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Flat knot 6.1063

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,1,2,3,1,1,2,2,1,0,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1063']
Arrow polynomial of the knot is: -2K12 - 4K1K2 + 2K1 + K2 + 2*K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.120', '6.213', '6.216', '6.320', '6.322', '6.615', '6.617', '6.891', '6.951', '6.955', '6.1001', '6.1012', '6.1022', '6.1043', '6.1047', '6.1063', '6.1074', '6.1249', '6.1544', '6.1546', '6.1555', '6.1573', '6.1574', '6.1585', '6.1756', '6.1757', '6.1762', '6.1802', '6.1803', '6.1824', '6.1881', '6.1935']
Outer characteristic polynomial of the knot is: t^7+43t^5+107t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1063']
2-strand cable arrow polynomial of the knot is: -64K14K2*2 + 320K1*4K2 - 912K14 + 64K1*3K2K3 + 64K1*3K3K4 - 1120K1*3K3 + 352K12K2*3 - 2016K1*2K2*2 - 480K1*2K2K4 + 6496K1*2K2 - 368K12K3*2 - 32K1*2K3K5 - 112K1*2K4*2 - 5920K1*2 + 160K1K23K3 - 1184K1K2*2K3 - 256K1K2K3K4 + 5304K1K2K3 + 1640K1K3K4 + 240K1K4K5 - 632K2*4 - 160K2*2K3*2 - 16K2*2K4*2 + 1600K2*2K4 - 4236K22 + 296K2K3K5 + 32K2K4K6 - 2156K3*2 - 1038K4*2 - 156K5*2 - 12K6**2 + 4300
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1063']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73331', 'vk6.73341', 'vk6.73494', 'vk6.73503', 'vk6.75251', 'vk6.75264', 'vk6.75501', 'vk6.75509', 'vk6.78215', 'vk6.78231', 'vk6.78455', 'vk6.78474', 'vk6.80039', 'vk6.80054', 'vk6.80189', 'vk6.80203', 'vk6.81935', 'vk6.81939', 'vk6.82195', 'vk6.82211', 'vk6.82662', 'vk6.82663', 'vk6.84725', 'vk6.84727', 'vk6.85025', 'vk6.85031', 'vk6.85753', 'vk6.86502', 'vk6.87328', 'vk6.87688', 'vk6.89626', 'vk6.90087']
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,-2,0,1,1,2,0,0,1,2,3,1,1,2,2,1,0,2,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1063']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.120', '6.213', '6.216', '6.320', '6.322', '6.615', '6.617', '6.891', '6.951', '6.955', '6.1001', '6.1012', '6.1022', '6.1043', '6.1047', '6.1063', '6.1074', '6.1249', '6.1544', '6.1546', '6.1555', '6.1573', '6.1574', '6.1585', '6.1756', '6.1757', '6.1762', '6.1802', '6.1803', '6.1824', '6.1881', '6.1935']

Outer characteristic polynomial of the knot is: t^7+43t^5+107t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1063']

2-strand cable arrow polynomial of the knot is: -64*K1**4*K2**2 + 320*K1**4*K2 - 912*K1**4 + 64*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1120*K1**3*K3 + 352*K1**2*K2**3 - 2016*K1**2*K2**2 - 480*K1**2*K2*K4 + 6496*K1**2*K2 - 368*K1**2*K3**2 - 32*K1**2*K3*K5 - 112*K1**2*K4**2 - 5920*K1**2 + 160*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 256*K1*K2*K3*K4 + 5304*K1*K2*K3 + 1640*K1*K3*K4 + 240*K1*K4*K5 - 632*K2**4 - 160*K2**2*K3**2 - 16*K2**2*K4**2 + 1600*K2**2*K4 - 4236*K2**2 + 296*K2*K3*K5 + 32*K2*K4*K6 - 2156*K3**2 - 1038*K4**2 - 156*K5**2 - 12*K6**2 + 4300

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1063']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73331', 'vk6.73341', 'vk6.73494', 'vk6.73503', 'vk6.75251', 'vk6.75264', 'vk6.75501', 'vk6.75509', 'vk6.78215', 'vk6.78231', 'vk6.78455', 'vk6.78474', 'vk6.80039', 'vk6.80054', 'vk6.80189', 'vk6.80203', 'vk6.81935', 'vk6.81939', 'vk6.82195', 'vk6.82211', 'vk6.82662', 'vk6.82663', 'vk6.84725', 'vk6.84727', 'vk6.85025', 'vk6.85031', 'vk6.85753', 'vk6.86502', 'vk6.87328', 'vk6.87688', 'vk6.89626', 'vk6.90087']

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	O1O2O3O4U5U2O6U3O5U1U4U6
R3 orbit	{'O1O2O3O4U5U2O6U3O5U1U4U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U1U4O6U2O5U3U6
Gauss code of K*	O1O2O3U1U4U5U2O6O4U3O5U6
Gauss code of -K*	O1O2O3U4O5U1O6O4U2U5U6U3
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 0 2 -1 2],[ 2 0 0 2 3 0 2],[ 1 0 0 1 1 0 1],[ 0 -2 -1 0 0 0 1],[-2 -3 -1 0 0 -2 0],[ 1 0 0 0 2 0 2],[-2 -2 -1 -1 0 -2 0]]
Primitive based matrix	[[ 0 2 2 0 -1 -1 -2],[-2 0 0 0 -1 -2 -3],[-2 0 0 -1 -1 -2 -2],[ 0 0 1 0 -1 0 -2],[ 1 1 1 1 0 0 0],[ 1 2 2 0 0 0 0],[ 2 3 2 2 0 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,1,1,2,0,0,1,2,3,1,1,2,2,1,0,2,0,0,0]
Phi over symmetry	[-2,-2,0,1,1,2,0,0,1,2,3,1,1,2,2,1,0,2,0,0,0]
Phi of -K	[-2,-1,-1,0,2,2,1,1,0,1,2,0,0,2,2,1,1,1,2,1,0]
Phi of K*	[-2,-2,0,1,1,2,0,1,1,2,2,2,1,2,1,1,0,0,0,1,1]
Phi of -K*	[-2,-1,-1,0,2,2,0,0,2,2,3,0,0,2,2,1,1,1,1,0,0]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	5z^2+24z+29
Enhanced Jones-Krushkal polynomial	5w^3z^2+24w^2z+29w
Inner characteristic polynomial	t^6+29t^4+60t^2+4
Outer characteristic polynomial	t^7+43t^5+107t^3+9t
Flat arrow polynomial	-2K12 - 4K1K2 + 2K1 + K2 + 2*K3 + 2
2-strand cable arrow polynomial	-64K14K2*2 + 320K1*4K2 - 912K14 + 64K1*3K2K3 + 64K1*3K3K4 - 1120K1*3K3 + 352K12K2*3 - 2016K1*2K2*2 - 480K1*2K2K4 + 6496K1*2K2 - 368K12K3*2 - 32K1*2K3K5 - 112K1*2K4*2 - 5920K1*2 + 160K1K23K3 - 1184K1K2*2K3 - 256K1K2K3K4 + 5304K1K2K3 + 1640K1K3K4 + 240K1K4K5 - 632K2*4 - 160K2*2K3*2 - 16K2*2K4*2 + 1600K2*2K4 - 4236K22 + 296K2K3K5 + 32K2K4K6 - 2156K3*2 - 1038K4*2 - 156K5*2 - 12K6**2 + 4300
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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {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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