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Flat knot 6.1023

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,1,1,0,1,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1023', '7.35855']
Arrow polynomial of the knot is: 4K13 - 14K1*2 - 4K1K2 - K1 + 7K2 + K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1023', '6.1530', '6.1662', '6.1668', '6.1801']
Outer characteristic polynomial of the knot is: t^7+22t^5+38t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1023', '7.35855']
2-strand cable arrow polynomial of the knot is: -768K16 - 1216K1*4K2*2 + 3744K1*4K2 - 6064K14 + 992K1*3K2K3 - 928K1*3K3 - 576K12K2*4 + 2880K1*2K2*3 + 128K1*2K2*2K4 - 10816K12K2*2 - 640K1*2K2K4 + 10232K1*2K2 - 528K12K3*2 - 1532K1*2 + 736K1K23K3 - 1600K1K2*2K3 - 128K1K2*2K5 - 160K1K2K3K4 + 6840K1K2K3 + 496K1K3K4 + 24K1K4K5 - 32K2*6 + 64K2*4K4 - 2280K24 - 432K2*2K3*2 - 48K2*2K4*2 + 1664K2*2K4 - 1870K22 + 256K2K3K5 + 16K2K4K6 - 908K3*2 - 222K4*2 - 32K5*2 - 2K6**2 + 2708
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1023']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.68', 'vk6.125', 'vk6.222', 'vk6.271', 'vk6.292', 'vk6.678', 'vk6.1220', 'vk6.1269', 'vk6.1360', 'vk6.1409', 'vk6.1438', 'vk6.1922', 'vk6.2378', 'vk6.2442', 'vk6.2928', 'vk6.2984', 'vk6.5730', 'vk6.5761', 'vk6.7799', 'vk6.7830', 'vk6.13284', 'vk6.13315', 'vk6.14785', 'vk6.14813', 'vk6.15943', 'vk6.15971', 'vk6.18048', 'vk6.24492', 'vk6.33033', 'vk6.33388', 'vk6.43914', 'vk6.50496']
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: [-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,1,1,0,1,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1023', '7.35855']

Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 4*K1*K2 - K1 + 7*K2 + K3 + 8

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1023', '6.1530', '6.1662', '6.1668', '6.1801']

Outer characteristic polynomial of the knot is: t^7+22t^5+38t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1023', '7.35855']

2-strand cable arrow polynomial of the knot is: -768*K1**6 - 1216*K1**4*K2**2 + 3744*K1**4*K2 - 6064*K1**4 + 992*K1**3*K2*K3 - 928*K1**3*K3 - 576*K1**2*K2**4 + 2880*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 10816*K1**2*K2**2 - 640*K1**2*K2*K4 + 10232*K1**2*K2 - 528*K1**2*K3**2 - 1532*K1**2 + 736*K1*K2**3*K3 - 1600*K1*K2**2*K3 - 128*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6840*K1*K2*K3 + 496*K1*K3*K4 + 24*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2280*K2**4 - 432*K2**2*K3**2 - 48*K2**2*K4**2 + 1664*K2**2*K4 - 1870*K2**2 + 256*K2*K3*K5 + 16*K2*K4*K6 - 908*K3**2 - 222*K4**2 - 32*K5**2 - 2*K6**2 + 2708

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1023']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.68', 'vk6.125', 'vk6.222', 'vk6.271', 'vk6.292', 'vk6.678', 'vk6.1220', 'vk6.1269', 'vk6.1360', 'vk6.1409', 'vk6.1438', 'vk6.1922', 'vk6.2378', 'vk6.2442', 'vk6.2928', 'vk6.2984', 'vk6.5730', 'vk6.5761', 'vk6.7799', 'vk6.7830', 'vk6.13284', 'vk6.13315', 'vk6.14785', 'vk6.14813', 'vk6.15943', 'vk6.15971', 'vk6.18048', 'vk6.24492', 'vk6.33033', 'vk6.33388', 'vk6.43914', 'vk6.50496']

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	O1O2O3O4U3U4O5U1O6U5U6U2
R3 orbit	{'O1O2O3O4U3U4O5U1O6U5U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U5U6O5U4O6U1U2
Gauss code of K*	O1O2O3U4U3U5U6O5O6U1O4U2
Gauss code of -K*	O1O2O3U2O4U3O5O6U5U6U1U4
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 -2 1 -1 1 0 1],[ 2 0 2 -1 1 1 1],[-1 -2 0 -1 1 -1 1],[ 1 1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[ 0 -1 1 0 0 0 1],[-1 -1 -1 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 0 -1 -2],[-1 0 1 1 -1 -1 -2],[-1 -1 0 0 0 -1 -1],[-1 -1 0 0 -1 0 -1],[ 0 1 0 1 0 0 -1],[ 1 1 1 0 0 0 1],[ 2 2 1 1 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,1,2,-1,-1,1,1,2,0,0,1,1,1,0,1,0,1,-1]
Phi over symmetry	[-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,1,1,0,1,0,-1,-1]
Phi of -K	[-2,-1,0,1,1,1,2,1,1,2,2,1,1,1,2,0,1,0,-1,-1,0]
Phi of K*	[-1,-1,-1,0,1,2,-1,0,0,2,2,1,0,1,1,1,1,2,1,1,2]
Phi of -K*	[-2,-1,0,1,1,1,-1,1,1,1,2,0,0,1,1,1,0,1,0,-1,-1]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	3z^2+22z+33
Enhanced Jones-Krushkal polynomial	3w^3z^2+22w^2z+33w
Inner characteristic polynomial	t^6+14t^4+21t^2
Outer characteristic polynomial	t^7+22t^5+38t^3+5t
Flat arrow polynomial	4K13 - 14K1*2 - 4K1K2 - K1 + 7K2 + K3 + 8
2-strand cable arrow polynomial	-768K16 - 1216K1*4K2*2 + 3744K1*4K2 - 6064K14 + 992K1*3K2K3 - 928K1*3K3 - 576K12K2*4 + 2880K1*2K2*3 + 128K1*2K2*2K4 - 10816K12K2*2 - 640K1*2K2K4 + 10232K1*2K2 - 528K12K3*2 - 1532K1*2 + 736K1K23K3 - 1600K1K2*2K3 - 128K1K2*2K5 - 160K1K2K3K4 + 6840K1K2K3 + 496K1K3K4 + 24K1K4K5 - 32K2*6 + 64K2*4K4 - 2280K24 - 432K2*2K3*2 - 48K2*2K4*2 + 1664K2*2K4 - 1870K22 + 256K2K3K5 + 16K2K4K6 - 908K3*2 - 222K4*2 - 32K5*2 - 2K6**2 + 2708
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}]]
If K is slice	False

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