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Flat knot 6.1754

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,1,2,2,1,0,0,1,1,0,1,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1754']
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+32t^5+60t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1754']
2-strand cable arrow polynomial of the knot is: -64K16 - 896K1*4K2*2 + 3264K1*4K2 - 5296K14 - 384K1*3K2*2K3 + 640K13K2K3 - 896K1*3K3 + 384K12K2*5 - 960K1*2K2*4 - 384K1*2K2*3K4 + 2752K12K2*3 + 576K1*2K2*2K4 - 8176K12K2*2 + 384K1*2K2K32 + 96K1*2K2K42 - 928K1*2K2K4 + 8440K1*2K2 - 1264K12K3*2 - 96K1*2K3K5 - 144K1*2K4*2 - 3056K1*2 + 1056K1K23K3 - 896K1K2*2K3 - 320K1K2*2K5 - 160K1K2K3K4 + 6504K1K2K3 + 1360K1K3K4 + 88K1K4K5 - 288K2*6 + 352K2*4K4 - 1288K24 - 32K2*3K6 - 448K22K3*2 - 112K2*2K4*2 + 896K2*2K4 - 2582K22 + 240K2K3K5 + 24K2K4K6 - 1376K3*2 - 386K4*2 - 40K5*2 - 2K6**2 + 3416
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1754']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16343', 'vk6.16344', 'vk6.16385', 'vk6.16387', 'vk6.19185', 'vk6.19191', 'vk6.19478', 'vk6.19484', 'vk6.22770', 'vk6.22772', 'vk6.25986', 'vk6.25990', 'vk6.26375', 'vk6.26377', 'vk6.34634', 'vk6.34635', 'vk6.34721', 'vk6.34723', 'vk6.38083', 'vk6.38089', 'vk6.42355', 'vk6.42356', 'vk6.44571', 'vk6.44579', 'vk6.54610', 'vk6.54611', 'vk6.56537', 'vk6.56541', 'vk6.59138', 'vk6.59140', 'vk6.66261', 'vk6.66263']
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,-1,0,1,2,2,1,0,0,1,1,0,1,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1754']

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+32t^5+60t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1754']

2-strand cable arrow polynomial of the knot is: -64*K1**6 - 896*K1**4*K2**2 + 3264*K1**4*K2 - 5296*K1**4 - 384*K1**3*K2**2*K3 + 640*K1**3*K2*K3 - 896*K1**3*K3 + 384*K1**2*K2**5 - 960*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 2752*K1**2*K2**3 + 576*K1**2*K2**2*K4 - 8176*K1**2*K2**2 + 384*K1**2*K2*K3**2 + 96*K1**2*K2*K4**2 - 928*K1**2*K2*K4 + 8440*K1**2*K2 - 1264*K1**2*K3**2 - 96*K1**2*K3*K5 - 144*K1**2*K4**2 - 3056*K1**2 + 1056*K1*K2**3*K3 - 896*K1*K2**2*K3 - 320*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 6504*K1*K2*K3 + 1360*K1*K3*K4 + 88*K1*K4*K5 - 288*K2**6 + 352*K2**4*K4 - 1288*K2**4 - 32*K2**3*K6 - 448*K2**2*K3**2 - 112*K2**2*K4**2 + 896*K2**2*K4 - 2582*K2**2 + 240*K2*K3*K5 + 24*K2*K4*K6 - 1376*K3**2 - 386*K4**2 - 40*K5**2 - 2*K6**2 + 3416

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1754']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16343', 'vk6.16344', 'vk6.16385', 'vk6.16387', 'vk6.19185', 'vk6.19191', 'vk6.19478', 'vk6.19484', 'vk6.22770', 'vk6.22772', 'vk6.25986', 'vk6.25990', 'vk6.26375', 'vk6.26377', 'vk6.34634', 'vk6.34635', 'vk6.34721', 'vk6.34723', 'vk6.38083', 'vk6.38089', 'vk6.42355', 'vk6.42356', 'vk6.44571', 'vk6.44579', 'vk6.54610', 'vk6.54611', 'vk6.56537', 'vk6.56541', 'vk6.59138', 'vk6.59140', 'vk6.66261', 'vk6.66263']

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	O1O2O3U4U5O4O6U1U3O5U6U2
R3 orbit	{'O1O2O3U4U5O4O6U1U3O5U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3U2U4O5U1U3O4O6U5U6
Gauss code of K*	O1O2U3O4O5U1U5U2O6O3U6U4
Gauss code of -K*	O1O2U3O4O5U2U6O3O6U4U1U5
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 1 -1 0 1],[ 2 0 2 1 2 2 1],[-1 -2 0 0 -2 0 0],[-1 -1 0 0 -1 0 0],[ 1 -2 2 1 0 0 2],[ 0 -2 0 0 0 0 1],[-1 -1 0 0 -2 -1 0]]
Primitive based matrix	[[ 0 1 1 1 0 -1 -2],[-1 0 0 0 0 -1 -1],[-1 0 0 0 0 -2 -2],[-1 0 0 0 -1 -2 -1],[ 0 0 0 1 0 0 -2],[ 1 1 2 2 0 0 -2],[ 2 1 2 1 2 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,1,2,0,0,0,1,1,0,0,2,2,1,2,1,0,2,2]
Phi over symmetry	[-2,-1,0,1,1,1,-1,0,1,2,2,1,0,0,1,1,0,1,0,0,0]
Phi of -K	[-2,-1,0,1,1,1,-1,0,1,2,2,1,0,0,1,1,0,1,0,0,0]
Phi of K*	[-1,-1,-1,0,1,2,0,0,0,0,2,0,1,0,1,1,1,2,1,0,-1]
Phi of -K*	[-2,-1,0,1,1,1,2,2,1,1,2,0,1,2,2,0,1,0,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	4z^2+21z+27
Enhanced Jones-Krushkal polynomial	-2w^4z^2+6w^3z^2-2w^3z+23w^2z+27w
Inner characteristic polynomial	t^6+24t^4+39t^2
Outer characteristic polynomial	t^7+32t^5+60t^3+8t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-64K16 - 896K1*4K2*2 + 3264K1*4K2 - 5296K14 - 384K1*3K2*2K3 + 640K13K2K3 - 896K1*3K3 + 384K12K2*5 - 960K1*2K2*4 - 384K1*2K2*3K4 + 2752K12K2*3 + 576K1*2K2*2K4 - 8176K12K2*2 + 384K1*2K2K32 + 96K1*2K2K42 - 928K1*2K2K4 + 8440K1*2K2 - 1264K12K3*2 - 96K1*2K3K5 - 144K1*2K4*2 - 3056K1*2 + 1056K1K23K3 - 896K1K2*2K3 - 320K1K2*2K5 - 160K1K2K3K4 + 6504K1K2K3 + 1360K1K3K4 + 88K1K4K5 - 288K2*6 + 352K2*4K4 - 1288K24 - 32K2*3K6 - 448K22K3*2 - 112K2*2K4*2 + 896K2*2K4 - 2582K22 + 240K2K3K5 + 24K2K4K6 - 1376K3*2 - 386K4*2 - 40K5*2 - 2K6**2 + 3416
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}, {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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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