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Flat knot 6.1781

Min(phi) over symmetries of the knot is: [-2,0,0,0,1,1,0,0,1,2,2,-1,0,0,1,-1,-1,0,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.1781']
Arrow polynomial of the knot is: 4K13 - 14K1*2 - 8K1K2 + K1 + 7K2 + 3*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.474', '6.1684', '6.1716', '6.1749', '6.1781']
Outer characteristic polynomial of the knot is: t^7+27t^5+88t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1781']
2-strand cable arrow polynomial of the knot is: -128K14K2*2 + 1216K1*4K2 - 4624K14 + 512K1*3K2K3 - 1664K1*3K3 + 320K12K2*3 + 32K1*2K2*2K4 - 5600K12K2*2 + 32K1*2K2K32 - 672K1*2K2K4 + 12968K1*2K2 - 1296K12K3*2 - 32K1*2K3K5 - 96K1*2K4*2 - 8952K1*2 + 128K1K23K3 - 768K1K2*2K3 - 128K1K2*2K5 - 320K1K2K3K4 + 10144K1K2K3 + 2224K1K3K4 + 280K1K4K5 - 32K2*6 + 96K2*4K4 - 616K24 - 32K2*3K6 - 320K22K3*2 - 128K2*2K4*2 + 1352K2*2K4 - 6954K22 + 520K2K3K5 + 104K2K4K6 - 3576K3*2 - 1042K4*2 - 200K5*2 - 22K6**2 + 7272
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1781']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73291', 'vk6.73293', 'vk6.73434', 'vk6.73436', 'vk6.74082', 'vk6.74086', 'vk6.74653', 'vk6.74657', 'vk6.75436', 'vk6.75438', 'vk6.76120', 'vk6.76124', 'vk6.78172', 'vk6.78174', 'vk6.78404', 'vk6.78406', 'vk6.79088', 'vk6.79092', 'vk6.79995', 'vk6.79997', 'vk6.80148', 'vk6.80150', 'vk6.80596', 'vk6.80600', 'vk6.83809', 'vk6.83825', 'vk6.85106', 'vk6.85137', 'vk6.86601', 'vk6.86614', 'vk6.87373', 'vk6.87375']
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: [-2,0,0,0,1,1,0,0,1,2,2,-1,0,0,1,-1,-1,0,1,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.1781']

Arrow polynomial of the knot is: 4*K1**3 - 14*K1**2 - 8*K1*K2 + K1 + 7*K2 + 3*K3 + 8

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.474', '6.1684', '6.1716', '6.1749', '6.1781']

Outer characteristic polynomial of the knot is: t^7+27t^5+88t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1781']

2-strand cable arrow polynomial of the knot is: -128*K1**4*K2**2 + 1216*K1**4*K2 - 4624*K1**4 + 512*K1**3*K2*K3 - 1664*K1**3*K3 + 320*K1**2*K2**3 + 32*K1**2*K2**2*K4 - 5600*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 672*K1**2*K2*K4 + 12968*K1**2*K2 - 1296*K1**2*K3**2 - 32*K1**2*K3*K5 - 96*K1**2*K4**2 - 8952*K1**2 + 128*K1*K2**3*K3 - 768*K1*K2**2*K3 - 128*K1*K2**2*K5 - 320*K1*K2*K3*K4 + 10144*K1*K2*K3 + 2224*K1*K3*K4 + 280*K1*K4*K5 - 32*K2**6 + 96*K2**4*K4 - 616*K2**4 - 32*K2**3*K6 - 320*K2**2*K3**2 - 128*K2**2*K4**2 + 1352*K2**2*K4 - 6954*K2**2 + 520*K2*K3*K5 + 104*K2*K4*K6 - 3576*K3**2 - 1042*K4**2 - 200*K5**2 - 22*K6**2 + 7272

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1781']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73291', 'vk6.73293', 'vk6.73434', 'vk6.73436', 'vk6.74082', 'vk6.74086', 'vk6.74653', 'vk6.74657', 'vk6.75436', 'vk6.75438', 'vk6.76120', 'vk6.76124', 'vk6.78172', 'vk6.78174', 'vk6.78404', 'vk6.78406', 'vk6.79088', 'vk6.79092', 'vk6.79995', 'vk6.79997', 'vk6.80148', 'vk6.80150', 'vk6.80596', 'vk6.80600', 'vk6.83809', 'vk6.83825', 'vk6.85106', 'vk6.85137', 'vk6.86601', 'vk6.86614', 'vk6.87373', 'vk6.87375']

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	O1O2O3U4U5O6O4U1U3O5U2U6
R3 orbit	{'O1O2O3U4U5O6O4U1U3O5U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3U4U2O5U1U3O6O4U5U6
Gauss code of K*	O1O2U3O4O5U1U4U2O6O3U5U6
Gauss code of -K*	O1O2U3O4O5U6U1O3O6U4U2U5
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 -2 0 1 0 0 1],[ 2 0 1 1 2 2 1],[ 0 -1 0 1 -1 0 0],[-1 -1 -1 0 -1 0 -1],[ 0 -2 1 1 0 -1 2],[ 0 -2 0 0 1 0 1],[-1 -1 0 1 -2 -1 0]]
Primitive based matrix	[[ 0 1 1 0 0 0 -2],[-1 0 1 0 -1 -2 -1],[-1 -1 0 -1 0 -1 -1],[ 0 0 1 0 0 -1 -1],[ 0 1 0 0 0 1 -2],[ 0 2 1 1 -1 0 -2],[ 2 1 1 1 2 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,0,0,0,2,-1,0,1,2,1,1,0,1,1,0,1,1,-1,2,2]
Phi over symmetry	[-2,0,0,0,1,1,0,0,1,2,2,-1,0,0,1,-1,-1,0,1,0,-1]
Phi of -K	[-2,0,0,0,1,1,0,0,1,2,2,-1,0,0,1,-1,-1,0,1,0,-1]
Phi of K*	[-1,-1,0,0,0,2,-1,0,0,1,2,-1,1,0,2,1,-1,0,0,1,0]
Phi of -K*	[-2,0,0,0,1,1,1,2,2,1,1,-1,0,0,1,-1,2,1,1,0,1]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	z^2+22z+41
Enhanced Jones-Krushkal polynomial	w^3z^2+22w^2z+41w
Inner characteristic polynomial	t^6+21t^4+59t^2
Outer characteristic polynomial	t^7+27t^5+88t^3+4t
Flat arrow polynomial	4K13 - 14K1*2 - 8K1K2 + K1 + 7K2 + 3*K3 + 8
2-strand cable arrow polynomial	-128K14K2*2 + 1216K1*4K2 - 4624K14 + 512K1*3K2K3 - 1664K1*3K3 + 320K12K2*3 + 32K1*2K2*2K4 - 5600K12K2*2 + 32K1*2K2K32 - 672K1*2K2K4 + 12968K1*2K2 - 1296K12K3*2 - 32K1*2K3K5 - 96K1*2K4*2 - 8952K1*2 + 128K1K23K3 - 768K1K2*2K3 - 128K1K2*2K5 - 320K1K2K3K4 + 10144K1K2K3 + 2224K1K3K4 + 280K1K4K5 - 32K2*6 + 96K2*4K4 - 616K24 - 32K2*3K6 - 320K22K3*2 - 128K2*2K4*2 + 1352K2*2K4 - 6954K22 + 520K2K3K5 + 104K2K4K6 - 3576K3*2 - 1042K4*2 - 200K5*2 - 22K6**2 + 7272
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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