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Flat knot 6.1242

Min(phi) over symmetries of the knot is: [-2,-1,-1,1,1,2,-1,1,1,2,2,0,0,1,1,1,2,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.1242']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 4K12 - 8K1K2 - 4K1K3 - 2K1 + 2K2 + 2K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1242', '6.1916']
Outer characteristic polynomial of the knot is: t^7+38t^5+65t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1242']
2-strand cable arrow polynomial of the knot is: -1152K14K2*2 + 6656K1*4K2 - 9664K14 - 768K1*3K2*2K3 + 3072K13K2K3 - 1664K1*3K3 - 256K12K2*4 + 3136K1*2K2*3 - 128K1*2K2*2K3*2 + 512K1*2K2*2K4 - 14432K12K2*2 + 768K1*2K2K32 - 1408K1*2K2K4 + 11680K1*2K2 - 2624K12K3*2 - 64K1*2K4*2 - 1248K1*2 + 1344K1K23K3 + 256K1K2*2K3K4 - 3456K1K22K3 - 768K1K2*2K5 + 256K1K2K33 - 768K1K2K3K4 - 256K1K2K3K6 + 11120K1K2K3 + 2336K1K3K4 + 256K1K4K5 + 48K1K5K6 - 64K2*6 - 32K2*4K4*2 + 256K2*4K4 - 2176K24 + 128K2*3K3K5 + 64K2*3K4K6 - 192K2*3K6 - 1760K22K3*2 - 64K2*2K3K7 - 352K2*2K4*2 + 2720K2*2K4 - 128K22K5*2 - 48K2*2K6*2 - 3708K2*2 - 64K2K32K4 + 1408K2K3K5 + 384K2K4K6 + 96K2K5K7 + 16K2K6K8 - 128K34 + 64K3*2K6 - 2056K32 - 812K4*2 - 264K5*2 - 84K6*2 - 16K7*2 - 2K8**2 + 4028
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1242']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3215', 'vk6.3223', 'vk6.3237', 'vk6.3322', 'vk6.3335', 'vk6.3353', 'vk6.3445', 'vk6.3502', 'vk6.15224', 'vk6.15244', 'vk6.15254', 'vk6.33876', 'vk6.33892', 'vk6.33900', 'vk6.34336', 'vk6.34338', 'vk6.48088', 'vk6.48153', 'vk6.48160', 'vk6.54442']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is r.
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,-1,1,1,2,-1,1,1,2,2,0,0,1,1,1,2,2,0,0,2]

Flat knots (up to 7 crossings) with same phi are :['6.1242']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 4*K1**2 - 8*K1*K2 - 4*K1*K3 - 2*K1 + 2*K2 + 2*K3 + K4 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1242', '6.1916']

Outer characteristic polynomial of the knot is: t^7+38t^5+65t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1242']

2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 6656*K1**4*K2 - 9664*K1**4 - 768*K1**3*K2**2*K3 + 3072*K1**3*K2*K3 - 1664*K1**3*K3 - 256*K1**2*K2**4 + 3136*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 512*K1**2*K2**2*K4 - 14432*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 1408*K1**2*K2*K4 + 11680*K1**2*K2 - 2624*K1**2*K3**2 - 64*K1**2*K4**2 - 1248*K1**2 + 1344*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 3456*K1*K2**2*K3 - 768*K1*K2**2*K5 + 256*K1*K2*K3**3 - 768*K1*K2*K3*K4 - 256*K1*K2*K3*K6 + 11120*K1*K2*K3 + 2336*K1*K3*K4 + 256*K1*K4*K5 + 48*K1*K5*K6 - 64*K2**6 - 32*K2**4*K4**2 + 256*K2**4*K4 - 2176*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 192*K2**3*K6 - 1760*K2**2*K3**2 - 64*K2**2*K3*K7 - 352*K2**2*K4**2 + 2720*K2**2*K4 - 128*K2**2*K5**2 - 48*K2**2*K6**2 - 3708*K2**2 - 64*K2*K3**2*K4 + 1408*K2*K3*K5 + 384*K2*K4*K6 + 96*K2*K5*K7 + 16*K2*K6*K8 - 128*K3**4 + 64*K3**2*K6 - 2056*K3**2 - 812*K4**2 - 264*K5**2 - 84*K6**2 - 16*K7**2 - 2*K8**2 + 4028

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1242']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3215', 'vk6.3223', 'vk6.3237', 'vk6.3322', 'vk6.3335', 'vk6.3353', 'vk6.3445', 'vk6.3502', 'vk6.15224', 'vk6.15244', 'vk6.15254', 'vk6.33876', 'vk6.33892', 'vk6.33900', 'vk6.34336', 'vk6.34338', 'vk6.48088', 'vk6.48153', 'vk6.48160', 'vk6.54442']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is r.

