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Flat knot 6.1278

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,1,2,1,-1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1278']
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+49t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1278']
2-strand cable arrow polynomial of the knot is: 2784K14K2 - 5856K14 + 960K1*3K2K3 - 1408K1*3K3 - 128K12K2*4 + 928K1*2K2*3 + 128K1*2K2*2K4 - 7776K12K2*2 - 576K1*2K2K4 + 10672K1*2K2 - 1152K12K3*2 - 32K1*2K4*2 - 4268K1*2 + 416K1K23K3 - 1504K1K2*2K3 - 288K1K2*2K5 - 160K1K2K3K4 + 7792K1K2K3 + 1344K1K3K4 + 104K1K4K5 - 32K2*6 + 64K2*4K4 - 888K24 - 32K2*3K6 - 176K22K3*2 - 16K2*2K4*2 + 1328K2*2K4 - 4286K22 + 240K2K3K5 + 16K2K4K6 - 1976K3*2 - 530K4*2 - 68K5*2 - 2K6**2 + 4376
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1278']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4374', 'vk6.4405', 'vk6.5692', 'vk6.5723', 'vk6.7757', 'vk6.7788', 'vk6.9235', 'vk6.9266', 'vk6.10472', 'vk6.10546', 'vk6.10643', 'vk6.10695', 'vk6.10726', 'vk6.10828', 'vk6.14610', 'vk6.15322', 'vk6.15449', 'vk6.16229', 'vk6.17987', 'vk6.24425', 'vk6.30159', 'vk6.30233', 'vk6.30330', 'vk6.30455', 'vk6.33964', 'vk6.34369', 'vk6.34425', 'vk6.43852', 'vk6.50443', 'vk6.50474', 'vk6.54198', 'vk6.63433']
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,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,1,2,1,-1,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1278']

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+49t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1278']

2-strand cable arrow polynomial of the knot is: 2784*K1**4*K2 - 5856*K1**4 + 960*K1**3*K2*K3 - 1408*K1**3*K3 - 128*K1**2*K2**4 + 928*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 7776*K1**2*K2**2 - 576*K1**2*K2*K4 + 10672*K1**2*K2 - 1152*K1**2*K3**2 - 32*K1**2*K4**2 - 4268*K1**2 + 416*K1*K2**3*K3 - 1504*K1*K2**2*K3 - 288*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 7792*K1*K2*K3 + 1344*K1*K3*K4 + 104*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 888*K2**4 - 32*K2**3*K6 - 176*K2**2*K3**2 - 16*K2**2*K4**2 + 1328*K2**2*K4 - 4286*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 1976*K3**2 - 530*K4**2 - 68*K5**2 - 2*K6**2 + 4376

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1278']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4374', 'vk6.4405', 'vk6.5692', 'vk6.5723', 'vk6.7757', 'vk6.7788', 'vk6.9235', 'vk6.9266', 'vk6.10472', 'vk6.10546', 'vk6.10643', 'vk6.10695', 'vk6.10726', 'vk6.10828', 'vk6.14610', 'vk6.15322', 'vk6.15449', 'vk6.16229', 'vk6.17987', 'vk6.24425', 'vk6.30159', 'vk6.30233', 'vk6.30330', 'vk6.30455', 'vk6.33964', 'vk6.34369', 'vk6.34425', 'vk6.43852', 'vk6.50443', 'vk6.50474', 'vk6.54198', 'vk6.63433']

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	O1O2O3U1O4O5U4U3O6U5U6U2
R3 orbit	{'O1O2O3U1O4O5U4U3O6U5U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3U2U4U5O4U1U6O5O6U3
Gauss code of K*	O1O2O3U4U3U5O4U6U1O6O5U2
Gauss code of -K*	O1O2O3U2O4O5U3U5O6U4U1U6
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 -1 1 1],[ 2 0 2 1 0 1 0],[-1 -2 0 -1 -1 1 1],[ 0 -1 1 0 0 2 1],[ 1 0 1 0 0 1 1],[-1 -1 -1 -2 -1 0 1],[-1 0 -1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 1 1 1 0 -1 -2],[-1 0 1 1 -1 -1 -2],[-1 -1 0 1 -2 -1 -1],[-1 -1 -1 0 -1 -1 0],[ 0 1 2 1 0 0 -1],[ 1 1 1 1 0 0 0],[ 2 2 1 0 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,0,1,2,-1,-1,1,1,2,-1,2,1,1,1,1,0,0,1,0]
Phi over symmetry	[-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,1,2,1,-1,-1,-1]
Phi of -K	[-2,-1,0,1,1,1,1,1,1,2,3,1,1,1,1,0,-1,0,-1,-1,-1]
Phi of K*	[-1,-1,-1,0,1,2,-1,-1,0,1,3,-1,-1,1,2,0,1,1,1,1,1]
Phi of -K*	[-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,1,2,1,-1,-1,-1]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	6z^2+27z+31
Enhanced Jones-Krushkal polynomial	6w^3z^2+27w^2z+31w
Inner characteristic polynomial	t^6+18t^4+18t^2+1
Outer characteristic polynomial	t^7+26t^5+49t^3+8t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	2784K14K2 - 5856K14 + 960K1*3K2K3 - 1408K1*3K3 - 128K12K2*4 + 928K1*2K2*3 + 128K1*2K2*2K4 - 7776K12K2*2 - 576K1*2K2K4 + 10672K1*2K2 - 1152K12K3*2 - 32K1*2K4*2 - 4268K1*2 + 416K1K23K3 - 1504K1K2*2K3 - 288K1K2*2K5 - 160K1K2K3K4 + 7792K1K2K3 + 1344K1K3K4 + 104K1K4K5 - 32K2*6 + 64K2*4K4 - 888K24 - 32K2*3K6 - 176K22K3*2 - 16K2*2K4*2 + 1328K2*2K4 - 4286K22 + 240K2K3K5 + 16K2K4K6 - 1976K3*2 - 530K4*2 - 68K5*2 - 2K6**2 + 4376
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}], [{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}, {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}]]
If K is slice	False

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