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Flat knot 6.786

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,1,3,3,1,1,2,1,1,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.786']
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+48t^5+66t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.786']
2-strand cable arrow polynomial of the knot is: -64K16 + 352K1*4K2 - 1472K14 + 192K1*3K2K3 - 224K1*3K3 - 1472K12K2*2 - 64K1*2K2K4 + 3184K1*2K2 - 480K12K3*2 - 48K1*2K4*2 - 1824K1*2 - 192K1K22K3 - 128K1K2K3K4 + 2264K1K2K3 + 640K1K3K4 + 136K1K4K5 + 24K1K5K6 - 152K24 - 48K2*2K3*2 - 8K2*2K4*2 + 336K2*2K4 - 1614K22 + 128K2K3K5 + 16K2K4K6 - 816K3*2 - 306K4*2 - 88K5*2 - 18K6**2 + 1744
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.786']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11250', 'vk6.11328', 'vk6.12511', 'vk6.12622', 'vk6.13877', 'vk6.13974', 'vk6.14142', 'vk6.14367', 'vk6.14948', 'vk6.15071', 'vk6.15594', 'vk6.16066', 'vk6.17432', 'vk6.22589', 'vk6.22620', 'vk6.23940', 'vk6.24081', 'vk6.24173', 'vk6.26145', 'vk6.26562', 'vk6.30924', 'vk6.31047', 'vk6.33688', 'vk6.33767', 'vk6.34581', 'vk6.36236', 'vk6.37648', 'vk6.37699', 'vk6.42273', 'vk6.44798', 'vk6.52016', 'vk6.52107', 'vk6.54099', 'vk6.54392', 'vk6.54577', 'vk6.56484', 'vk6.56666', 'vk6.59047', 'vk6.60074', 'vk6.64563']
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,-1,0,1,1,2,0,1,1,3,3,1,1,2,1,1,1,1,-1,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.786']

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+48t^5+66t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.786']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 352*K1**4*K2 - 1472*K1**4 + 192*K1**3*K2*K3 - 224*K1**3*K3 - 1472*K1**2*K2**2 - 64*K1**2*K2*K4 + 3184*K1**2*K2 - 480*K1**2*K3**2 - 48*K1**2*K4**2 - 1824*K1**2 - 192*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 2264*K1*K2*K3 + 640*K1*K3*K4 + 136*K1*K4*K5 + 24*K1*K5*K6 - 152*K2**4 - 48*K2**2*K3**2 - 8*K2**2*K4**2 + 336*K2**2*K4 - 1614*K2**2 + 128*K2*K3*K5 + 16*K2*K4*K6 - 816*K3**2 - 306*K4**2 - 88*K5**2 - 18*K6**2 + 1744

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.786']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11250', 'vk6.11328', 'vk6.12511', 'vk6.12622', 'vk6.13877', 'vk6.13974', 'vk6.14142', 'vk6.14367', 'vk6.14948', 'vk6.15071', 'vk6.15594', 'vk6.16066', 'vk6.17432', 'vk6.22589', 'vk6.22620', 'vk6.23940', 'vk6.24081', 'vk6.24173', 'vk6.26145', 'vk6.26562', 'vk6.30924', 'vk6.31047', 'vk6.33688', 'vk6.33767', 'vk6.34581', 'vk6.36236', 'vk6.37648', 'vk6.37699', 'vk6.42273', 'vk6.44798', 'vk6.52016', 'vk6.52107', 'vk6.54099', 'vk6.54392', 'vk6.54577', 'vk6.56484', 'vk6.56666', 'vk6.59047', 'vk6.60074', 'vk6.64563']

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	O1O2O3O4U3O5U6U4U2O6U1U5
R3 orbit	{'O1O2O3U2O4O5U6U3U4O6U1U5', 'O1O2O3O4U3O5U6U4U2O6U1U5'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4U5U4O6U3U1U6O5U2
Gauss code of K*	O1O2O3U1O4O5U4U3U6U2O6U5
Gauss code of -K*	O1O2O3U4O5U2U5U1U6O4O6U3
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 0 -1 1 3 -2],[ 1 0 1 -1 2 3 -1],[ 0 -1 0 -1 1 1 -1],[ 1 1 1 0 1 1 0],[-1 -2 -1 -1 0 0 -1],[-3 -3 -1 -1 0 0 -3],[ 2 1 1 0 1 3 0]]
Primitive based matrix	[[ 0 3 1 0 -1 -1 -2],[-3 0 0 -1 -1 -3 -3],[-1 0 0 -1 -1 -2 -1],[ 0 1 1 0 -1 -1 -1],[ 1 1 1 1 0 1 0],[ 1 3 2 1 -1 0 -1],[ 2 3 1 1 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,0,1,1,2,0,1,1,3,3,1,1,2,1,1,1,1,-1,0,1]
Phi over symmetry	[-3,-1,0,1,1,2,0,1,1,3,3,1,1,2,1,1,1,1,-1,0,1]
Phi of -K	[-2,-1,-1,0,1,3,0,1,1,2,2,1,0,0,1,0,1,3,0,2,2]
Phi of K*	[-3,-1,0,1,1,2,2,2,1,3,2,0,0,1,2,0,0,1,-1,0,1]
Phi of -K*	[-2,-1,-1,0,1,3,0,1,1,1,3,1,1,1,1,1,2,3,1,1,0]
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+32t^4+29t^2
Outer characteristic polynomial	t^7+48t^5+66t^3+3t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-64K16 + 352K1*4K2 - 1472K14 + 192K1*3K2K3 - 224K1*3K3 - 1472K12K2*2 - 64K1*2K2K4 + 3184K1*2K2 - 480K12K3*2 - 48K1*2K4*2 - 1824K1*2 - 192K1K22K3 - 128K1K2K3K4 + 2264K1K2K3 + 640K1K3K4 + 136K1K4K5 + 24K1K5K6 - 152K24 - 48K2*2K3*2 - 8K2*2K4*2 + 336K2*2K4 - 1614K22 + 128K2K3K5 + 16K2K4K6 - 816K3*2 - 306K4*2 - 88K5*2 - 18K6**2 + 1744
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}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {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}, {3, 4}, {2}, {1}]]
If K is slice	False

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