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Flat knot 6.1173

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,0,1,3,4,0,0,1,1,0,0,1,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1173']
Arrow polynomial of the knot is: 4K12K2 - 2K12 - 2K1K2 - 2K1K3 + K1 - 2K2**2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']
Outer characteristic polynomial of the knot is: t^7+57t^5+48t^3+5t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1173']
2-strand cable arrow polynomial of the knot is: 640K12K2*3 - 1824K1*2K2*2 + 32K1*2K2K3K5 - 256K12K2K4 + 2664K1*2K2 - 64K12K3*2 - 64K1*2K5*2 - 2888K1*2 + 160K1K22K3K4 - 928K1K22K3 + 128K1K2*2K4K5 - 384K1K22K5 - 320K1K2K3K4 + 3096K1K2K3 - 160K1K2K4K5 - 32K1K2K4K7 + 968K1K3K4 + 560K1K4K5 + 112K1K5K6 - 32K24K4*2 + 128K2*4K4 - 760K24 + 32K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 224K22K3*2 + 32K2*2K4*3 - 424K2*2K4*2 - 32K2*2K4K8 + 1816K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 2698K2*2 - 192K2K32K4 - 64K2K3K4K5 + 968K2K3K5 + 360K2K4K6 + 56K2K5K7 + 8K2K6K8 + 32K32K6 - 1416K32 + 16K3K4K7 - 8K44 + 8K4*2K8 - 1044K42 - 544K5*2 - 78K6*2 - 8K7*2 - 2K8**2 + 2732
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1173']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4236', 'vk6.4315', 'vk6.5507', 'vk6.5625', 'vk6.7616', 'vk6.7703', 'vk6.9105', 'vk6.9184', 'vk6.18373', 'vk6.18711', 'vk6.24826', 'vk6.25283', 'vk6.37014', 'vk6.37462', 'vk6.44187', 'vk6.44506', 'vk6.48548', 'vk6.48603', 'vk6.49251', 'vk6.49367', 'vk6.50341', 'vk6.50398', 'vk6.51078', 'vk6.51109', 'vk6.56146', 'vk6.56373', 'vk6.60671', 'vk6.61018', 'vk6.65810', 'vk6.66062', 'vk6.68807', 'vk6.69015']
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,-1,0,0,2,2,0,0,1,3,4,0,0,1,1,0,0,1,1,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.1173']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.128', '6.408', '6.452', '6.532', '6.867', '6.917', '6.938', '6.1164', '6.1173', '6.1174']

Outer characteristic polynomial of the knot is: t^7+57t^5+48t^3+5t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1173']

2-strand cable arrow polynomial of the knot is: 640*K1**2*K2**3 - 1824*K1**2*K2**2 + 32*K1**2*K2*K3*K5 - 256*K1**2*K2*K4 + 2664*K1**2*K2 - 64*K1**2*K3**2 - 64*K1**2*K5**2 - 2888*K1**2 + 160*K1*K2**2*K3*K4 - 928*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 3096*K1*K2*K3 - 160*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 968*K1*K3*K4 + 560*K1*K4*K5 + 112*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 760*K2**4 + 32*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 224*K2**2*K3**2 + 32*K2**2*K4**3 - 424*K2**2*K4**2 - 32*K2**2*K4*K8 + 1816*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2698*K2**2 - 192*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 968*K2*K3*K5 + 360*K2*K4*K6 + 56*K2*K5*K7 + 8*K2*K6*K8 + 32*K3**2*K6 - 1416*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1044*K4**2 - 544*K5**2 - 78*K6**2 - 8*K7**2 - 2*K8**2 + 2732

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1173']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4236', 'vk6.4315', 'vk6.5507', 'vk6.5625', 'vk6.7616', 'vk6.7703', 'vk6.9105', 'vk6.9184', 'vk6.18373', 'vk6.18711', 'vk6.24826', 'vk6.25283', 'vk6.37014', 'vk6.37462', 'vk6.44187', 'vk6.44506', 'vk6.48548', 'vk6.48603', 'vk6.49251', 'vk6.49367', 'vk6.50341', 'vk6.50398', 'vk6.51078', 'vk6.51109', 'vk6.56146', 'vk6.56373', 'vk6.60671', 'vk6.61018', 'vk6.65810', 'vk6.66062', 'vk6.68807', 'vk6.69015']

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	O1O2O3O4U1U5U4O5O6U3U2U6
R3 orbit	{'O1O2O3O4U1U5U4O5O6U3U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U3U2O5O6U1U6U4
Gauss code of K*	O1O2O3U2U4O5O6O4U1U6U5U3
Gauss code of -K*	O1O2O3U1U4O5O4O6U5U3U2U6
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 -3 0 0 2 -1 2],[ 3 0 3 2 1 2 2],[ 0 -3 0 0 1 -1 2],[ 0 -2 0 0 1 -1 1],[-2 -1 -1 -1 0 -2 0],[ 1 -2 1 1 2 0 2],[-2 -2 -2 -1 0 -2 0]]
Primitive based matrix	[[ 0 2 2 0 0 -1 -3],[-2 0 0 -1 -1 -2 -1],[-2 0 0 -1 -2 -2 -2],[ 0 1 1 0 0 -1 -2],[ 0 1 2 0 0 -1 -3],[ 1 2 2 1 1 0 -2],[ 3 1 2 2 3 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,0,1,3,0,1,1,2,1,1,2,2,2,0,1,2,1,3,2]
Phi over symmetry	[-3,-1,0,0,2,2,0,0,1,3,4,0,0,1,1,0,0,1,1,1,0]
Phi of -K	[-3,-1,0,0,2,2,0,0,1,3,4,0,0,1,1,0,0,1,1,1,0]
Phi of K*	[-2,-2,0,0,1,3,0,0,1,1,3,1,1,1,4,0,0,0,0,1,0]
Phi of -K*	[-3,-1,0,0,2,2,2,2,3,1,2,1,1,2,2,0,1,1,1,2,0]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	7z^2+24z+21
Enhanced Jones-Krushkal polynomial	7w^3z^2+24w^2z+21w
Inner characteristic polynomial	t^6+39t^4+24t^2+1
Outer characteristic polynomial	t^7+57t^5+48t^3+5t
Flat arrow polynomial	4K12K2 - 2K12 - 2K1K2 - 2K1K3 + K1 - 2K2**2 + K3 + K4 + 2
2-strand cable arrow polynomial	640K12K2*3 - 1824K1*2K2*2 + 32K1*2K2K3K5 - 256K12K2K4 + 2664K1*2K2 - 64K12K3*2 - 64K1*2K5*2 - 2888K1*2 + 160K1K22K3K4 - 928K1K22K3 + 128K1K2*2K4K5 - 384K1K22K5 - 320K1K2K3K4 + 3096K1K2K3 - 160K1K2K4K5 - 32K1K2K4K7 + 968K1K3K4 + 560K1K4K5 + 112K1K5K6 - 32K24K4*2 + 128K2*4K4 - 760K24 + 32K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 224K22K3*2 + 32K2*2K4*3 - 424K2*2K4*2 - 32K2*2K4K8 + 1816K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 2698K2*2 - 192K2K32K4 - 64K2K3K4K5 + 968K2K3K5 + 360K2K4K6 + 56K2K5K7 + 8K2K6K8 + 32K32K6 - 1416K32 + 16K3K4K7 - 8K44 + 8K4*2K8 - 1044K42 - 544K5*2 - 78K6*2 - 8K7*2 - 2K8**2 + 2732
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice	False

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