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Flat knot 6.1077

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,1,2,2,1,0,1,1,0,1,1,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1077']
Arrow polynomial of the knot is: 4K13 - 10K1*2 - 4K1K2 - K1 + 5K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874']
Outer characteristic polynomial of the knot is: t^7+24t^5+37t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1077']
2-strand cable arrow polynomial of the knot is: -192K16 - 256K1*4K2*2 + 2976K1*4K2 - 6032K14 - 128K1*3K2*2K3 + 896K13K2K3 + 32K1*3K3K4 - 1536K1*3K3 - 192K12K2*4 + 1664K1*2K2*3 + 224K1*2K2*2K4 - 8848K12K2*2 + 128K1*2K2K32 - 928K1*2K2K4 + 12256K1*2K2 - 1072K12K3*2 - 112K1*2K4*2 - 5376K1*2 + 448K1K23K3 - 1184K1K2*2K3 - 256K1K2*2K5 - 96K1K2K3K4 + 8240K1K2K3 + 1240K1K3K4 + 96K1K4K5 - 32K2*6 + 64K2*4K4 - 1688K24 - 32K2*3K6 - 256K22K3*2 - 16K2*2K4*2 + 1480K2*2K4 - 4230K22 + 176K2K3K5 + 16K2K4K6 - 1872K3*2 - 454K4*2 - 40K5*2 - 2K6**2 + 4892
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1077']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11479', 'vk6.11782', 'vk6.12797', 'vk6.13132', 'vk6.17053', 'vk6.17294', 'vk6.20859', 'vk6.20947', 'vk6.22267', 'vk6.22359', 'vk6.23773', 'vk6.28325', 'vk6.31236', 'vk6.31585', 'vk6.32805', 'vk6.35568', 'vk6.36017', 'vk6.39949', 'vk6.40109', 'vk6.42027', 'vk6.42969', 'vk6.43264', 'vk6.46491', 'vk6.46627', 'vk6.52244', 'vk6.53077', 'vk6.53397', 'vk6.55460', 'vk6.58855', 'vk6.59941', 'vk6.64411', 'vk6.69719']
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,-1,0,1,2,2,1,0,1,1,0,1,1,0,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1077']

Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874']

Outer characteristic polynomial of the knot is: t^7+24t^5+37t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1077']

2-strand cable arrow polynomial of the knot is: -192*K1**6 - 256*K1**4*K2**2 + 2976*K1**4*K2 - 6032*K1**4 - 128*K1**3*K2**2*K3 + 896*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1536*K1**3*K3 - 192*K1**2*K2**4 + 1664*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 8848*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 928*K1**2*K2*K4 + 12256*K1**2*K2 - 1072*K1**2*K3**2 - 112*K1**2*K4**2 - 5376*K1**2 + 448*K1*K2**3*K3 - 1184*K1*K2**2*K3 - 256*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 8240*K1*K2*K3 + 1240*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 1688*K2**4 - 32*K2**3*K6 - 256*K2**2*K3**2 - 16*K2**2*K4**2 + 1480*K2**2*K4 - 4230*K2**2 + 176*K2*K3*K5 + 16*K2*K4*K6 - 1872*K3**2 - 454*K4**2 - 40*K5**2 - 2*K6**2 + 4892

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1077']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11479', 'vk6.11782', 'vk6.12797', 'vk6.13132', 'vk6.17053', 'vk6.17294', 'vk6.20859', 'vk6.20947', 'vk6.22267', 'vk6.22359', 'vk6.23773', 'vk6.28325', 'vk6.31236', 'vk6.31585', 'vk6.32805', 'vk6.35568', 'vk6.36017', 'vk6.39949', 'vk6.40109', 'vk6.42027', 'vk6.42969', 'vk6.43264', 'vk6.46491', 'vk6.46627', 'vk6.52244', 'vk6.53077', 'vk6.53397', 'vk6.55460', 'vk6.58855', 'vk6.59941', 'vk6.64411', 'vk6.69719']

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	O1O2O3O4U5U4O6U2O5U3U1U6
R3 orbit	{'O1O2O3O4U5U4O6U2O5U3U1U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U4U2O6U3O5U1U6
Gauss code of K*	O1O2O3U2U4U1U5O6O5U3O4U6
Gauss code of -K*	O1O2O3U4O5U1O6O4U6U3U5U2
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 -1 0 1 -1 2],[ 1 0 0 1 1 -1 2],[ 1 0 0 0 0 0 1],[ 0 -1 0 0 1 -1 0],[-1 -1 0 -1 0 -1 -1],[ 1 1 0 1 1 0 2],[-2 -2 -1 0 1 -2 0]]
Primitive based matrix	[[ 0 2 1 0 -1 -1 -1],[-2 0 1 0 -1 -2 -2],[-1 -1 0 -1 0 -1 -1],[ 0 0 1 0 0 -1 -1],[ 1 1 0 0 0 0 0],[ 1 2 1 1 0 0 1],[ 1 2 1 1 0 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,1,1,1,-1,0,1,2,2,1,0,1,1,0,1,1,0,0,-1]
Phi over symmetry	[-2,-1,0,1,1,1,-1,0,1,2,2,1,0,1,1,0,1,1,0,0,-1]
Phi of -K	[-1,-1,-1,0,1,2,-1,0,0,1,1,0,0,1,1,1,2,2,0,2,2]
Phi of K*	[-2,-1,0,1,1,1,2,2,1,1,2,0,1,1,2,0,0,1,-1,0,0]
Phi of -K*	[-1,-1,-1,0,1,2,-1,0,1,1,2,0,1,1,2,0,0,1,1,0,-1]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	4z^2+25z+35
Enhanced Jones-Krushkal polynomial	4w^3z^2+25w^2z+35w
Inner characteristic polynomial	t^6+16t^4+10t^2+1
Outer characteristic polynomial	t^7+24t^5+37t^3+8t
Flat arrow polynomial	4K13 - 10K1*2 - 4K1K2 - K1 + 5K2 + K3 + 6
2-strand cable arrow polynomial	-192K16 - 256K1*4K2*2 + 2976K1*4K2 - 6032K14 - 128K1*3K2*2K3 + 896K13K2K3 + 32K1*3K3K4 - 1536K1*3K3 - 192K12K2*4 + 1664K1*2K2*3 + 224K1*2K2*2K4 - 8848K12K2*2 + 128K1*2K2K32 - 928K1*2K2K4 + 12256K1*2K2 - 1072K12K3*2 - 112K1*2K4*2 - 5376K1*2 + 448K1K23K3 - 1184K1K2*2K3 - 256K1K2*2K5 - 96K1K2K3K4 + 8240K1K2K3 + 1240K1K3K4 + 96K1K4K5 - 32K2*6 + 64K2*4K4 - 1688K24 - 32K2*3K6 - 256K22K3*2 - 16K2*2K4*2 + 1480K2*2K4 - 4230K22 + 176K2K3K5 + 16K2K4K6 - 1872K3*2 - 454K4*2 - 40K5*2 - 2K6**2 + 4892
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}], [{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}, {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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