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Flat knot 6.38

Min(phi) over symmetries of the knot is: [-5,-1,-1,2,2,3,1,3,2,5,4,1,1,2,2,1,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.38']
Arrow polynomial of the knot is: 8K13 + 4K1*2K3 - 4K12 - 10K1K2 - K1 - 2K2K3 + 2K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.38']
Outer characteristic polynomial of the knot is: t^7+120t^5+95t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.38']
2-strand cable arrow polynomial of the knot is: -800K14 + 256K1*2K2*3 - 3168K1*2K2*2 - 192K1*2K2K4 + 4944K1*2K2 - 32K12K3*2 - 4096K1*2 + 1152K1K23K3 + 192K1K2*2K3K4 - 1088K1K22K3 - 128K1K2*2K5 - 832K1K2K3K4 - 64K1K2K3K6 + 5040K1K2K3 + 1248K1K3K4 + 208K1K4K5 + 16K1K5K6 - 64K26 - 256K2*4K3*2 + 128K2*4K4 - 32K24K6*2 - 1408K2*4 + 128K2*3K3K5 + 128K2*3K4K6 - 128K2*3K6 + 320K22K3*2K4 - 1504K22K3*2 - 64K2*2K3K7 - 360K2*2K4*2 + 32K2*2K4K62 + 2392K2*2K4 - 96K22K6*2 - 3530K2*2 - 64K2K32K4 - 64K2K3K4K5 + 992K2K3K5 - 96K2K42K6 + 536K2K4K6 + 64K2K6K8 - 128K32K4*2 + 16K3*2K6 - 1920K32 + 48K3K4K7 - 8K42K6*2 + 16K4*2K8 - 1228K42 - 160K5*2 - 166K6*2 - 8K8**2 + 3826
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.38']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71591', 'vk6.71715', 'vk6.72133', 'vk6.72327', 'vk6.74061', 'vk6.74623', 'vk6.76811', 'vk6.77208', 'vk6.77517', 'vk6.77664', 'vk6.79061', 'vk6.79628', 'vk6.80582', 'vk6.81033', 'vk6.81346', 'vk6.81393', 'vk6.85415', 'vk6.85493', 'vk6.87986', 'vk6.89314']
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,-1,-1,2,2,3,1,3,2,5,4,1,1,2,2,1,2,2,-1,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.38']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K3 - 4*K1**2 - 10*K1*K2 - K1 - 2*K2*K3 + 2*K2 + K3 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.38']

Outer characteristic polynomial of the knot is: t^7+120t^5+95t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.38']

2-strand cable arrow polynomial of the knot is: -800*K1**4 + 256*K1**2*K2**3 - 3168*K1**2*K2**2 - 192*K1**2*K2*K4 + 4944*K1**2*K2 - 32*K1**2*K3**2 - 4096*K1**2 + 1152*K1*K2**3*K3 + 192*K1*K2**2*K3*K4 - 1088*K1*K2**2*K3 - 128*K1*K2**2*K5 - 832*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5040*K1*K2*K3 + 1248*K1*K3*K4 + 208*K1*K4*K5 + 16*K1*K5*K6 - 64*K2**6 - 256*K2**4*K3**2 + 128*K2**4*K4 - 32*K2**4*K6**2 - 1408*K2**4 + 128*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 + 320*K2**2*K3**2*K4 - 1504*K2**2*K3**2 - 64*K2**2*K3*K7 - 360*K2**2*K4**2 + 32*K2**2*K4*K6**2 + 2392*K2**2*K4 - 96*K2**2*K6**2 - 3530*K2**2 - 64*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 992*K2*K3*K5 - 96*K2*K4**2*K6 + 536*K2*K4*K6 + 64*K2*K6*K8 - 128*K3**2*K4**2 + 16*K3**2*K6 - 1920*K3**2 + 48*K3*K4*K7 - 8*K4**2*K6**2 + 16*K4**2*K8 - 1228*K4**2 - 160*K5**2 - 166*K6**2 - 8*K8**2 + 3826

