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Flat knot 6.1313

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,2,3,2,0,1,2,1,1,0,1,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1313']
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+44t^5+103t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1313']
2-strand cable arrow polynomial of the knot is: -768K14K2*2 + 1280K1*4K2 - 3584K14 + 864K1*3K2K3 - 544K1*3K3 - 128K12K2*4 + 1440K1*2K2*3 + 128K1*2K2*2K4 - 7312K12K2*2 - 608K1*2K2K4 + 9624K1*2K2 - 512K12K3*2 - 32K1*2K4*2 - 4668K1*2 + 288K1K23K3 - 640K1K2*2K3 - 128K1K2*2K5 - 128K1K2K3K4 + 6304K1K2K3 + 504K1K3K4 + 64K1K4K5 - 1048K2*4 - 224K2*2K3*2 - 8K2*2K4*2 + 912K2*2K4 - 3534K22 + 216K2K3K5 + 8K2K4K6 - 1380K3*2 - 250K4*2 - 56K5*2 - 2K6**2 + 3920
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1313']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81560', 'vk6.81634', 'vk6.81640', 'vk6.81830', 'vk6.81835', 'vk6.82050', 'vk6.82222', 'vk6.82224', 'vk6.82332', 'vk6.82334', 'vk6.82536', 'vk6.82541', 'vk6.82993', 'vk6.83132', 'vk6.83139', 'vk6.83548', 'vk6.83555', 'vk6.83928', 'vk6.84074', 'vk6.84088', 'vk6.84526', 'vk6.84884', 'vk6.84897', 'vk6.85910', 'vk6.85917', 'vk6.86399', 'vk6.86414', 'vk6.86438', 'vk6.86449', 'vk6.88828', 'vk6.89766', 'vk6.89878']
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,-1,0,1,1,2,1,1,2,3,2,0,1,2,1,1,0,1,1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.1313']

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+44t^5+103t^3+10t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1313']

2-strand cable arrow polynomial of the knot is: -768*K1**4*K2**2 + 1280*K1**4*K2 - 3584*K1**4 + 864*K1**3*K2*K3 - 544*K1**3*K3 - 128*K1**2*K2**4 + 1440*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7312*K1**2*K2**2 - 608*K1**2*K2*K4 + 9624*K1**2*K2 - 512*K1**2*K3**2 - 32*K1**2*K4**2 - 4668*K1**2 + 288*K1*K2**3*K3 - 640*K1*K2**2*K3 - 128*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6304*K1*K2*K3 + 504*K1*K3*K4 + 64*K1*K4*K5 - 1048*K2**4 - 224*K2**2*K3**2 - 8*K2**2*K4**2 + 912*K2**2*K4 - 3534*K2**2 + 216*K2*K3*K5 + 8*K2*K4*K6 - 1380*K3**2 - 250*K4**2 - 56*K5**2 - 2*K6**2 + 3920

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1313']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81560', 'vk6.81634', 'vk6.81640', 'vk6.81830', 'vk6.81835', 'vk6.82050', 'vk6.82222', 'vk6.82224', 'vk6.82332', 'vk6.82334', 'vk6.82536', 'vk6.82541', 'vk6.82993', 'vk6.83132', 'vk6.83139', 'vk6.83548', 'vk6.83555', 'vk6.83928', 'vk6.84074', 'vk6.84088', 'vk6.84526', 'vk6.84884', 'vk6.84897', 'vk6.85910', 'vk6.85917', 'vk6.86399', 'vk6.86414', 'vk6.86438', 'vk6.86449', 'vk6.88828', 'vk6.89766', 'vk6.89878']

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	O1O2O3U4O5O6U1U2O4U6U3U5
R3 orbit	{'O1O2O3U4O5O6U1U2O4U6U3U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U1U5O6U2U3O5O4U6
Gauss code of K*	O1O2O3U4U5U2O6U3U1O4O5U6
Gauss code of -K*	O1O2O3U4O5O6U3U1O4U2U5U6
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 0 2 1],[ 3 0 1 2 2 3 1],[ 1 -1 0 1 1 2 0],[-1 -2 -1 0 0 0 -1],[ 0 -2 -1 0 0 1 1],[-2 -3 -2 0 -1 0 0],[-1 -1 0 1 -1 0 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 0 0 -1 -2 -3],[-1 0 0 1 -1 0 -1],[-1 0 -1 0 0 -1 -2],[ 0 1 1 0 0 -1 -2],[ 1 2 0 1 1 0 -1],[ 3 3 1 2 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,0,0,1,2,3,-1,1,0,1,0,1,2,1,2,1]
Phi over symmetry	[-3,-1,0,1,1,2,1,1,2,3,2,0,1,2,1,1,0,1,1,1,1]
Phi of -K	[-3,-1,0,1,1,2,1,1,2,3,2,0,1,2,1,1,0,1,1,1,1]
Phi of K*	[-2,-1,-1,0,1,3,1,1,1,1,2,-1,1,1,2,0,2,3,0,1,1]
Phi of -K*	[-3,-1,0,1,1,2,1,2,1,2,3,1,0,1,2,1,0,1,1,0,0]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	5z^2+26z+33
Enhanced Jones-Krushkal polynomial	5w^3z^2+26w^2z+33w
Inner characteristic polynomial	t^6+28t^4+56t^2+4
Outer characteristic polynomial	t^7+44t^5+103t^3+10t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-768K14K2*2 + 1280K1*4K2 - 3584K14 + 864K1*3K2K3 - 544K1*3K3 - 128K12K2*4 + 1440K1*2K2*3 + 128K1*2K2*2K4 - 7312K12K2*2 - 608K1*2K2K4 + 9624K1*2K2 - 512K12K3*2 - 32K1*2K4*2 - 4668K1*2 + 288K1K23K3 - 640K1K2*2K3 - 128K1K2*2K5 - 128K1K2K3K4 + 6304K1K2K3 + 504K1K3K4 + 64K1K4K5 - 1048K2*4 - 224K2*2K3*2 - 8K2*2K4*2 + 912K2*2K4 - 3534K22 + 216K2K3K5 + 8K2K4K6 - 1380K3*2 - 250K4*2 - 56K5*2 - 2K6**2 + 3920
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}], [{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}, {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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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