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Flat knot 6.268

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,3,1,2,4,1,1,1,2,1,0,2,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.268']
Arrow polynomial of the knot is: -2K12 - 2K1K2 - 2K1K3 + K1 + 2K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.62', '6.176', '6.181', '6.194', '6.228', '6.267', '6.268', '6.449']
Outer characteristic polynomial of the knot is: t^7+74t^5+65t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.268']
2-strand cable arrow polynomial of the knot is: -16K14 + 32K1*3K2K3 - 1344K1*3K3 - 288K12K2*2 + 192K1*2K2K32 + 3640K1*2K2 - 1552K12K3*2 - 96K1*2K3K5 - 96K1*2K4K6 - 48K1*2K6*2 - 4796K1*2 + 96K1K23K3 - 736K1K2*2K3 + 32K1K2K33 - 576K1K2K3K4 + 5944K1K2K3 + 1944K1K3K4 + 272K1K4K5 + 136K1K5K6 + 40K1K6K7 - 72K24 - 752K2*2K3*2 - 8K2*2K4*2 + 472K2*2K4 - 8K22K6*2 - 3162K2*2 - 96K2K32K4 + 792K2K3K5 + 112K2K4K6 + 8K2K6K8 - 96K3*4 + 104K3*2K6 - 2504K32 + 8K3K4K7 - 636K42 - 224K5*2 - 102K6*2 - 12K7*2 - 2K8**2 + 3396
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.268']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11005', 'vk6.11084', 'vk6.12171', 'vk6.12278', 'vk6.18208', 'vk6.18544', 'vk6.24671', 'vk6.25094', 'vk6.30574', 'vk6.30669', 'vk6.31844', 'vk6.31891', 'vk6.36796', 'vk6.37251', 'vk6.44040', 'vk6.44381', 'vk6.51816', 'vk6.51883', 'vk6.52680', 'vk6.52774', 'vk6.56002', 'vk6.56276', 'vk6.60542', 'vk6.60884', 'vk6.63500', 'vk6.63544', 'vk6.63978', 'vk6.64022', 'vk6.65661', 'vk6.65944', 'vk6.68710', 'vk6.68919']
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: [-4,-1,0,1,1,3,1,3,1,2,4,1,1,1,2,1,0,2,-1,-1,1]

Flat knots (up to 7 crossings) with same phi are :['6.268']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.62', '6.176', '6.181', '6.194', '6.228', '6.267', '6.268', '6.449']

Outer characteristic polynomial of the knot is: t^7+74t^5+65t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.268']

2-strand cable arrow polynomial of the knot is: -16*K1**4 + 32*K1**3*K2*K3 - 1344*K1**3*K3 - 288*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 3640*K1**2*K2 - 1552*K1**2*K3**2 - 96*K1**2*K3*K5 - 96*K1**2*K4*K6 - 48*K1**2*K6**2 - 4796*K1**2 + 96*K1*K2**3*K3 - 736*K1*K2**2*K3 + 32*K1*K2*K3**3 - 576*K1*K2*K3*K4 + 5944*K1*K2*K3 + 1944*K1*K3*K4 + 272*K1*K4*K5 + 136*K1*K5*K6 + 40*K1*K6*K7 - 72*K2**4 - 752*K2**2*K3**2 - 8*K2**2*K4**2 + 472*K2**2*K4 - 8*K2**2*K6**2 - 3162*K2**2 - 96*K2*K3**2*K4 + 792*K2*K3*K5 + 112*K2*K4*K6 + 8*K2*K6*K8 - 96*K3**4 + 104*K3**2*K6 - 2504*K3**2 + 8*K3*K4*K7 - 636*K4**2 - 224*K5**2 - 102*K6**2 - 12*K7**2 - 2*K8**2 + 3396

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.268']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11005', 'vk6.11084', 'vk6.12171', 'vk6.12278', 'vk6.18208', 'vk6.18544', 'vk6.24671', 'vk6.25094', 'vk6.30574', 'vk6.30669', 'vk6.31844', 'vk6.31891', 'vk6.36796', 'vk6.37251', 'vk6.44040', 'vk6.44381', 'vk6.51816', 'vk6.51883', 'vk6.52680', 'vk6.52774', 'vk6.56002', 'vk6.56276', 'vk6.60542', 'vk6.60884', 'vk6.63500', 'vk6.63544', 'vk6.63978', 'vk6.64022', 'vk6.65661', 'vk6.65944', 'vk6.68710', 'vk6.68919']

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	O1O2O3O4O5U1U3O6U5U6U2U4
R3 orbit	{'O1O2O3O4O5U1U3O6U5U6U2U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U4U6U1O6U3U5
Gauss code of K*	O1O2O3O4U5U3U6U4U1O5O6U2
Gauss code of -K*	O1O2O3O4U3O5O6U4U1U5U2U6
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 -4 0 -1 3 1 1],[ 4 0 3 1 4 2 1],[ 0 -3 0 -1 2 0 1],[ 1 -1 1 0 2 1 1],[-3 -4 -2 -2 0 -1 1],[-1 -2 0 -1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 3 1 1 0 -1 -4],[-3 0 1 -1 -2 -2 -4],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 0 -1 -2],[ 0 2 1 0 0 -1 -3],[ 1 2 1 1 1 0 -1],[ 4 4 1 2 3 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,0,1,4,-1,1,2,2,4,1,1,1,1,0,1,2,1,3,1]
Phi over symmetry	[-4,-1,0,1,1,3,1,3,1,2,4,1,1,1,2,1,0,2,-1,-1,1]
Phi of -K	[-4,-1,0,1,1,3,2,1,3,4,3,0,1,1,2,1,0,1,-1,1,3]
Phi of K*	[-3,-1,-1,0,1,4,1,3,1,2,3,1,1,1,3,0,1,4,0,1,2]
Phi of -K*	[-4,-1,0,1,1,3,1,3,1,2,4,1,1,1,2,1,0,2,-1,-1,1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	3z^2+20z+29
Enhanced Jones-Krushkal polynomial	3w^3z^2+20w^2z+29w
Inner characteristic polynomial	t^6+46t^4+11t^2
Outer characteristic polynomial	t^7+74t^5+65t^3+4t
Flat arrow polynomial	-2K12 - 2K1K2 - 2K1K3 + K1 + 2K2 + K3 + K4 + 2
2-strand cable arrow polynomial	-16K14 + 32K1*3K2K3 - 1344K1*3K3 - 288K12K2*2 + 192K1*2K2K32 + 3640K1*2K2 - 1552K12K3*2 - 96K1*2K3K5 - 96K1*2K4K6 - 48K1*2K6*2 - 4796K1*2 + 96K1K23K3 - 736K1K2*2K3 + 32K1K2K33 - 576K1K2K3K4 + 5944K1K2K3 + 1944K1K3K4 + 272K1K4K5 + 136K1K5K6 + 40K1K6K7 - 72K24 - 752K2*2K3*2 - 8K2*2K4*2 + 472K2*2K4 - 8K22K6*2 - 3162K2*2 - 96K2K32K4 + 792K2K3K5 + 112K2K4K6 + 8K2K6K8 - 96K3*4 + 104K3*2K6 - 2504K32 + 8K3K4K7 - 636K42 - 224K5*2 - 102K6*2 - 12K7*2 - 2K8**2 + 3396
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}]]
If K is slice	False

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