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Flat knot 6.1727

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,0,0,0,-1,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1727', '7.44672']
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+20t^5+29t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1727']
2-strand cable arrow polynomial of the knot is: -512K16 - 2048K1*4K2*2 + 3328K1*4K2 - 4640K14 + 1472K1*3K2K3 - 928K1*3K3 - 1344K12K2*4 + 3712K1*2K2*3 + 256K1*2K2*2K4 - 9792K12K2*2 - 1024K1*2K2K4 + 8264K1*2K2 - 288K12K3*2 - 1316K1*2 + 1536K1K23K3 - 1472K1K2*2K3 - 416K1K2*2K5 - 96K1K2K3K4 + 5848K1K2K3 + 344K1K3K4 + 16K1K4K5 - 32K2*6 + 64K2*4K4 - 2504K24 - 32K2*3K6 - 464K22K3*2 - 16K2*2K4*2 + 1648K2*2K4 - 1022K22 + 240K2K3K5 + 16K2K4K6 - 652K3*2 - 158K4*2 - 24K5*2 - 2K6**2 + 2036
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1727']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.487', 'vk6.555', 'vk6.614', 'vk6.961', 'vk6.1056', 'vk6.1114', 'vk6.1652', 'vk6.1763', 'vk6.1831', 'vk6.2145', 'vk6.2240', 'vk6.2299', 'vk6.2581', 'vk6.2841', 'vk6.3058', 'vk6.3175', 'vk6.12044', 'vk6.13035', 'vk6.20486', 'vk6.20997', 'vk6.21839', 'vk6.22420', 'vk6.27883', 'vk6.28448', 'vk6.29391', 'vk6.32686', 'vk6.39324', 'vk6.40217', 'vk6.41502', 'vk6.46716', 'vk6.46869', 'vk6.57348']
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,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,0,0,0,-1,-1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1727', '7.44672']

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+20t^5+29t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1727']

2-strand cable arrow polynomial of the knot is: -512*K1**6 - 2048*K1**4*K2**2 + 3328*K1**4*K2 - 4640*K1**4 + 1472*K1**3*K2*K3 - 928*K1**3*K3 - 1344*K1**2*K2**4 + 3712*K1**2*K2**3 + 256*K1**2*K2**2*K4 - 9792*K1**2*K2**2 - 1024*K1**2*K2*K4 + 8264*K1**2*K2 - 288*K1**2*K3**2 - 1316*K1**2 + 1536*K1*K2**3*K3 - 1472*K1*K2**2*K3 - 416*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 5848*K1*K2*K3 + 344*K1*K3*K4 + 16*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 2504*K2**4 - 32*K2**3*K6 - 464*K2**2*K3**2 - 16*K2**2*K4**2 + 1648*K2**2*K4 - 1022*K2**2 + 240*K2*K3*K5 + 16*K2*K4*K6 - 652*K3**2 - 158*K4**2 - 24*K5**2 - 2*K6**2 + 2036

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1727']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.487', 'vk6.555', 'vk6.614', 'vk6.961', 'vk6.1056', 'vk6.1114', 'vk6.1652', 'vk6.1763', 'vk6.1831', 'vk6.2145', 'vk6.2240', 'vk6.2299', 'vk6.2581', 'vk6.2841', 'vk6.3058', 'vk6.3175', 'vk6.12044', 'vk6.13035', 'vk6.20486', 'vk6.20997', 'vk6.21839', 'vk6.22420', 'vk6.27883', 'vk6.28448', 'vk6.29391', 'vk6.32686', 'vk6.39324', 'vk6.40217', 'vk6.41502', 'vk6.46716', 'vk6.46869', 'vk6.57348']

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	O1O2O3U2U3O4O5U6U1O6U4U5
R3 orbit	{'O1O2O3U2U3O4O5U6U1O6U4U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U5O6U3U6O4O5U1U2
Gauss code of K*	O1O2U1O3O4U2U5U6O5O6U3U4
Gauss code of -K*	O1O2U3O4O3U1U2O5O6U5U6U4
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 1 1],[ 1 1 0 1 0 0 1],[-1 -1 -1 0 0 0 -1],[ 0 0 0 0 0 1 0],[-2 -1 0 0 -1 0 -2],[ 1 -1 -1 1 0 2 0]]
Primitive based matrix	[[ 0 2 1 0 -1 -1 -1],[-2 0 0 -1 0 -1 -2],[-1 0 0 0 -1 -1 -1],[ 0 1 0 0 0 0 0],[ 1 0 1 0 0 1 1],[ 1 1 1 0 -1 0 1],[ 1 2 1 0 -1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,0,0,0,-1,-1,-1]
Phi over symmetry	[-2,-1,0,1,1,1,0,1,0,1,2,0,1,1,1,0,0,0,-1,-1,-1]
Phi of -K	[-1,-1,-1,0,1,2,-1,-1,1,1,3,-1,1,1,2,1,1,1,1,1,1]
Phi of K*	[-2,-1,0,1,1,1,1,1,1,2,3,1,1,1,1,1,1,1,-1,-1,-1]
Phi of -K*	[-1,-1,-1,0,1,2,-1,-1,0,1,2,-1,0,1,1,0,1,0,0,1,0]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	6z^2+25z+27
Enhanced Jones-Krushkal polynomial	6w^3z^2+25w^2z+27w
Inner characteristic polynomial	t^6+12t^4+14t^2+1
Outer characteristic polynomial	t^7+20t^5+29t^3+6t
Flat arrow polynomial	4K13 - 10K1*2 - 4K1K2 - K1 + 5K2 + K3 + 6
2-strand cable arrow polynomial	-512K16 - 2048K1*4K2*2 + 3328K1*4K2 - 4640K14 + 1472K1*3K2K3 - 928K1*3K3 - 1344K12K2*4 + 3712K1*2K2*3 + 256K1*2K2*2K4 - 9792K12K2*2 - 1024K1*2K2K4 + 8264K1*2K2 - 288K12K3*2 - 1316K1*2 + 1536K1K23K3 - 1472K1K2*2K3 - 416K1K2*2K5 - 96K1K2K3K4 + 5848K1K2K3 + 344K1K3K4 + 16K1K4K5 - 32K2*6 + 64K2*4K4 - 2504K24 - 32K2*3K6 - 464K22K3*2 - 16K2*2K4*2 + 1648K2*2K4 - 1022K22 + 240K2K3K5 + 16K2K4K6 - 652K3*2 - 158K4*2 - 24K5*2 - 2K6**2 + 2036
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {5}, {3, 4}, {1}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {5}, {2, 3}, {1}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice	False

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