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Flat knot 6.1533

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,2,0,1,1,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1533']
Arrow polynomial of the knot is: 4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845']
Outer characteristic polynomial of the knot is: t^7+26t^5+26t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1533']
2-strand cable arrow polynomial of the knot is: -576K16 - 320K1*4K2*2 + 2688K1*4K2 - 5440K14 + 864K1*3K2K3 - 1056K1*3K3 - 192K12K2*4 + 736K1*2K2*3 + 192K1*2K2*2K4 - 5984K12K2*2 - 864K1*2K2K4 + 9312K1*2K2 - 704K12K3*2 - 80K1*2K4*2 - 3556K1*2 + 256K1K23K3 - 736K1K2*2K3 - 96K1K2*2K5 - 192K1K2K3K4 + 5984K1K2K3 + 1304K1K3K4 + 144K1K4K5 - 32K2*6 + 64K2*4K4 - 488K24 - 96K2*2K3*2 - 48K2*2K4*2 + 736K2*2K4 - 3526K22 + 96K2K3K5 + 16K2K4K6 - 1636K3*2 - 522K4*2 - 48K5*2 - 2K6**2 + 3800
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1533']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4140', 'vk6.4171', 'vk6.5382', 'vk6.5413', 'vk6.7500', 'vk6.7527', 'vk6.9005', 'vk6.9036', 'vk6.12425', 'vk6.12456', 'vk6.13346', 'vk6.13571', 'vk6.13602', 'vk6.14269', 'vk6.14716', 'vk6.14737', 'vk6.15205', 'vk6.15876', 'vk6.15895', 'vk6.30826', 'vk6.30857', 'vk6.32014', 'vk6.32045', 'vk6.33064', 'vk6.33095', 'vk6.33864', 'vk6.34323', 'vk6.48496', 'vk6.50275', 'vk6.53518', 'vk6.53937', 'vk6.54256']
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,-1,0,1,1,1,0,1,1,1,2,1,1,1,2,0,1,1,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1533']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.361', '6.460', '6.555', '6.651', '6.753', '6.782', '6.1029', '6.1197', '6.1200', '6.1232', '6.1236', '6.1278', '6.1281', '6.1343', '6.1380', '6.1385', '6.1389', '6.1484', '6.1492', '6.1493', '6.1527', '6.1533', '6.1550', '6.1553', '6.1557', '6.1576', '6.1578', '6.1582', '6.1586', '6.1674', '6.1698', '6.1754', '6.1759', '6.1775', '6.1791', '6.1798', '6.1800', '6.1805', '6.1822', '6.1826', '6.1839', '6.1844', '6.1845']

Outer characteristic polynomial of the knot is: t^7+26t^5+26t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1533']

2-strand cable arrow polynomial of the knot is: -576*K1**6 - 320*K1**4*K2**2 + 2688*K1**4*K2 - 5440*K1**4 + 864*K1**3*K2*K3 - 1056*K1**3*K3 - 192*K1**2*K2**4 + 736*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 5984*K1**2*K2**2 - 864*K1**2*K2*K4 + 9312*K1**2*K2 - 704*K1**2*K3**2 - 80*K1**2*K4**2 - 3556*K1**2 + 256*K1*K2**3*K3 - 736*K1*K2**2*K3 - 96*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5984*K1*K2*K3 + 1304*K1*K3*K4 + 144*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 488*K2**4 - 96*K2**2*K3**2 - 48*K2**2*K4**2 + 736*K2**2*K4 - 3526*K2**2 + 96*K2*K3*K5 + 16*K2*K4*K6 - 1636*K3**2 - 522*K4**2 - 48*K5**2 - 2*K6**2 + 3800

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1533']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4140', 'vk6.4171', 'vk6.5382', 'vk6.5413', 'vk6.7500', 'vk6.7527', 'vk6.9005', 'vk6.9036', 'vk6.12425', 'vk6.12456', 'vk6.13346', 'vk6.13571', 'vk6.13602', 'vk6.14269', 'vk6.14716', 'vk6.14737', 'vk6.15205', 'vk6.15876', 'vk6.15895', 'vk6.30826', 'vk6.30857', 'vk6.32014', 'vk6.32045', 'vk6.33064', 'vk6.33095', 'vk6.33864', 'vk6.34323', 'vk6.48496', 'vk6.50275', 'vk6.53518', 'vk6.53937', 'vk6.54256']

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	O1O2O3U2O4U5U3O6O5U6U1U4
R3 orbit	{'O1O2O3U2O4U5U3O6O5U6U1U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U3U5O6O5U1U6O4U2
Gauss code of K*	O1O2O3U2U4U5O4U3O6O5U1U6
Gauss code of -K*	O1O2O3U4U3O5O4U1O6U5U6U2
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 1 2 0 -1],[ 1 0 -1 2 2 1 -1],[ 1 1 0 1 1 0 0],[-1 -2 -1 0 0 -1 -1],[-2 -2 -1 0 0 -1 -1],[ 0 -1 0 1 1 0 -1],[ 1 1 0 1 1 1 0]]
Primitive based matrix	[[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 -1 -1 -2],[-1 0 0 -1 -1 -1 -2],[ 0 1 1 0 0 -1 -1],[ 1 1 1 0 0 0 1],[ 1 1 1 1 0 0 1],[ 1 2 2 1 -1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,2,0,1,1,0,-1,-1]
Phi over symmetry	[-2,-1,0,1,1,1,0,1,1,1,2,1,1,1,2,0,1,1,0,-1,-1]
Phi of -K	[-1,-1,-1,0,1,2,-1,0,0,1,2,1,0,0,1,1,1,2,0,1,1]
Phi of K*	[-2,-1,0,1,1,1,1,1,1,2,2,0,0,1,1,0,0,1,-1,-1,0]
Phi of -K*	[-1,-1,-1,0,1,2,-1,-1,1,2,2,0,0,1,1,1,1,1,1,1,0]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	2z^2+23z+39
Enhanced Jones-Krushkal polynomial	2w^3z^2+23w^2z+39w
Inner characteristic polynomial	t^6+18t^4+11t^2
Outer characteristic polynomial	t^7+26t^5+26t^3+6t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-576K16 - 320K1*4K2*2 + 2688K1*4K2 - 5440K14 + 864K1*3K2K3 - 1056K1*3K3 - 192K12K2*4 + 736K1*2K2*3 + 192K1*2K2*2K4 - 5984K12K2*2 - 864K1*2K2K4 + 9312K1*2K2 - 704K12K3*2 - 80K1*2K4*2 - 3556K1*2 + 256K1K23K3 - 736K1K2*2K3 - 96K1K2*2K5 - 192K1K2K3K4 + 5984K1K2K3 + 1304K1K3K4 + 144K1K4K5 - 32K2*6 + 64K2*4K4 - 488K24 - 96K2*2K3*2 - 48K2*2K4*2 + 736K2*2K4 - 3526K22 + 96K2K3K5 + 16K2K4K6 - 1636K3*2 - 522K4*2 - 48K5*2 - 2K6**2 + 3800
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}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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