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Flat knot 6.1916

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,0,1,2,2,1,0,2,2,0,-1,0,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1916']
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+32t^5+62t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1916']
2-strand cable arrow polynomial of the knot is: -1152K14K2*2 + 7680K1*4K2 - 9408K14 - 768K1*3K2*2K3 + 3072K13K2K3 - 2688K1*3K3 - 256K12K2*4 + 4672K1*2K2*3 - 128K1*2K2*2K3*2 + 256K1*2K2*2K4 - 14624K12K2*2 + 768K1*2K2K32 - 2336K1*2K2K4 + 11664K1*2K2 - 2240K12K3*2 - 128K1*2K3K5 - 64K1*2K4*2 - 2160K1*2 + 1856K1K23K3 + 256K1K2*2K3K4 - 3840K1K22K3 - 704K1K2*2K5 + 256K1K2K33 - 704K1K2K3K4 - 64K1K2K3K6 + 11152K1K2K3 - 192K1K2K4K5 + 2496K1K3K4 + 256K1K4K5 + 48K1K5K6 - 64K2*6 - 32K2*4K4*2 + 256K2*4K4 - 2752K24 + 128K2*3K3K5 + 64K2*3K4K6 - 128K2*3K6 - 1952K22K3*2 - 64K2*2K3K7 - 416K2*2K4*2 - 32K2*2K4K8 + 2992K2*2K4 - 128K22K5*2 - 16K2*2K6*2 - 3404K2*2 - 96K2K32K4 + 1440K2K3K5 + 384K2K4K6 + 96K2K5K7 + 16K2K6K8 - 128K34 + 80K3*2K6 - 2072K32 - 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 :['6.1916']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3213', 'vk6.3221', 'vk6.3239', 'vk6.3324', 'vk6.3337', 'vk6.3351', 'vk6.3443', 'vk6.3504', 'vk6.15228', 'vk6.15240', 'vk6.15258', 'vk6.15262', 'vk6.33873', 'vk6.33888', 'vk6.33902', 'vk6.34332', 'vk6.48087', 'vk6.48093', 'vk6.48155', 'vk6.54446']
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,-2,1,1,1,1,0,0,1,2,2,1,0,2,2,0,-1,0,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1916']

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+32t^5+62t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1916']

2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 7680*K1**4*K2 - 9408*K1**4 - 768*K1**3*K2**2*K3 + 3072*K1**3*K2*K3 - 2688*K1**3*K3 - 256*K1**2*K2**4 + 4672*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 256*K1**2*K2**2*K4 - 14624*K1**2*K2**2 + 768*K1**2*K2*K3**2 - 2336*K1**2*K2*K4 + 11664*K1**2*K2 - 2240*K1**2*K3**2 - 128*K1**2*K3*K5 - 64*K1**2*K4**2 - 2160*K1**2 + 1856*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 3840*K1*K2**2*K3 - 704*K1*K2**2*K5 + 256*K1*K2*K3**3 - 704*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 11152*K1*K2*K3 - 192*K1*K2*K4*K5 + 2496*K1*K3*K4 + 256*K1*K4*K5 + 48*K1*K5*K6 - 64*K2**6 - 32*K2**4*K4**2 + 256*K2**4*K4 - 2752*K2**4 + 128*K2**3*K3*K5 + 64*K2**3*K4*K6 - 128*K2**3*K6 - 1952*K2**2*K3**2 - 64*K2**2*K3*K7 - 416*K2**2*K4**2 - 32*K2**2*K4*K8 + 2992*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 3404*K2**2 - 96*K2*K3**2*K4 + 1440*K2*K3*K5 + 384*K2*K4*K6 + 96*K2*K5*K7 + 16*K2*K6*K8 - 128*K3**4 + 80*K3**2*K6 - 2072*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 :['6.1916']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3213', 'vk6.3221', 'vk6.3239', 'vk6.3324', 'vk6.3337', 'vk6.3351', 'vk6.3443', 'vk6.3504', 'vk6.15228', 'vk6.15240', 'vk6.15258', 'vk6.15262', 'vk6.33873', 'vk6.33888', 'vk6.33902', 'vk6.34332', 'vk6.48087', 'vk6.48093', 'vk6.48155', 'vk6.54446']

