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Flat knot 6.800

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,1,0,2,3,3,0,1,1,2,1,0,0,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.800']
Arrow polynomial of the knot is: -8K12 - 2K1K2 + K1 + 4K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.377', '6.638', '6.646', '6.647', '6.657', '6.658', '6.662', '6.692', '6.695', '6.714', '6.720', '6.726', '6.730', '6.731', '6.749', '6.756', '6.772', '6.779', '6.781', '6.800', '6.829', '6.1085', '6.1089', '6.1302', '6.1349', '6.1350', '6.1360', '6.1362', '6.1375', '6.1384']
Outer characteristic polynomial of the knot is: t^7+53t^5+85t^3+19t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.800']
2-strand cable arrow polynomial of the knot is: 448K14K2 - 2560K14 + 224K1*3K2K3 - 224K1*3K3 + 256K12K2*3 + 32K1*2K2*2K4 - 3584K12K2*2 - 576K1*2K2K4 + 7552K1*2K2 - 1056K12K3*2 - 32K1*2K3K5 - 224K1*2K4*2 - 5816K1*2 - 768K1K22K3 - 32K1K2*2K5 - 128K1K2K3K4 + 6696K1K2K3 + 2080K1K3K4 + 304K1K4K5 - 96K2*4 - 32K2*2K3*2 - 8K2*2K4*2 + 640K2*2K4 - 4566K22 + 120K2K3K5 + 8K2K4K6 - 2576K3*2 - 788K4*2 - 104K5*2 - 2K6**2 + 4810
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.800']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73294', 'vk6.73306', 'vk6.73435', 'vk6.73447', 'vk6.74088', 'vk6.74097', 'vk6.74659', 'vk6.74666', 'vk6.75435', 'vk6.75447', 'vk6.76122', 'vk6.76131', 'vk6.78175', 'vk6.78179', 'vk6.78405', 'vk6.78409', 'vk6.79090', 'vk6.79099', 'vk6.79996', 'vk6.80000', 'vk6.80147', 'vk6.80151', 'vk6.80594', 'vk6.80603', 'vk6.83794', 'vk6.83801', 'vk6.85107', 'vk6.85121', 'vk6.86593', 'vk6.86615', 'vk6.87374', 'vk6.87402']
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: [-3,-1,0,0,2,2,1,0,2,3,3,0,1,1,2,1,0,0,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.800']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.377', '6.638', '6.646', '6.647', '6.657', '6.658', '6.662', '6.692', '6.695', '6.714', '6.720', '6.726', '6.730', '6.731', '6.749', '6.756', '6.772', '6.779', '6.781', '6.800', '6.829', '6.1085', '6.1089', '6.1302', '6.1349', '6.1350', '6.1360', '6.1362', '6.1375', '6.1384']

Outer characteristic polynomial of the knot is: t^7+53t^5+85t^3+19t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.800']

2-strand cable arrow polynomial of the knot is: 448*K1**4*K2 - 2560*K1**4 + 224*K1**3*K2*K3 - 224*K1**3*K3 + 256*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 3584*K1**2*K2**2 - 576*K1**2*K2*K4 + 7552*K1**2*K2 - 1056*K1**2*K3**2 - 32*K1**2*K3*K5 - 224*K1**2*K4**2 - 5816*K1**2 - 768*K1*K2**2*K3 - 32*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 6696*K1*K2*K3 + 2080*K1*K3*K4 + 304*K1*K4*K5 - 96*K2**4 - 32*K2**2*K3**2 - 8*K2**2*K4**2 + 640*K2**2*K4 - 4566*K2**2 + 120*K2*K3*K5 + 8*K2*K4*K6 - 2576*K3**2 - 788*K4**2 - 104*K5**2 - 2*K6**2 + 4810

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.800']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73294', 'vk6.73306', 'vk6.73435', 'vk6.73447', 'vk6.74088', 'vk6.74097', 'vk6.74659', 'vk6.74666', 'vk6.75435', 'vk6.75447', 'vk6.76122', 'vk6.76131', 'vk6.78175', 'vk6.78179', 'vk6.78405', 'vk6.78409', 'vk6.79090', 'vk6.79099', 'vk6.79996', 'vk6.80000', 'vk6.80147', 'vk6.80151', 'vk6.80594', 'vk6.80603', 'vk6.83794', 'vk6.83801', 'vk6.85107', 'vk6.85121', 'vk6.86593', 'vk6.86615', 'vk6.87374', 'vk6.87402']

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	O1O2O3O4U5O6U4U1U3O5U2U6
R3 orbit	{'O1O2O3O4U5O6U4U1U3O5U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U3O6U2U4U1O5U6
Gauss code of K*	O1O2O3U4O5O6U2U5U3U1O4U6
Gauss code of -K*	O1O2O3U4O5U3U1U6U2O4O6U5
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 0 1 0 -2 3],[ 2 0 1 1 0 0 3],[ 0 -1 0 1 1 -2 2],[-1 -1 -1 0 0 -2 1],[ 0 0 -1 0 0 0 0],[ 2 0 2 2 0 0 3],[-3 -3 -2 -1 0 -3 0]]
Primitive based matrix	[[ 0 3 1 0 0 -2 -2],[-3 0 -1 0 -2 -3 -3],[-1 1 0 0 -1 -1 -2],[ 0 0 0 0 -1 0 0],[ 0 2 1 1 0 -1 -2],[ 2 3 1 0 1 0 0],[ 2 3 2 0 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,0,0,2,2,1,0,2,3,3,0,1,1,2,1,0,0,1,2,0]
Phi over symmetry	[-3,-1,0,0,2,2,1,0,2,3,3,0,1,1,2,1,0,0,1,2,0]
Phi of -K	[-2,-2,0,0,1,3,0,0,2,1,2,1,2,2,2,-1,0,1,1,3,1]
Phi of K*	[-3,-1,0,0,2,2,1,1,3,2,2,0,1,1,2,1,0,1,2,2,0]
Phi of -K*	[-2,-2,0,0,1,3,0,0,1,1,3,0,2,2,3,-1,0,0,1,2,1]
Symmetry type of based matrix	c
u-polynomial	-t^3+2t^2-t
Normalized Jones-Krushkal polynomial	z^2+22z+41
Enhanced Jones-Krushkal polynomial	w^3z^2+22w^2z+41w
Inner characteristic polynomial	t^6+35t^4+47t^2+9
Outer characteristic polynomial	t^7+53t^5+85t^3+19t
Flat arrow polynomial	-8K12 - 2K1K2 + K1 + 4K2 + K3 + 5
2-strand cable arrow polynomial	448K14K2 - 2560K14 + 224K1*3K2K3 - 224K1*3K3 + 256K12K2*3 + 32K1*2K2*2K4 - 3584K12K2*2 - 576K1*2K2K4 + 7552K1*2K2 - 1056K12K3*2 - 32K1*2K3K5 - 224K1*2K4*2 - 5816K1*2 - 768K1K22K3 - 32K1K2*2K5 - 128K1K2K3K4 + 6696K1K2K3 + 2080K1K3K4 + 304K1K4K5 - 96K2*4 - 32K2*2K3*2 - 8K2*2K4*2 + 640K2*2K4 - 4566K22 + 120K2K3K5 + 8K2K4K6 - 2576K3*2 - 788K4*2 - 104K5*2 - 2K6**2 + 4810
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice	False

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