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	O1O2O3O4U3U5U4O5O6U1U6U2
R3 orbit	{'O1O2O3O4U3U5U4O5O6U1U6U2'}
R3 orbit length	1
Gauss code of -K	Same
Gauss code of K*	O1O2O3U2U4O5O4O6U5U6U1U3
Gauss code of -K*	O1O2O3U2U4O5O4O6U5U6U1U3
Diagrammatic symmetry type	r
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 2 -1 1],[ 2 0 2 -1 2 1 1],[-1 -2 0 -1 2 -2 0],[ 1 1 1 0 1 0 0],[-2 -2 -2 -1 0 -2 0],[ 1 -1 2 0 2 0 1],[-1 -1 0 0 0 -1 0]]
Primitive based matrix	[[ 0 2 1 1 -1 -1 -2],[-2 0 0 -2 -1 -2 -2],[-1 0 0 0 0 -1 -1],[-1 2 0 0 -1 -2 -2],[ 1 1 0 1 0 0 1],[ 1 2 1 2 0 0 -1],[ 2 2 1 2 -1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,1,1,2,0,2,1,2,2,0,0,1,1,1,2,2,0,-1,1]
Phi over symmetry	[-2,-1,-1,1,1,2,-1,1,1,2,2,0,0,1,1,1,2,2,0,0,2]
Phi of -K	[-2,-1,-1,1,1,2,0,2,1,2,2,0,0,1,1,1,2,2,0,-1,1]
Phi of K*	[-2,-1,-1,1,1,2,-1,1,1,2,2,0,0,1,1,1,2,2,0,0,2]
Phi of -K*	[-2,-1,-1,1,1,2,-1,1,1,2,2,0,0,1,1,1,2,2,0,0,2]
Symmetry type of based matrix	r
u-polynomial	0
Normalized Jones-Krushkal polynomial	9z^2+30z+25
Enhanced Jones-Krushkal polynomial	9w^3z^2+30w^2z+25w
Inner characteristic polynomial	t^6+26t^4+39t^2+4
Outer characteristic polynomial	t^7+38t^5+65t^3+8t
Flat arrow polynomial	8K13 + 4K1*2K2 - 4K12 - 8K1K2 - 4K1K3 - 2K1 + 2K2 + 2K3 + K4 + 2
2-strand cable arrow polynomial	-1152K14K2*2 + 6656K1*4K2 - 9664K14 - 768K1*3K2*2K3 + 3072K13K2K3 - 1664K1*3K3 - 256K12K2*4 + 3136K1*2K2*3 - 128K1*2K2*2K3*2 + 512K1*2K2*2K4 - 14432K12K2*2 + 768K1*2K2K32 - 1408K1*2K2K4 + 11680K1*2K2 - 2624K12K3*2 - 64K1*2K4*2 - 1248K1*2 + 1344K1K23K3 + 256K1K2*2K3K4 - 3456K1K22K3 - 768K1K2*2K5 + 256K1K2K33 - 768K1K2K3K4 - 256K1K2K3K6 + 11120K1K2K3 + 2336K1K3K4 + 256K1K4K5 + 48K1K5K6 - 64K2*6 - 32K2*4K4*2 + 256K2*4K4 - 2176K24 + 128K2*3K3K5 + 64K2*3K4K6 - 192K2*3K6 - 1760K22K3*2 - 64K2*2K3K7 - 352K2*2K4*2 + 2720K2*2K4 - 128K22K5*2 - 48K2*2K6*2 - 3708K2*2 - 64K2K32K4 + 1408K2K3K5 + 384K2K4K6 + 96K2K5K7 + 16K2K6K8 - 128K34 + 64K3*2K6 - 2056K32 - 812K4*2 - 264K5*2 - 84K6*2 - 16K7*2 - 2K8**2 + 4028
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice	False

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