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Flat knot 6.33

Min(phi) over symmetries of the knot is: [-5,-1,-1,1,3,3,1,2,5,3,4,0,2,1,2,2,2,3,2,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.33']
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+135t^5+163t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.33']
2-strand cable arrow polynomial of the knot is: 192K12K2*2K4 - 704K12K2*2 - 480K1*2K2K4 + 2112K1*2K2 - 128K12K3*2 - 352K1*2K4*2 - 3464K1*2 - 320K1K22K3 - 64K1K2*2K5 - 352K1K2K3K4 + 2976K1K2K3 - 32K1K2K4K5 - 32K1K2K4K7 + 1976K1K3K4 + 664K1K4K5 + 24K1K5K6 + 72K1K6K7 - 2K10*2 + 8K10K4K6 - 128K24 - 128K2*2K3*2 - 96K2*2K4*2 + 1200K2*2K4 - 3010K22 - 192K2K32K4 - 32K2K3K4K5 + 488K2K3K5 - 32K2K42K6 + 512K2K4K6 + 48K2K5K7 + 16K2K6K8 - 128K3*4 + 328K3*2K6 - 2088K32 + 88K3K4K7 + 8K3K5K8 - 8K4*2K6*2 + 24K4*2K8 - 1612K42 - 392K5*2 - 348K6*2 - 88K7*2 - 16K8**2 + 3490
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.33']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19970', 'vk6.19974', 'vk6.21233', 'vk6.21241', 'vk6.26977', 'vk6.26985', 'vk6.28707', 'vk6.28711', 'vk6.38387', 'vk6.38395', 'vk6.40556', 'vk6.40572', 'vk6.45269', 'vk6.45285', 'vk6.47061', 'vk6.47069', 'vk6.56772', 'vk6.56780', 'vk6.57892', 'vk6.57908', 'vk6.61251', 'vk6.61265', 'vk6.62453', 'vk6.62461', 'vk6.66474', 'vk6.66478', 'vk6.67260', 'vk6.67268', 'vk6.69126', 'vk6.69134', 'vk6.69891', 'vk6.69895']
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: [-5,-1,-1,1,3,3,1,2,5,3,4,0,2,1,2,2,2,3,2,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.33']

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+135t^5+163t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.33']

2-strand cable arrow polynomial of the knot is: 192*K1**2*K2**2*K4 - 704*K1**2*K2**2 - 480*K1**2*K2*K4 + 2112*K1**2*K2 - 128*K1**2*K3**2 - 352*K1**2*K4**2 - 3464*K1**2 - 320*K1*K2**2*K3 - 64*K1*K2**2*K5 - 352*K1*K2*K3*K4 + 2976*K1*K2*K3 - 32*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1976*K1*K3*K4 + 664*K1*K4*K5 + 24*K1*K5*K6 + 72*K1*K6*K7 - 2*K10**2 + 8*K10*K4*K6 - 128*K2**4 - 128*K2**2*K3**2 - 96*K2**2*K4**2 + 1200*K2**2*K4 - 3010*K2**2 - 192*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 488*K2*K3*K5 - 32*K2*K4**2*K6 + 512*K2*K4*K6 + 48*K2*K5*K7 + 16*K2*K6*K8 - 128*K3**4 + 328*K3**2*K6 - 2088*K3**2 + 88*K3*K4*K7 + 8*K3*K5*K8 - 8*K4**2*K6**2 + 24*K4**2*K8 - 1612*K4**2 - 392*K5**2 - 348*K6**2 - 88*K7**2 - 16*K8**2 + 3490

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.33']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19970', 'vk6.19974', 'vk6.21233', 'vk6.21241', 'vk6.26977', 'vk6.26985', 'vk6.28707', 'vk6.28711', 'vk6.38387', 'vk6.38395', 'vk6.40556', 'vk6.40572', 'vk6.45269', 'vk6.45285', 'vk6.47061', 'vk6.47069', 'vk6.56772', 'vk6.56780', 'vk6.57892', 'vk6.57908', 'vk6.61251', 'vk6.61265', 'vk6.62453', 'vk6.62461', 'vk6.66474', 'vk6.66478', 'vk6.67260', 'vk6.67268', 'vk6.69126', 'vk6.69134', 'vk6.69891', 'vk6.69895']

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	O1O2O3O4O5O6U1U4U3U6U5U2
R3 orbit	{'O1O2O3O4O5O6U1U4U3U6U5U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U5U2U1U4U3U6
Gauss code of K*	O1O2O3O4O5O6U1U6U3U2U5U4
Gauss code of -K*	O1O2O3O4O5O6U3U2U5U4U1U6
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 -5 1 -1 -1 3 3],[ 5 0 5 2 1 4 3],[-1 -5 0 -2 -2 2 2],[ 1 -2 2 0 0 3 2],[ 1 -1 2 0 0 2 1],[-3 -4 -2 -3 -2 0 0],[-3 -3 -2 -2 -1 0 0]]
Primitive based matrix	[[ 0 3 3 1 -1 -1 -5],[-3 0 0 -2 -1 -2 -3],[-3 0 0 -2 -2 -3 -4],[-1 2 2 0 -2 -2 -5],[ 1 1 2 2 0 0 -1],[ 1 2 3 2 0 0 -2],[ 5 3 4 5 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,-1,1,1,5,0,2,1,2,3,2,2,3,4,2,2,5,0,1,2]
Phi over symmetry	[-5,-1,-1,1,3,3,1,2,5,3,4,0,2,1,2,2,2,3,2,2,0]
Phi of -K	[-5,-1,-1,1,3,3,2,3,1,4,5,0,0,1,2,0,2,3,0,0,0]
Phi of K*	[-3,-3,-1,1,1,5,0,0,1,2,4,0,2,3,5,0,0,1,0,2,3]
Phi of -K*	[-5,-1,-1,1,3,3,1,2,5,3,4,0,2,1,2,2,2,3,2,2,0]
Symmetry type of based matrix	c
u-polynomial	t^5-2t^3+t
Normalized Jones-Krushkal polynomial	7z^2+24z+21
Enhanced Jones-Krushkal polynomial	-2w^4z^2+9w^3z^2-2w^3z+26w^2z+21w
Inner characteristic polynomial	t^6+89t^4+49t^2+1
Outer characteristic polynomial	t^7+135t^5+163t^3+9t
Flat arrow polynomial	K1 - 2K2K3 + K5 + 1
2-strand cable arrow polynomial	192K12K2*2K4 - 704K12K2*2 - 480K1*2K2K4 + 2112K1*2K2 - 128K12K3*2 - 352K1*2K4*2 - 3464K1*2 - 320K1K22K3 - 64K1K2*2K5 - 352K1K2K3K4 + 2976K1K2K3 - 32K1K2K4K5 - 32K1K2K4K7 + 1976K1K3K4 + 664K1K4K5 + 24K1K5K6 + 72K1K6K7 - 2K10*2 + 8K10K4K6 - 128K24 - 128K2*2K3*2 - 96K2*2K4*2 + 1200K2*2K4 - 3010K22 - 192K2K32K4 - 32K2K3K4K5 + 488K2K3K5 - 32K2K42K6 + 512K2K4K6 + 48K2K5K7 + 16K2K6K8 - 128K3*4 + 328K3*2K6 - 2088K32 + 88K3K4K7 + 8K3K5K8 - 8K4*2K6*2 + 24K4*2K8 - 1612K42 - 392K5*2 - 348K6*2 - 88K7*2 - 16K8**2 + 3490
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice	False

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