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Flat knot 6.1650

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,1,0,2,2,1,1,1,2,1,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1650']
Arrow polynomial of the knot is: 4K13 - 2K1*2 - 4K1*K2 - K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878']
Outer characteristic polynomial of the knot is: t^7+35t^5+46t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1650']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 96K1*4K2 - 176K14 + 128K1*3K2*3K3 + 416K13K2K3 - 832K1*2K2*4 + 1952K1*2K2*3 - 128K1*2K2*2K3*2 - 5872K1*2K2*2 - 672K1*2K2K4 + 3824K1*2K2 - 272K12K3*2 - 2388K1*2 - 640K1K24K3 + 1536K1K2*3K3 + 512K1K2*2K3K4 - 1152K1K22K3 - 64K1K2*2K5 + 32K1K2K33 + 5144K1K2K3 - 96K1K2K4K5 + 920K1K3K4 + 8K1K4K5 - 288K26 + 544K2*4K4 - 2344K24 - 736K2*2K3*2 - 336K2*2K4*2 + 1832K2*2K4 - 1022K22 - 96K2K32K4 + 120K2K3K5 + 120K2K4K6 - 1304K32 - 602K4*2 - 4K5*2 - 2K6**2 + 2096
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1650']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17498', 'vk6.17503', 'vk6.17555', 'vk6.17558', 'vk6.24022', 'vk6.24031', 'vk6.24094', 'vk6.24098', 'vk6.36282', 'vk6.36287', 'vk6.36347', 'vk6.36350', 'vk6.43435', 'vk6.43440', 'vk6.43464', 'vk6.43467', 'vk6.55612', 'vk6.55625', 'vk6.55644', 'vk6.55652', 'vk6.60122', 'vk6.60142', 'vk6.60160', 'vk6.60171', 'vk6.65319', 'vk6.65329', 'vk6.65351', 'vk6.65362', 'vk6.68491', 'vk6.68500', 'vk6.68514', 'vk6.68523']
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,-2,0,1,1,2,0,1,0,2,2,1,1,1,2,1,1,1,-1,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1650']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878']

Outer characteristic polynomial of the knot is: t^7+35t^5+46t^3+11t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1650']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 176*K1**4 + 128*K1**3*K2**3*K3 + 416*K1**3*K2*K3 - 832*K1**2*K2**4 + 1952*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 5872*K1**2*K2**2 - 672*K1**2*K2*K4 + 3824*K1**2*K2 - 272*K1**2*K3**2 - 2388*K1**2 - 640*K1*K2**4*K3 + 1536*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 1152*K1*K2**2*K3 - 64*K1*K2**2*K5 + 32*K1*K2*K3**3 + 5144*K1*K2*K3 - 96*K1*K2*K4*K5 + 920*K1*K3*K4 + 8*K1*K4*K5 - 288*K2**6 + 544*K2**4*K4 - 2344*K2**4 - 736*K2**2*K3**2 - 336*K2**2*K4**2 + 1832*K2**2*K4 - 1022*K2**2 - 96*K2*K3**2*K4 + 120*K2*K3*K5 + 120*K2*K4*K6 - 1304*K3**2 - 602*K4**2 - 4*K5**2 - 2*K6**2 + 2096

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1650']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17498', 'vk6.17503', 'vk6.17555', 'vk6.17558', 'vk6.24022', 'vk6.24031', 'vk6.24094', 'vk6.24098', 'vk6.36282', 'vk6.36287', 'vk6.36347', 'vk6.36350', 'vk6.43435', 'vk6.43440', 'vk6.43464', 'vk6.43467', 'vk6.55612', 'vk6.55625', 'vk6.55644', 'vk6.55652', 'vk6.60122', 'vk6.60142', 'vk6.60160', 'vk6.60171', 'vk6.65319', 'vk6.65329', 'vk6.65351', 'vk6.65362', 'vk6.68491', 'vk6.68500', 'vk6.68514', 'vk6.68523']

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	O1O2O3U1O4U5U2U3O5O6U4U6
R3 orbit	{'O1O2O3U1O4U5U2U3O5O6U4U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U5O4O6U1U2U6O5U3
Gauss code of K*	O1O2O3U1U4O5O4U6U2U3O6U5
Gauss code of -K*	O1O2O3U4O5U1U2U5O6O4U6U3
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 0 2 1 -2 1],[ 2 0 1 2 2 0 0],[ 0 -1 0 1 1 -1 1],[-2 -2 -1 0 0 -2 1],[-1 -2 -1 0 0 -1 1],[ 2 0 1 2 1 0 1],[-1 0 -1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -2 -2],[-2 0 1 0 -1 -2 -2],[-1 -1 0 -1 -1 0 -1],[-1 0 1 0 -1 -2 -1],[ 0 1 1 1 0 -1 -1],[ 2 2 0 2 1 0 0],[ 2 2 1 1 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,2,2,-1,0,1,2,2,1,1,0,1,1,2,1,1,1,0]
Phi over symmetry	[-2,-2,0,1,1,2,0,1,0,2,2,1,1,1,2,1,1,1,-1,-1,0]
Phi of -K	[-2,-2,0,1,1,2,0,1,1,3,2,1,2,2,2,0,0,1,-1,1,2]
Phi of K*	[-2,-1,-1,0,2,2,1,2,1,2,2,1,0,1,2,0,3,2,1,1,0]
Phi of -K*	[-2,-2,0,1,1,2,0,1,0,2,2,1,1,1,2,1,1,1,-1,-1,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	z^2+6z+9
Enhanced Jones-Krushkal polynomial	-4w^4z^2+5w^3z^2-12w^3z+18w^2z+9w
Inner characteristic polynomial	t^6+21t^4+19t^2+1
Outer characteristic polynomial	t^7+35t^5+46t^3+11t
Flat arrow polynomial	4K13 - 2K1*2 - 4K1*K2 - K1 + K2 + K3 + 2
2-strand cable arrow polynomial	-256K14K2*2 + 96K1*4K2 - 176K14 + 128K1*3K2*3K3 + 416K13K2K3 - 832K1*2K2*4 + 1952K1*2K2*3 - 128K1*2K2*2K3*2 - 5872K1*2K2*2 - 672K1*2K2K4 + 3824K1*2K2 - 272K12K3*2 - 2388K1*2 - 640K1K24K3 + 1536K1K2*3K3 + 512K1K2*2K3K4 - 1152K1K22K3 - 64K1K2*2K5 + 32K1K2K33 + 5144K1K2K3 - 96K1K2K4K5 + 920K1K3K4 + 8K1K4K5 - 288K26 + 544K2*4K4 - 2344K24 - 736K2*2K3*2 - 336K2*2K4*2 + 1832K2*2K4 - 1022K22 - 96K2K32K4 + 120K2K3K5 + 120K2K4K6 - 1304K32 - 602K4*2 - 4K5*2 - 2K6**2 + 2096
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}], [{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}, {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}]]
If K is slice	False

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