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Flat knot 6.153

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,1,2,1,4,4,1,0,2,2,0,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.153']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 6K12 - 6K1K2 - 2K1K3 - 3K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.153', '6.279']
Outer characteristic polynomial of the knot is: t^7+105t^5+73t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.153']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 672K1*4K2 - 1168K14 + 128K1*3K2*3K3 - 768K13K2*2K3 + 1568K13K2K3 - 704K1*3K3 - 448K12K2*4 + 256K1*2K2*3K3*2 + 3520K1*2K2*3 - 640K1*2K2*2K3*2 + 128K1*2K2*2K4 - 10928K12K2*2 + 384K1*2K2K32 - 992K1*2K2K4 + 8904K1*2K2 - 848K12K3*2 - 5548K1*2 - 256K1K24K3 - 128K1K2*3K3K4 + 3360K1K23K3 + 448K1K2*2K3K4 - 2464K1K22K3 - 288K1K2*2K5 + 96K1K2K33 - 512K1K2K3K4 + 9096K1K2K3 + 992K1K3K4 - 64K2*6 - 256K2*4K3*2 - 32K2*4K4*2 + 352K2*4K4 - 3432K24 + 192K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 2048K22K3*2 - 296K2*2K4*2 + 2136K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 2406K2*2 - 32K2K32K4 + 744K2K3K5 + 64K2K4K6 + 8K32K6 - 2128K32 - 388K4*2 - 52K5*2 - 10K6**2 + 4082
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.153']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16919', 'vk6.17161', 'vk6.20515', 'vk6.21899', 'vk6.23305', 'vk6.23604', 'vk6.27956', 'vk6.29433', 'vk6.35333', 'vk6.35765', 'vk6.39364', 'vk6.41542', 'vk6.42827', 'vk6.43109', 'vk6.45929', 'vk6.47616', 'vk6.55070', 'vk6.55319', 'vk6.57380', 'vk6.58539', 'vk6.59461', 'vk6.59752', 'vk6.62033', 'vk6.63027', 'vk6.64907', 'vk6.65120', 'vk6.66925', 'vk6.67774', 'vk6.68212', 'vk6.68356', 'vk6.69527', 'vk6.70233']
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,-2,0,1,2,3,1,2,1,4,4,1,0,2,2,0,1,1,-1,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.153']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 6*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 2*K2 + K3 + 3

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.153', '6.279']

Outer characteristic polynomial of the knot is: t^7+105t^5+73t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.153']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 672*K1**4*K2 - 1168*K1**4 + 128*K1**3*K2**3*K3 - 768*K1**3*K2**2*K3 + 1568*K1**3*K2*K3 - 704*K1**3*K3 - 448*K1**2*K2**4 + 256*K1**2*K2**3*K3**2 + 3520*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 10928*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 992*K1**2*K2*K4 + 8904*K1**2*K2 - 848*K1**2*K3**2 - 5548*K1**2 - 256*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 3360*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 2464*K1*K2**2*K3 - 288*K1*K2**2*K5 + 96*K1*K2*K3**3 - 512*K1*K2*K3*K4 + 9096*K1*K2*K3 + 992*K1*K3*K4 - 64*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 352*K2**4*K4 - 3432*K2**4 + 192*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 2048*K2**2*K3**2 - 296*K2**2*K4**2 + 2136*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K6**2 - 2406*K2**2 - 32*K2*K3**2*K4 + 744*K2*K3*K5 + 64*K2*K4*K6 + 8*K3**2*K6 - 2128*K3**2 - 388*K4**2 - 52*K5**2 - 10*K6**2 + 4082