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.38']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71591', 'vk6.71715', 'vk6.72133', 'vk6.72327', 'vk6.74061', 'vk6.74623', 'vk6.76811', 'vk6.77208', 'vk6.77517', 'vk6.77664', 'vk6.79061', 'vk6.79628', 'vk6.80582', 'vk6.81033', 'vk6.81346', 'vk6.81393', 'vk6.85415', 'vk6.85493', 'vk6.87986', 'vk6.89314']

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	O1O2O3O4O5O6U1U4U6U2U5U3
R3 orbit	{'O1O2O3O4O5O6U1U4U6U2U5U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U4U2U5U1U3U6
Gauss code of K*	Same
Gauss code of -K*	O1O2O3O4O5O6U4U2U5U1U3U6
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 -1 2 -1 3 2],[ 5 0 3 5 1 4 2],[ 1 -3 0 2 -1 2 1],[-2 -5 -2 0 -2 1 1],[ 1 -1 1 2 0 2 1],[-3 -4 -2 -1 -2 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 3 2 2 -1 -1 -5],[-3 0 0 -1 -2 -2 -4],[-2 0 0 -1 -1 -1 -2],[-2 1 1 0 -2 -2 -5],[ 1 2 1 2 0 1 -1],[ 1 2 1 2 -1 0 -3],[ 5 4 2 5 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-2,1,1,5,0,1,2,2,4,1,1,1,2,2,2,5,-1,1,3]
Phi over symmetry	[-5,-1,-1,2,2,3,1,3,2,5,4,1,1,2,2,1,2,2,-1,0,1]
Phi of -K	[-5,-1,-1,2,2,3,1,3,2,5,4,1,1,2,2,1,2,2,-1,0,1]
Phi of K*	[-3,-2,-2,1,1,5,0,1,2,2,4,1,1,1,2,2,2,5,-1,1,3]
Phi of -K*	[-5,-1,-1,2,2,3,1,3,2,5,4,1,1,2,2,1,2,2,-1,0,1]
Symmetry type of based matrix	+
u-polynomial	t^5-t^3-2t^2+2t
Normalized Jones-Krushkal polynomial	4z^2+21z+27
Enhanced Jones-Krushkal polynomial	4w^3z^2+21w^2z+27w
Inner characteristic polynomial	t^6+76t^4+15t^2
Outer characteristic polynomial	t^7+120t^5+95t^3+4t
Flat arrow polynomial	8K13 + 4K1*2K3 - 4K12 - 10K1K2 - K1 - 2K2K3 + 2K2 + K3 + 3
2-strand cable arrow polynomial	-800K14 + 256K1*2K2*3 - 3168K1*2K2*2 - 192K1*2K2K4 + 4944K1*2K2 - 32K12K3*2 - 4096K1*2 + 1152K1K23K3 + 192K1K2*2K3K4 - 1088K1K22K3 - 128K1K2*2K5 - 832K1K2K3K4 - 64K1K2K3K6 + 5040K1K2K3 + 1248K1K3K4 + 208K1K4K5 + 16K1K5K6 - 64K26 - 256K2*4K3*2 + 128K2*4K4 - 32K24K6*2 - 1408K2*4 + 128K2*3K3K5 + 128K2*3K4K6 - 128K2*3K6 + 320K22K3*2K4 - 1504K22K3*2 - 64K2*2K3K7 - 360K2*2K4*2 + 32K2*2K4K62 + 2392K2*2K4 - 96K22K6*2 - 3530K2*2 - 64K2K32K4 - 64K2K3K4K5 + 992K2K3K5 - 96K2K42K6 + 536K2K4K6 + 64K2K6K8 - 128K32K4*2 + 16K3*2K6 - 1920K32 + 48K3K4K7 - 8K42K6*2 + 16K4*2K8 - 1228K42 - 160K5*2 - 166K6*2 - 8K8**2 + 3826
Genus of based matrix	1
Fillings of based matrix	[[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {5}, {2, 4}, {1}]]
If K is slice	False

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