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Flat knot 6.1137

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,2,1,3,0,0,1,0,1,1,2,1,1,-2]
Flat knots (up to 7 crossings) with same phi are :['6.1137']
Arrow polynomial of the knot is: -8K14 + 4K1*3 + 8K1*2K2 + 2K12 - 2K1K2 - 2K1K3 - 2K1 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1137']
Outer characteristic polynomial of the knot is: t^7+48t^5+84t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1137']
2-strand cable arrow polynomial of the knot is: -336K14 - 1792K1*2K2*6 + 1792K1*2K2*5 - 2624K1*2K2*4 + 640K1*2K2*3 - 1024K1*2K2*2 + 1264K1*2K2 - 112K12K3*2 - 1112K1*2 + 2304K1K25K3 + 1088K1K2*3K3 + 1424K1K2K3 + 224K1K3K4 + 16K1K4K5 + 16K1K5K6 - 128K28 + 256K2*6K4 - 1568K26 - 1216K2*4K3*2 - 192K2*4K4*2 + 256K2*4K4 + 984K24 + 256K2*3K3K5 + 64K2*3K4K6 - 352K2*2K3*2 - 72K2*2K4*2 + 216K2*2K4 - 8K22K6*2 - 676K2*2 + 144K2K3K5 + 40K2K4K6 + 16K3*2K6 - 576K32 - 176K4*2 - 40K5*2 - 28K6**2 + 1030
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1137']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4633', 'vk6.4900', 'vk6.6049', 'vk6.6564', 'vk6.8088', 'vk6.8466', 'vk6.9464', 'vk6.9841', 'vk6.20285', 'vk6.21614', 'vk6.27565', 'vk6.29127', 'vk6.38978', 'vk6.41223', 'vk6.45749', 'vk6.47442', 'vk6.48667', 'vk6.48848', 'vk6.49383', 'vk6.49634', 'vk6.50673', 'vk6.50844', 'vk6.51146', 'vk6.51371', 'vk6.57134', 'vk6.58324', 'vk6.61744', 'vk6.62883', 'vk6.66759', 'vk6.67641', 'vk6.69413', 'vk6.70135']
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,-1,1,2,2,0,1,2,1,3,0,0,1,0,1,1,2,1,1,-2]

Flat knots (up to 7 crossings) with same phi are :['6.1137']

Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 8*K1**2*K2 + 2*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1137']

Outer characteristic polynomial of the knot is: t^7+48t^5+84t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1137']

2-strand cable arrow polynomial of the knot is: -336*K1**4 - 1792*K1**2*K2**6 + 1792*K1**2*K2**5 - 2624*K1**2*K2**4 + 640*K1**2*K2**3 - 1024*K1**2*K2**2 + 1264*K1**2*K2 - 112*K1**2*K3**2 - 1112*K1**2 + 2304*K1*K2**5*K3 + 1088*K1*K2**3*K3 + 1424*K1*K2*K3 + 224*K1*K3*K4 + 16*K1*K4*K5 + 16*K1*K5*K6 - 128*K2**8 + 256*K2**6*K4 - 1568*K2**6 - 1216*K2**4*K3**2 - 192*K2**4*K4**2 + 256*K2**4*K4 + 984*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 352*K2**2*K3**2 - 72*K2**2*K4**2 + 216*K2**2*K4 - 8*K2**2*K6**2 - 676*K2**2 + 144*K2*K3*K5 + 40*K2*K4*K6 + 16*K3**2*K6 - 576*K3**2 - 176*K4**2 - 40*K5**2 - 28*K6**2 + 1030

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1137']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4633', 'vk6.4900', 'vk6.6049', 'vk6.6564', 'vk6.8088', 'vk6.8466', 'vk6.9464', 'vk6.9841', 'vk6.20285', 'vk6.21614', 'vk6.27565', 'vk6.29127', 'vk6.38978', 'vk6.41223', 'vk6.45749', 'vk6.47442', 'vk6.48667', 'vk6.48848', 'vk6.49383', 'vk6.49634', 'vk6.50673', 'vk6.50844', 'vk6.51146', 'vk6.51371', 'vk6.57134', 'vk6.58324', 'vk6.61744', 'vk6.62883', 'vk6.66759', 'vk6.67641', 'vk6.69413', 'vk6.70135']

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	O1O2O3O4U1U2U3O5O6U5U4U6
R3 orbit	{'O1O2O3O4U1U2U3O5O6U5U4U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U1U6O5O6U2U3U4
Gauss code of K*	O1O2O3U4U5O4O6O5U1U2U3U6
Gauss code of -K*	O1O2O3U1U3O4O5O6U2U4U5U6
Diagrammatic symmetry type	c
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -3 -1 1 2 -1 2],[ 3 0 1 2 3 0 1],[ 1 -1 0 1 2 0 1],[-1 -2 -1 0 1 0 1],[-2 -3 -2 -1 0 0 2],[ 1 0 0 0 0 0 1],[-2 -1 -1 -1 -2 -1 0]]
Primitive based matrix	[[ 0 2 2 1 -1 -1 -3],[-2 0 2 -1 0 -2 -3],[-2 -2 0 -1 -1 -1 -1],[-1 1 1 0 0 -1 -2],[ 1 0 1 0 0 0 0],[ 1 2 1 1 0 0 -1],[ 3 3 1 2 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,1,1,3,-2,1,0,2,3,1,1,1,1,0,1,2,0,0,1]
Phi over symmetry	[-3,-1,-1,1,2,2,0,1,2,1,3,0,0,1,0,1,1,2,1,1,-2]
Phi of -K	[-3,-1,-1,1,2,2,1,2,2,2,4,0,1,1,2,2,3,2,0,0,-2]
Phi of K*	[-2,-2,-1,1,1,3,-2,0,2,2,4,0,1,3,2,1,2,2,0,1,2]
Phi of -K*	[-3,-1,-1,1,2,2,0,1,2,1,3,0,0,1,0,1,1,2,1,1,-2]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	2z+5
Enhanced Jones-Krushkal polynomial	12w^4z-16w^3z+6w^2z+5w
Inner characteristic polynomial	t^6+28t^4+26t^2
Outer characteristic polynomial	t^7+48t^5+84t^3
Flat arrow polynomial	-8K14 + 4K1*3 + 8K1*2K2 + 2K12 - 2K1K2 - 2K1K3 - 2K1 + 1
2-strand cable arrow polynomial	-336K14 - 1792K1*2K2*6 + 1792K1*2K2*5 - 2624K1*2K2*4 + 640K1*2K2*3 - 1024K1*2K2*2 + 1264K1*2K2 - 112K12K3*2 - 1112K1*2 + 2304K1K25K3 + 1088K1K2*3K3 + 1424K1K2K3 + 224K1K3K4 + 16K1K4K5 + 16K1K5K6 - 128K28 + 256K2*6K4 - 1568K26 - 1216K2*4K3*2 - 192K2*4K4*2 + 256K2*4K4 + 984K24 + 256K2*3K3K5 + 64K2*3K4K6 - 352K2*2K3*2 - 72K2*2K4*2 + 216K2*2K4 - 8K22K6*2 - 676K2*2 + 144K2K3K5 + 40K2K4K6 + 16K3*2K6 - 576K32 - 176K4*2 - 40K5*2 - 28K6**2 + 1030
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {4, 5}, {1, 3}, {2}]]
If K is slice	False

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