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Flat knot 6.13

Min(phi) over symmetries of the knot is: [-5,-2,-2,1,4,4,1,2,3,4,5,0,1,2,3,2,3,4,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.13']
Arrow polynomial of the knot is: K1 - 2K2K3 + K5 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.13', '6.30', '6.33', '6.42', '6.56']
Outer characteristic polynomial of the knot is: t^7+169t^5+174t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.13']
2-strand cable arrow polynomial of the knot is: -192K13K3 + 1584K12K2 - 1216K12K3*2 - 192K1*2K3K5 - 64K1*2K6*2 - 2488K1*2 - 448K1K22K3 + 512K1K2K33 - 256K1K2K3K4 + 64K1K2K3K62 - 192K1K2K3K6 + 2400K1K2K3 - 192K1K3*2K5 - 64K1K3K4K6 + 1680K1K3K4 + 416K1K4K5 + 48K1K5K6 + 80K1K6K7 - 2K102 + 8K10K4K6 - 128K24 - 384K2*2K3*2 + 480K2*2K4 - 32K22K6*2 - 1666K2*2 - 64K2K32K4 - 64K2K3K4K5 + 656K2K3K5 + 72K2K4K6 + 32K2K5K7 - 640K3*4 - 64K3*2K4*2 - 64K3*2K6*2 + 528K3*2K6 - 1168K32 + 96K3K4K7 + 16K3K6K9 - 8K4*2K6*2 - 700K4*2 - 312K5*2 - 84K6*2 - 48K7**2 + 2058
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.13']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.82816', 'vk6.82818', 'vk6.82932', 'vk6.82933', 'vk6.82935', 'vk6.82939', 'vk6.83251', 'vk6.83253', 'vk6.83255', 'vk6.83263', 'vk6.86072', 'vk6.86074', 'vk6.86799', 'vk6.86800', 'vk6.86801', 'vk6.86803', 'vk6.89665', 'vk6.89667', 'vk6.90001', 'vk6.90005']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is +.
The reverse -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: [-5,-2,-2,1,4,4,1,2,3,4,5,0,1,2,3,2,3,4,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.13']

Arrow polynomial of the knot is: K1 - 2*K2*K3 + K5 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.13', '6.30', '6.33', '6.42', '6.56']

Outer characteristic polynomial of the knot is: t^7+169t^5+174t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.13']

2-strand cable arrow polynomial of the knot is: -192*K1**3*K3 + 1584*K1**2*K2 - 1216*K1**2*K3**2 - 192*K1**2*K3*K5 - 64*K1**2*K6**2 - 2488*K1**2 - 448*K1*K2**2*K3 + 512*K1*K2*K3**3 - 256*K1*K2*K3*K4 + 64*K1*K2*K3*K6**2 - 192*K1*K2*K3*K6 + 2400*K1*K2*K3 - 192*K1*K3**2*K5 - 64*K1*K3*K4*K6 + 1680*K1*K3*K4 + 416*K1*K4*K5 + 48*K1*K5*K6 + 80*K1*K6*K7 - 2*K10**2 + 8*K10*K4*K6 - 128*K2**4 - 384*K2**2*K3**2 + 480*K2**2*K4 - 32*K2**2*K6**2 - 1666*K2**2 - 64*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 656*K2*K3*K5 + 72*K2*K4*K6 + 32*K2*K5*K7 - 640*K3**4 - 64*K3**2*K4**2 - 64*K3**2*K6**2 + 528*K3**2*K6 - 1168*K3**2 + 96*K3*K4*K7 + 16*K3*K6*K9 - 8*K4**2*K6**2 - 700*K4**2 - 312*K5**2 - 84*K6**2 - 48*K7**2 + 2058

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.13']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.82816', 'vk6.82818', 'vk6.82932', 'vk6.82933', 'vk6.82935', 'vk6.82939', 'vk6.83251', 'vk6.83253', 'vk6.83255', 'vk6.83263', 'vk6.86072', 'vk6.86074', 'vk6.86799', 'vk6.86800', 'vk6.86801', 'vk6.86803', 'vk6.89665', 'vk6.89667', 'vk6.90001', 'vk6.90005']

The R3 orbit of minmal crossing diagrams contains:

The diagrammatic symmetry type of this knot is +.

The reverse -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	O1O2O3O4O5O6U1U3U2U4U6U5
R3 orbit	{'O1O2O3O4O5O6U1U3U2U4U6U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U2U1U3U5U4U6
Gauss code of K*	Same
Gauss code of -K*	O1O2O3O4O5O6U2U1U3U5U4U6
Diagrammatic symmetry type	+
Flat genus of the diagram	3
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -5 -2 -2 1 4 4],[ 5 0 2 1 3 5 4],[ 2 -2 0 0 2 4 3],[ 2 -1 0 0 1 3 2],[-1 -3 -2 -1 0 2 1],[-4 -5 -4 -3 -2 0 0],[-4 -4 -3 -2 -1 0 0]]
Primitive based matrix	[[ 0 4 4 1 -2 -2 -5],[-4 0 0 -1 -2 -3 -4],[-4 0 0 -2 -3 -4 -5],[-1 1 2 0 -1 -2 -3],[ 2 2 3 1 0 0 -1],[ 2 3 4 2 0 0 -2],[ 5 4 5 3 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-4,-1,2,2,5,0,1,2,3,4,2,3,4,5,1,2,3,0,1,2]
Phi over symmetry	[-5,-2,-2,1,4,4,1,2,3,4,5,0,1,2,3,2,3,4,1,2,0]
Phi of -K	[-5,-2,-2,1,4,4,1,2,3,4,5,0,1,2,3,2,3,4,1,2,0]
Phi of K*	[-4,-4,-1,2,2,5,0,1,2,3,4,2,3,4,5,1,2,3,0,1,2]
Phi of -K*	[-5,-2,-2,1,4,4,1,2,3,4,5,0,1,2,3,2,3,4,1,2,0]
Symmetry type of based matrix	+
u-polynomial	t^5-2t^4+2t^2-t
Normalized Jones-Krushkal polynomial	6z^2+23z+23
Enhanced Jones-Krushkal polynomial	6w^3z^2+23w^2z+23w
Inner characteristic polynomial	t^6+103t^4+40t^2
Outer characteristic polynomial	t^7+169t^5+174t^3+4t
Flat arrow polynomial	K1 - 2K2K3 + K5 + 1
2-strand cable arrow polynomial	-192K13K3 + 1584K12K2 - 1216K12K3*2 - 192K1*2K3K5 - 64K1*2K6*2 - 2488K1*2 - 448K1K22K3 + 512K1K2K33 - 256K1K2K3K4 + 64K1K2K3K62 - 192K1K2K3K6 + 2400K1K2K3 - 192K1K3*2K5 - 64K1K3K4K6 + 1680K1K3K4 + 416K1K4K5 + 48K1K5K6 + 80K1K6K7 - 2K102 + 8K10K4K6 - 128K24 - 384K2*2K3*2 + 480K2*2K4 - 32K22K6*2 - 1666K2*2 - 64K2K32K4 - 64K2K3K4K5 + 656K2K3K5 + 72K2K4K6 + 32K2K5K7 - 640K3*4 - 64K3*2K4*2 - 64K3*2K6*2 + 528K3*2K6 - 1168K32 + 96K3K4K7 + 16K3K6K9 - 8K4*2K6*2 - 700K4*2 - 312K5*2 - 84K6*2 - 48K7**2 + 2058
Genus of based matrix	1
Fillings of based matrix	[[{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {4}, {2, 3}, {1}]]
If K is slice	False

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