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	O1O2O3U1U3U4O5O4O6U5U6U2
R3 orbit	{'O1O2O3U1U3U4O5O4O6U5U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3U2U4U5O4O6O5U6U1U3
Gauss code of K*	O1O2O3U4U3U5O4O5O6U1U6U2
Gauss code of -K*	O1O2O3U2U4U3O4O5O6U5U1U6
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 1 1 -2 1],[ 2 0 2 1 2 0 0],[-1 -2 0 0 0 -2 1],[-1 -1 0 0 -1 0 0],[-1 -2 0 1 0 -2 0],[ 2 0 2 0 2 0 1],[-1 0 -1 0 0 -1 0]]
Primitive based matrix	[[ 0 1 1 1 1 -2 -2],[-1 0 1 0 0 -2 -2],[-1 -1 0 0 0 0 -1],[-1 0 0 0 1 -2 -2],[-1 0 0 -1 0 -1 0],[ 2 2 0 2 1 0 0],[ 2 2 1 2 0 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,2,2,-1,0,0,2,2,0,0,0,1,-1,2,2,1,0,0]
Phi over symmetry	[-2,-2,1,1,1,1,0,0,1,2,2,1,0,2,2,0,-1,0,0,-1,0]
Phi of -K	[-2,-2,1,1,1,1,0,1,1,2,3,1,1,3,2,0,-1,0,0,-1,0]
Phi of K*	[-1,-1,-1,-1,2,2,-1,0,0,2,3,0,0,1,1,-1,3,2,1,1,0]
Phi of -K*	[-2,-2,1,1,1,1,0,0,1,2,2,1,0,2,2,0,-1,0,0,-1,0]
Symmetry type of based matrix	c
u-polynomial	2t^2-4t
Normalized Jones-Krushkal polynomial	9z^2+30z+25
Enhanced Jones-Krushkal polynomial	9w^3z^2+30w^2z+25w
Inner characteristic polynomial	t^6+20t^4+36t^2
Outer characteristic polynomial	t^7+32t^5+62t^3+4t
Flat arrow polynomial	8K13 + 4K1*2K2 - 4K12 - 8K1K2 - 4K1K3 - 2K1 + 2K2 + 2K3 + K4 + 2
2-strand cable arrow polynomial	-1152K14K2*2 + 7680K1*4K2 - 9408K14 - 768K1*3K2*2K3 + 3072K13K2K3 - 2688K1*3K3 - 256K12K2*4 + 4672K1*2K2*3 - 128K1*2K2*2K3*2 + 256K1*2K2*2K4 - 14624K12K2*2 + 768K1*2K2K32 - 2336K1*2K2K4 + 11664K1*2K2 - 2240K12K3*2 - 128K1*2K3K5 - 64K1*2K4*2 - 2160K1*2 + 1856K1K23K3 + 256K1K2*2K3K4 - 3840K1K22K3 - 704K1K2*2K5 + 256K1K2K33 - 704K1K2K3K4 - 64K1K2K3K6 + 11152K1K2K3 - 192K1K2K4K5 + 2496K1K3K4 + 256K1K4K5 + 48K1K5K6 - 64K2*6 - 32K2*4K4*2 + 256K2*4K4 - 2752K24 + 128K2*3K3K5 + 64K2*3K4K6 - 128K2*3K6 - 1952K22K3*2 - 64K2*2K3K7 - 416K2*2K4*2 - 32K2*2K4K8 + 2992K2*2K4 - 128K22K5*2 - 16K2*2K6*2 - 3404K2*2 - 96K2K32K4 + 1440K2K3K5 + 384K2K4K6 + 96K2K5K7 + 16K2K6K8 - 128K34 + 80K3*2K6 - 2072K32 - 812K4*2 - 264K5*2 - 84K6*2 - 16K7*2 - 2K8**2 + 4028
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice	False

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