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Flat knot 6.1233

Min(phi) over symmetries of the knot is: [-1,-1,-1,1,1,1,-1,-1,1,1,1,0,0,0,1,1,1,0,-1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1233', '7.24434', '7.41351', '7.41492']
Arrow polynomial of the knot is: 8K13 - 8K1*2 - 8K1K2 - 2K1 + 4K2 + 2K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923']
Outer characteristic polynomial of the knot is: t^7+15t^5+27t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1233', '7.41351']
2-strand cable arrow polynomial of the knot is: -2048K16 - 4352K1*4K2*2 + 7424K1*4K2 - 7072K14 + 3520K1*3K2K3 - 1056K1*3K3 - 3136K12K2*4 + 7552K1*2K2*3 + 1024K1*2K2*2K4 - 15872K12K2*2 - 1600K1*2K2K4 + 9200K1*2K2 - 896K12K3*2 - 224K1*2K4*2 + 1448K1*2 + 3648K1K23K3 - 3104K1K2*2K3 - 800K1K2*2K5 - 448K1K2K3K4 + 7104K1K2K3 + 592K1K3K4 + 144K1K4K5 - 704K2*6 + 1088K2*4K4 - 4256K24 - 192K2*3K6 - 960K22K3*2 - 384K2*2K4*2 + 2416K2*2K4 + 708K22 + 336K2K3K5 + 112K2K4K6 - 404K3*2 - 120K4*2 - 20K5*2 - 4K6**2 + 1334
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1233']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.76', 'vk6.78', 'vk6.137', 'vk6.141', 'vk6.234', 'vk6.238', 'vk6.277', 'vk6.279', 'vk6.388', 'vk6.392', 'vk6.808', 'vk6.816', 'vk6.1285', 'vk6.1289', 'vk6.1376', 'vk6.1380', 'vk6.1419', 'vk6.1421', 'vk6.1553', 'vk6.1561', 'vk6.2685', 'vk6.2693', 'vk6.2940', 'vk6.2944', 'vk6.14845', 'vk6.14849', 'vk6.16003', 'vk6.16007', 'vk6.25958', 'vk6.25966', 'vk6.33348', 'vk6.33364']
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: [-1,-1,-1,1,1,1,-1,-1,1,1,1,0,0,0,1,1,1,0,-1,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1233', '7.24434', '7.41351', '7.41492']

Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 8*K1*K2 - 2*K1 + 4*K2 + 2*K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.414', '6.594', '6.608', '6.790', '6.1233', '6.1285', '6.1293', '6.1513', '6.1752', '6.1787', '6.1810', '6.1818', '6.1867', '6.1868', '6.1923']

Outer characteristic polynomial of the knot is: t^7+15t^5+27t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1233', '7.41351']

2-strand cable arrow polynomial of the knot is: -2048*K1**6 - 4352*K1**4*K2**2 + 7424*K1**4*K2 - 7072*K1**4 + 3520*K1**3*K2*K3 - 1056*K1**3*K3 - 3136*K1**2*K2**4 + 7552*K1**2*K2**3 + 1024*K1**2*K2**2*K4 - 15872*K1**2*K2**2 - 1600*K1**2*K2*K4 + 9200*K1**2*K2 - 896*K1**2*K3**2 - 224*K1**2*K4**2 + 1448*K1**2 + 3648*K1*K2**3*K3 - 3104*K1*K2**2*K3 - 800*K1*K2**2*K5 - 448*K1*K2*K3*K4 + 7104*K1*K2*K3 + 592*K1*K3*K4 + 144*K1*K4*K5 - 704*K2**6 + 1088*K2**4*K4 - 4256*K2**4 - 192*K2**3*K6 - 960*K2**2*K3**2 - 384*K2**2*K4**2 + 2416*K2**2*K4 + 708*K2**2 + 336*K2*K3*K5 + 112*K2*K4*K6 - 404*K3**2 - 120*K4**2 - 20*K5**2 - 4*K6**2 + 1334

