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Flat knot 6.380

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,2,2,3,2,1,1,1,1,1,1,2,0,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.380']
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+25t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.380']
2-strand cable arrow polynomial of the knot is: -64K16 - 64K1*4K2*2 + 736K1*4K2 - 2640K14 + 192K1*3K2K3 - 288K1*3K3 - 2240K12K2*2 - 32K1*2K2K4 + 4176K1*2K2 - 368K12K3*2 - 1168K1*2 - 128K1K22K3 - 32K1K2K3K4 + 2128K1K2K3 + 280K1K3K4 + 16K1K4K5 - 216K2*4 - 112K2*2K3*2 - 8K2*2K4*2 + 216K2*2K4 - 1358K22 + 80K2K3K5 + 8K2K4K6 - 460K3*2 - 82K4*2 - 12K5*2 - 2K6**2 + 1440
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.380']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11254', 'vk6.11332', 'vk6.12515', 'vk6.12626', 'vk6.13894', 'vk6.13989', 'vk6.14144', 'vk6.14369', 'vk6.14965', 'vk6.15086', 'vk6.15600', 'vk6.16072', 'vk6.17424', 'vk6.22595', 'vk6.22626', 'vk6.23932', 'vk6.24073', 'vk6.24165', 'vk6.26142', 'vk6.26561', 'vk6.30936', 'vk6.31059', 'vk6.33705', 'vk6.33780', 'vk6.34577', 'vk6.36228', 'vk6.37665', 'vk6.37712', 'vk6.42271', 'vk6.44795', 'vk6.52004', 'vk6.52099', 'vk6.54101', 'vk6.54409', 'vk6.54579', 'vk6.56495', 'vk6.56668', 'vk6.59053', 'vk6.60058', 'vk6.64557']
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,-1,0,1,1,2,0,2,2,3,2,1,1,1,1,1,1,2,0,1,2]

Flat knots (up to 7 crossings) with same phi are :['6.380']

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+25t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.380']

2-strand cable arrow polynomial of the knot is: -64*K1**6 - 64*K1**4*K2**2 + 736*K1**4*K2 - 2640*K1**4 + 192*K1**3*K2*K3 - 288*K1**3*K3 - 2240*K1**2*K2**2 - 32*K1**2*K2*K4 + 4176*K1**2*K2 - 368*K1**2*K3**2 - 1168*K1**2 - 128*K1*K2**2*K3 - 32*K1*K2*K3*K4 + 2128*K1*K2*K3 + 280*K1*K3*K4 + 16*K1*K4*K5 - 216*K2**4 - 112*K2**2*K3**2 - 8*K2**2*K4**2 + 216*K2**2*K4 - 1358*K2**2 + 80*K2*K3*K5 + 8*K2*K4*K6 - 460*K3**2 - 82*K4**2 - 12*K5**2 - 2*K6**2 + 1440

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.380']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11254', 'vk6.11332', 'vk6.12515', 'vk6.12626', 'vk6.13894', 'vk6.13989', 'vk6.14144', 'vk6.14369', 'vk6.14965', 'vk6.15086', 'vk6.15600', 'vk6.16072', 'vk6.17424', 'vk6.22595', 'vk6.22626', 'vk6.23932', 'vk6.24073', 'vk6.24165', 'vk6.26142', 'vk6.26561', 'vk6.30936', 'vk6.31059', 'vk6.33705', 'vk6.33780', 'vk6.34577', 'vk6.36228', 'vk6.37665', 'vk6.37712', 'vk6.42271', 'vk6.44795', 'vk6.52004', 'vk6.52099', 'vk6.54101', 'vk6.54409', 'vk6.54579', 'vk6.56495', 'vk6.56668', 'vk6.59053', 'vk6.60058', 'vk6.64557']

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	O1O2O3O4O5U4U1O6U5U3U6U2
R3 orbit	{'O1O2O3O4U3O5U1O6U4U5U6U2', 'O1O2O3O4O5U4U1O6U5U3U6U2'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U4U6U3U1O6U5U2
Gauss code of K*	O1O2O3O4U5U4U2U6U1O6O5U3
Gauss code of -K*	O1O2O3O4U2O5O6U4U6U3U1U5
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 0 -1 1 2],[ 3 0 3 2 0 2 2],[-1 -3 0 -1 -1 0 2],[ 0 -2 1 0 -1 1 2],[ 1 0 1 1 0 1 1],[-1 -2 0 -1 -1 0 1],[-2 -2 -2 -2 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 -1 -2 -2 -1 -2],[-1 1 0 0 -1 -1 -2],[-1 2 0 0 -1 -1 -3],[ 0 2 1 1 0 -1 -2],[ 1 1 1 1 1 0 0],[ 3 2 2 3 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,1,2,2,1,2,0,1,1,2,1,1,3,1,2,0]
Phi over symmetry	[-3,-1,0,1,1,2,0,2,2,3,2,1,1,1,1,1,1,2,0,1,2]
Phi of -K	[-3,-1,0,1,1,2,2,1,1,2,3,0,1,1,2,0,0,0,0,-1,0]
Phi of K*	[-2,-1,-1,0,1,3,-1,0,0,2,3,0,0,1,1,0,1,2,0,1,2]
Phi of -K*	[-3,-1,0,1,1,2,0,2,2,3,2,1,1,1,1,1,1,2,0,1,2]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	z^2+14z+25
Enhanced Jones-Krushkal polynomial	w^3z^2+14w^2z+25w
Inner characteristic polynomial	t^6+36t^4+8t^2
Outer characteristic polynomial	t^7+52t^5+25t^3+3t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-64K16 - 64K1*4K2*2 + 736K1*4K2 - 2640K14 + 192K1*3K2K3 - 288K1*3K3 - 2240K12K2*2 - 32K1*2K2K4 + 4176K1*2K2 - 368K12K3*2 - 1168K1*2 - 128K1K22K3 - 32K1K2K3K4 + 2128K1K2K3 + 280K1K3K4 + 16K1K4K5 - 216K2*4 - 112K2*2K3*2 - 8K2*2K4*2 + 216K2*2K4 - 1358K22 + 80K2K3K5 + 8K2K4K6 - 460K3*2 - 82K4*2 - 12K5*2 - 2K6**2 + 1440
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}, {4, 5}, {3}, {2}, {1}], [{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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