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Flat knot 6.1874

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,0,1,1,2,0,1,1,0,1,1,0,0,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1874']
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+17t^5+44t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1874']
2-strand cable arrow polynomial of the knot is: -320K14K2*2 + 896K1*4K2 - 2224K14 + 224K1*3K2K3 + 32K1*3K3K4 + 512K1*2K2*3 - 3328K1*2K2*2 + 4480K1*2K2 - 464K12K3*2 - 64K1*2K4*2 - 2812K1*2 + 160K1K23K3 + 3776K1K2K3 + 664K1K3K4 + 48K1K4K5 - 32K2*6 + 64K2*4K4 - 696K24 - 368K2*2K3*2 - 48K2*2K4*2 + 568K2*2K4 - 2334K22 + 224K2K3K5 + 16K2K4K6 - 1384K3*2 - 426K4*2 - 52K5*2 - 2K6**2 + 2888
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1874']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17108', 'vk6.17349', 'vk6.20579', 'vk6.21987', 'vk6.23501', 'vk6.23838', 'vk6.28044', 'vk6.29502', 'vk6.35642', 'vk6.36081', 'vk6.39447', 'vk6.41648', 'vk6.43008', 'vk6.43318', 'vk6.46035', 'vk6.47703', 'vk6.55259', 'vk6.55509', 'vk6.57445', 'vk6.58615', 'vk6.59667', 'vk6.60013', 'vk6.62120', 'vk6.63088', 'vk6.65059', 'vk6.65252', 'vk6.66975', 'vk6.67839', 'vk6.68322', 'vk6.68470', 'vk6.69593', 'vk6.70285']
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: [-2,0,0,0,1,1,0,0,1,1,2,0,1,1,0,1,1,0,0,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1874']

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+17t^5+44t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1874']

2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 896*K1**4*K2 - 2224*K1**4 + 224*K1**3*K2*K3 + 32*K1**3*K3*K4 + 512*K1**2*K2**3 - 3328*K1**2*K2**2 + 4480*K1**2*K2 - 464*K1**2*K3**2 - 64*K1**2*K4**2 - 2812*K1**2 + 160*K1*K2**3*K3 + 3776*K1*K2*K3 + 664*K1*K3*K4 + 48*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 696*K2**4 - 368*K2**2*K3**2 - 48*K2**2*K4**2 + 568*K2**2*K4 - 2334*K2**2 + 224*K2*K3*K5 + 16*K2*K4*K6 - 1384*K3**2 - 426*K4**2 - 52*K5**2 - 2*K6**2 + 2888

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1874']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17108', 'vk6.17349', 'vk6.20579', 'vk6.21987', 'vk6.23501', 'vk6.23838', 'vk6.28044', 'vk6.29502', 'vk6.35642', 'vk6.36081', 'vk6.39447', 'vk6.41648', 'vk6.43008', 'vk6.43318', 'vk6.46035', 'vk6.47703', 'vk6.55259', 'vk6.55509', 'vk6.57445', 'vk6.58615', 'vk6.59667', 'vk6.60013', 'vk6.62120', 'vk6.63088', 'vk6.65059', 'vk6.65252', 'vk6.66975', 'vk6.67839', 'vk6.68322', 'vk6.68470', 'vk6.69593', 'vk6.70285']

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	O1O2O3U4U2O5U1O6O4U6U5U3
R3 orbit	{'O1O2O3U4U2O5U1O6O4U6U5U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3U1U4U5O6O5U3O4U2U6
Gauss code of K*	O1O2O3U4U5U3O6O5U2O4U1U6
Gauss code of -K*	O1O2O3U4U3O5U2O6O4U1U6U5
Diagrammatic symmetry type	c
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -1 0 2 0 0 -1],[ 1 0 0 2 0 0 -1],[ 0 0 0 0 0 -1 -1],[-2 -2 0 0 0 -1 -1],[ 0 0 0 0 0 -1 -1],[ 0 0 1 1 1 0 0],[ 1 1 1 1 1 0 0]]
Primitive based matrix	[[ 0 2 0 0 0 -1 -1],[-2 0 0 0 -1 -1 -2],[ 0 0 0 0 -1 -1 0],[ 0 0 0 0 -1 -1 0],[ 0 1 1 1 0 0 0],[ 1 1 1 1 0 0 1],[ 1 2 0 0 0 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,0,0,0,1,1,0,0,1,1,2,0,1,1,0,1,1,0,0,0,-1]
Phi over symmetry	[-2,0,0,0,1,1,0,0,1,1,2,0,1,1,0,1,1,0,0,0,-1]
Phi of -K	[-1,-1,0,0,0,2,-1,0,0,1,2,1,1,1,1,0,1,2,1,2,1]
Phi of K*	[-2,0,0,0,1,1,1,2,2,1,2,1,1,1,1,0,1,0,1,0,-1]
Phi of -K*	[-1,-1,0,0,0,2,-1,0,0,0,2,0,1,1,1,1,1,1,0,0,0]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	15z+31
Enhanced Jones-Krushkal polynomial	-2w^3z+17w^2z+31w
Inner characteristic polynomial	t^6+11t^4+19t^2
Outer characteristic polynomial	t^7+17t^5+44t^3
Flat arrow polynomial	4K13 - 10K1*2 - 4K1K2 - K1 + 5K2 + K3 + 6
2-strand cable arrow polynomial	-320K14K2*2 + 896K1*4K2 - 2224K14 + 224K1*3K2K3 + 32K1*3K3K4 + 512K1*2K2*3 - 3328K1*2K2*2 + 4480K1*2K2 - 464K12K3*2 - 64K1*2K4*2 - 2812K1*2 + 160K1K23K3 + 3776K1K2K3 + 664K1K3K4 + 48K1K4K5 - 32K2*6 + 64K2*4K4 - 696K24 - 368K2*2K3*2 - 48K2*2K4*2 + 568K2*2K4 - 2334K22 + 224K2K3K5 + 16K2K4K6 - 1384K3*2 - 426K4*2 - 52K5*2 - 2K6**2 + 2888
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}], [{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}], [{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}], [{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}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}], [{6}, {5}, {4}, {3}, {2}, {1}]]
If K is slice	False

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