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1233']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.76', 'vk6.78', 'vk6.137', 'vk6.141', 'vk6.234', 'vk6.238', 'vk6.277', 'vk6.279', 'vk6.388', 'vk6.392', 'vk6.808', 'vk6.816', 'vk6.1285', 'vk6.1289', 'vk6.1376', 'vk6.1380', 'vk6.1419', 'vk6.1421', 'vk6.1553', 'vk6.1561', 'vk6.2685', 'vk6.2693', 'vk6.2940', 'vk6.2944', 'vk6.14845', 'vk6.14849', 'vk6.16003', 'vk6.16007', 'vk6.25958', 'vk6.25966', 'vk6.33348', 'vk6.33364']

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	O1O2O3O4U3U4U1O5O6U5U6U2
R3 orbit	{'O1O2O3O4U3U4U1O5O6U5U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U3U5U6O5O6U4U1U2
Gauss code of K*	O1O2O3U4U5O4O5O6U3U6U1U2
Gauss code of -K*	O1O2O3U2U3O4O5O6U5U6U1U4
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 1 -1 1],[ 1 0 1 -1 1 0 0],[-1 -1 0 -1 1 -1 1],[ 1 1 1 0 1 0 0],[-1 -1 -1 -1 0 0 0],[ 1 0 1 0 0 0 1],[-1 0 -1 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 -1 -1 -1],[-1 0 1 1 -1 -1 -1],[-1 -1 0 0 0 0 -1],[-1 -1 0 0 -1 -1 0],[ 1 1 0 1 0 1 0],[ 1 1 0 1 -1 0 0],[ 1 1 1 0 0 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,1,1,1,-1,-1,1,1,1,0,0,0,1,1,1,0,-1,0,0]
Phi over symmetry	[-1,-1,-1,1,1,1,-1,-1,1,1,1,0,0,0,1,1,1,0,-1,0,0]
Phi of -K	[-1,-1,-1,1,1,1,-1,0,1,1,2,0,1,1,2,1,2,1,-1,-1,0]
Phi of K*	[-1,-1,-1,1,1,1,-1,0,1,1,2,1,1,1,1,2,2,1,-1,0,0]
Phi of -K*	[-1,-1,-1,1,1,1,-1,0,0,1,1,0,0,1,1,1,0,1,0,-1,-1]
Symmetry type of based matrix	c
u-polynomial	0
Normalized Jones-Krushkal polynomial	4z^2+21z+27
Enhanced Jones-Krushkal polynomial	4w^3z^2+21w^2z+27w
Inner characteristic polynomial	t^6+9t^4+13t^2+1
Outer characteristic polynomial	t^7+15t^5+27t^3+5t
Flat arrow polynomial	8K13 - 8K1*2 - 8K1K2 - 2K1 + 4K2 + 2K3 + 5
2-strand cable arrow polynomial	-2048K16 - 4352K1*4K2*2 + 7424K1*4K2 - 7072K14 + 3520K1*3K2K3 - 1056K1*3K3 - 3136K12K2*4 + 7552K1*2K2*3 + 1024K1*2K2*2K4 - 15872K12K2*2 - 1600K1*2K2K4 + 9200K1*2K2 - 896K12K3*2 - 224K1*2K4*2 + 1448K1*2 + 3648K1K23K3 - 3104K1K2*2K3 - 800K1K2*2K5 - 448K1K2K3K4 + 7104K1K2K3 + 592K1K3K4 + 144K1K4K5 - 704K2*6 + 1088K2*4K4 - 4256K24 - 192K2*3K6 - 960K22K3*2 - 384K2*2K4*2 + 2416K2*2K4 + 708K22 + 336K2K3K5 + 112K2K4K6 - 404K3*2 - 120K4*2 - 20K5*2 - 4K6**2 + 1334
Genus of based matrix	0
Fillings of based matrix	[[{5, 6}, {3, 4}, {1, 2}]]
If K is slice	True

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