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Flat knot 6.782

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.782', '6.1236', '7.38129', '7.41463']
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+38t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.782']
2-strand cable arrow polynomial of the knot is: -768K16 - 768K1*4K2*2 + 5216K1*4K2 - 9760K14 + 1184K1*3K2K3 - 1888K1*3K3 - 192K12K2*4 + 3936K1*2K2*3 + 192K1*2K2*2K4 - 12784K12K2*2 - 1568K1*2K2K4 + 11536K1*2K2 - 544K12K3*2 - 48K1*2K4*2 + 372K1*2 + 672K1K23K3 - 1440K1K2*2K3 - 224K1K2*2K5 - 192K1K2K3K4 + 7320K1K2K3 + 584K1K3K4 + 64K1K4K5 - 32K2*6 + 64K2*4K4 - 2296K24 - 352K2*2K3*2 - 48K2*2K4*2 + 1496K2*2K4 - 1286K22 + 200K2K3K5 + 16K2K4K6 - 724K3*2 - 186K4*2 - 24K5*2 - 2K6**2 + 2272
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.782']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20', 'vk6.30', 'vk6.39', 'vk6.149', 'vk6.156', 'vk6.164', 'vk6.173', 'vk6.1198', 'vk6.1207', 'vk6.1293', 'vk6.1302', 'vk6.1311', 'vk6.2359', 'vk6.2392', 'vk6.2399', 'vk6.2956', 'vk6.3529', 'vk6.3552', 'vk6.6905', 'vk6.6928', 'vk6.6936', 'vk6.6961', 'vk6.15381', 'vk6.15388', 'vk6.15500', 'vk6.33433', 'vk6.33452', 'vk6.33488', 'vk6.33509', 'vk6.33599', 'vk6.49934', 'vk6.53747']
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,-1,1,1,1,2,1,0,1,1,0,1,0,0,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.782', '6.1236', '7.38129', '7.41463']

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+38t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.782']

2-strand cable arrow polynomial of the knot is: -768*K1**6 - 768*K1**4*K2**2 + 5216*K1**4*K2 - 9760*K1**4 + 1184*K1**3*K2*K3 - 1888*K1**3*K3 - 192*K1**2*K2**4 + 3936*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 12784*K1**2*K2**2 - 1568*K1**2*K2*K4 + 11536*K1**2*K2 - 544*K1**2*K3**2 - 48*K1**2*K4**2 + 372*K1**2 + 672*K1*K2**3*K3 - 1440*K1*K2**2*K3 - 224*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 7320*K1*K2*K3 + 584*K1*K3*K4 + 64*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2296*K2**4 - 352*K2**2*K3**2 - 48*K2**2*K4**2 + 1496*K2**2*K4 - 1286*K2**2 + 200*K2*K3*K5 + 16*K2*K4*K6 - 724*K3**2 - 186*K4**2 - 24*K5**2 - 2*K6**2 + 2272

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.782']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20', 'vk6.30', 'vk6.39', 'vk6.149', 'vk6.156', 'vk6.164', 'vk6.173', 'vk6.1198', 'vk6.1207', 'vk6.1293', 'vk6.1302', 'vk6.1311', 'vk6.2359', 'vk6.2392', 'vk6.2399', 'vk6.2956', 'vk6.3529', 'vk6.3552', 'vk6.6905', 'vk6.6928', 'vk6.6936', 'vk6.6961', 'vk6.15381', 'vk6.15388', 'vk6.15500', 'vk6.33433', 'vk6.33452', 'vk6.33488', 'vk6.33509', 'vk6.33599', 'vk6.49934', 'vk6.53747']

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	O1O2O3O4U3O5U4U6U5O6U1U2
R3 orbit	{'O1O2O3O4U3O5U4U6U5O6U1U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U4O5U6U5U1O6U2
Gauss code of K*	O1O2O3U2O4O5U4U5U6U1O6U3
Gauss code of -K*	O1O2O3U1O4U3U4U5U6O5O6U2
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 0 2 -1],[ 1 0 1 -1 0 2 0],[-1 -1 0 -1 0 2 -2],[ 1 1 1 0 1 1 1],[ 0 0 0 -1 0 1 -1],[-2 -2 -2 -1 -1 0 -2],[ 1 0 2 -1 1 2 0]]
Primitive based matrix	[[ 0 2 1 0 -1 -1 -1],[-2 0 -2 -1 -1 -2 -2],[-1 2 0 0 -1 -1 -2],[ 0 1 0 0 -1 0 -1],[ 1 1 1 1 0 1 1],[ 1 2 1 0 -1 0 0],[ 1 2 2 1 -1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,1,1,1,2,1,1,2,2,0,1,1,2,1,0,1,-1,-1,0]
Phi over symmetry	[-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,0,-1,-1]
Phi of -K	[-1,-1,-1,0,1,2,-1,-1,0,1,2,0,0,0,1,1,1,1,1,1,-1]
Phi of K*	[-2,-1,0,1,1,1,-1,1,1,1,2,1,0,1,1,0,1,0,0,-1,-1]
Phi of -K*	[-1,-1,-1,0,1,2,-1,0,0,1,2,1,1,1,1,1,2,2,0,1,2]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	3z^2+22z+33
Enhanced Jones-Krushkal polynomial	3w^3z^2+22w^2z+33w
Inner characteristic polynomial	t^6+24t^4+21t^2+4
Outer characteristic polynomial	t^7+32t^5+38t^3+9t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-768K16 - 768K1*4K2*2 + 5216K1*4K2 - 9760K14 + 1184K1*3K2K3 - 1888K1*3K3 - 192K12K2*4 + 3936K1*2K2*3 + 192K1*2K2*2K4 - 12784K12K2*2 - 1568K1*2K2K4 + 11536K1*2K2 - 544K12K3*2 - 48K1*2K4*2 + 372K1*2 + 672K1K23K3 - 1440K1K2*2K3 - 224K1K2*2K5 - 192K1K2K3K4 + 7320K1K2K3 + 584K1K3K4 + 64K1K4K5 - 32K2*6 + 64K2*4K4 - 2296K24 - 352K2*2K3*2 - 48K2*2K4*2 + 1496K2*2K4 - 1286K22 + 200K2K3K5 + 16K2K4K6 - 724K3*2 - 186K4*2 - 24K5*2 - 2K6**2 + 2272
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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