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.153']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16919', 'vk6.17161', 'vk6.20515', 'vk6.21899', 'vk6.23305', 'vk6.23604', 'vk6.27956', 'vk6.29433', 'vk6.35333', 'vk6.35765', 'vk6.39364', 'vk6.41542', 'vk6.42827', 'vk6.43109', 'vk6.45929', 'vk6.47616', 'vk6.55070', 'vk6.55319', 'vk6.57380', 'vk6.58539', 'vk6.59461', 'vk6.59752', 'vk6.62033', 'vk6.63027', 'vk6.64907', 'vk6.65120', 'vk6.66925', 'vk6.67774', 'vk6.68212', 'vk6.68356', 'vk6.69527', 'vk6.70233']

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	O1O2O3O4O5U1O6U3U4U6U5U2
R3 orbit	{'O1O2O3O4O5U1O6U3U4U6U5U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U1U6U2U3O6U5
Gauss code of K*	O1O2O3O4O5U6U5U1U2U4O6U3
Gauss code of -K*	O1O2O3O4O5U3O6U2U4U5U1U6
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 1 -2 0 3 2],[ 4 0 4 1 2 3 2],[-1 -4 0 -3 -1 2 2],[ 2 -1 3 0 1 3 2],[ 0 -2 1 -1 0 2 1],[-3 -3 -2 -3 -2 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix	[[ 0 3 2 1 0 -2 -4],[-3 0 0 -2 -2 -3 -3],[-2 0 0 -2 -1 -2 -2],[-1 2 2 0 -1 -3 -4],[ 0 2 1 1 0 -1 -2],[ 2 3 2 3 1 0 -1],[ 4 3 2 4 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,0,2,4,0,2,2,3,3,2,1,2,2,1,3,4,1,2,1]
Phi over symmetry	[-4,-2,0,1,2,3,1,2,1,4,4,1,0,2,2,0,1,1,-1,0,1]
Phi of -K	[-4,-2,0,1,2,3,1,2,1,4,4,1,0,2,2,0,1,1,-1,0,1]
Phi of K*	[-3,-2,-1,0,2,4,1,0,1,2,4,-1,1,2,4,0,0,1,1,2,1]
Phi of -K*	[-4,-2,0,1,2,3,1,2,4,2,3,1,3,2,3,1,1,2,2,2,0]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	8z^2+29z+27
Enhanced Jones-Krushkal polynomial	8w^3z^2+29w^2z+27w
Inner characteristic polynomial	t^6+71t^4+25t^2+1
Outer characteristic polynomial	t^7+105t^5+73t^3+8t
Flat arrow polynomial	8K13 + 4K1*2K2 - 6K12 - 6K1K2 - 2K1K3 - 3K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial	-256K14K2*2 + 672K1*4K2 - 1168K14 + 128K1*3K2*3K3 - 768K13K2*2K3 + 1568K13K2K3 - 704K1*3K3 - 448K12K2*4 + 256K1*2K2*3K3*2 + 3520K1*2K2*3 - 640K1*2K2*2K3*2 + 128K1*2K2*2K4 - 10928K12K2*2 + 384K1*2K2K32 - 992K1*2K2K4 + 8904K1*2K2 - 848K12K3*2 - 5548K1*2 - 256K1K24K3 - 128K1K2*3K3K4 + 3360K1K23K3 + 448K1K2*2K3K4 - 2464K1K22K3 - 288K1K2*2K5 + 96K1K2K33 - 512K1K2K3K4 + 9096K1K2K3 + 992K1K3K4 - 64K2*6 - 256K2*4K3*2 - 32K2*4K4*2 + 352K2*4K4 - 3432K24 + 192K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 2048K22K3*2 - 296K2*2K4*2 + 2136K2*2K4 - 32K22K5*2 - 8K2*2K6*2 - 2406K2*2 - 32K2K32K4 + 744K2K3K5 + 64K2K4K6 + 8K32K6 - 2128K32 - 388K4*2 - 52K5*2 - 10K6**2 + 4082
Genus of based matrix	2
Fillings of based matrix	[[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{1, 6}, {5}, {4}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {3, 5}, {4}, {2}, {1}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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