,0,3,1,3,2,1,0,2,2,1,0,1,0,1,1]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	z^2+20z+37
Enhanced Jones-Krushkal polynomial	w^3z^2+20w^2z+37w
Inner characteristic polynomial	t^6+36t^4+86t^2
Outer characteristic polynomial	t^7+52t^5+163t^3+5t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-64K16 + 480K1*4K2 - 2736K14 + 544K1*3K2K3 - 512K1*3K3 - 2880K12K2*2 + 32K1*2K2K32 - 160K1*2K2K4 + 7624K1*2K2 - 720K12K3*2 - 64K1*2K4*2 - 5548K1*2 + 160K1K23K3 - 960K1K2*2K3 - 64K1K2*2K5 - 352K1K2K3K4 + 6128K1K2K3 + 1424K1K3K4 + 200K1K4K5 - 184K2*4 - 352K2*2K3*2 - 8K2*2K4*2 + 824K2*2K4 - 4710K22 + 488K2K3K5 + 8K2K4K6 - 2508K3*2 - 678K4*2 - 160K5*2 - 2K6**2 + 4748
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}], [{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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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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