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Flat knot 6.522

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,1,1,3,4,1,0,1,1,0,2,2,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.522']
Arrow polynomial of the knot is: 8K13 + 8K1*2K2 - 8K12 - 6K1K2 - 4K1K3 - 3K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.522', '6.559']
Outer characteristic polynomial of the knot is: t^7+64t^5+87t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.522']
2-strand cable arrow polynomial of the knot is: -512K14K2*2 + 608K1*4K2 - 976K14 + 128K1*3K2*3K3 + 1024K13K2K3 - 576K1*3K3 - 704K12K2*4 + 2752K1*2K2*3 - 448K1*2K2*2K3*2 - 128K1*2K2*2K4*2 + 288K1*2K2*2K4 - 9664K12K2*2 + 128K1*2K2K32 - 864K1*2K2K4 + 8496K1*2K2 - 688K12K3*2 - 96K1*2K4*2 - 5624K1*2 + 2624K1K23K3 + 672K1K2*2K3K4 - 2560K1K22K3 + 192K1K2*2K4K5 - 576K1K22K5 + 64K1K2K33 - 512K1K2K3K4 + 9376K1K2K3 + 1040K1K3K4 + 184K1K4K5 - 64K26 - 256K2*4K3*2 - 192K2*4K4*2 + 832K2*4K4 - 3888K24 + 256K2*3K3K5 + 128K2*3K4K6 - 128K2*3K6 - 1888K22K3*2 - 728K2*2K4*2 + 3008K2*2K4 - 128K22K5*2 - 16K2*2K6*2 - 2858K2*2 - 32K2K32K4 - 32K2K3K4K5 + 944K2K3K5 + 136K2K4K6 + 16K2K5K7 - 2356K3*2 - 660K4*2 - 172K5*2 - 14K6**2 + 4442
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.522']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4726', 'vk6.5045', 'vk6.6253', 'vk6.6701', 'vk6.8223', 'vk6.8663', 'vk6.9605', 'vk6.9930', 'vk6.20299', 'vk6.21632', 'vk6.27591', 'vk6.29143', 'vk6.39013', 'vk6.41261', 'vk6.45777', 'vk6.47454', 'vk6.48758', 'vk6.48959', 'vk6.49557', 'vk6.49771', 'vk6.50768', 'vk6.50972', 'vk6.51245', 'vk6.51452', 'vk6.57150', 'vk6.58334', 'vk6.61772', 'vk6.62891', 'vk6.66771', 'vk6.67647', 'vk6.69415', 'vk6.70137']
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: [-3,-1,-1,1,2,2,-1,1,1,3,4,1,0,1,1,0,2,2,1,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.522']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.522', '6.559']

Outer characteristic polynomial of the knot is: t^7+64t^5+87t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.522']

2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 608*K1**4*K2 - 976*K1**4 + 128*K1**3*K2**3*K3 + 1024*K1**3*K2*K3 - 576*K1**3*K3 - 704*K1**2*K2**4 + 2752*K1**2*K2**3 - 448*K1**2*K2**2*K3**2 - 128*K1**2*K2**2*K4**2 + 288*K1**2*K2**2*K4 - 9664*K1**2*K2**2 + 128*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 8496*K1**2*K2 - 688*K1**2*K3**2 - 96*K1**2*K4**2 - 5624*K1**2 + 2624*K1*K2**3*K3 + 672*K1*K2**2*K3*K4 - 2560*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 - 576*K1*K2**2*K5 + 64*K1*K2*K3**3 - 512*K1*K2*K3*K4 + 9376*K1*K2*K3 + 1040*K1*K3*K4 + 184*K1*K4*K5 - 64*K2**6 - 256*K2**4*K3**2 - 192*K2**4*K4**2 + 832*K2**4*K4 - 3888*K2**4 + 256*K2**3*K3*K5 + 128*K2**3*K4*K6 - 128*K2**3*K6 - 1888*K2**2*K3**2 - 728*K2**2*K4**2 + 3008*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 2858*K2**2 - 32*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 944*K2*K3*K5 + 136*K2*K4*K6 + 16*K2*K5*K7 - 2356*K3**2 - 660*K4**2 - 172*K5**2 - 14*K6**2 + 4442

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.522']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4726', 'vk6.5045', 'vk6.6253', 'vk6.6701', 'vk6.8223', 'vk6.8663', 'vk6.9605', 'vk6.9930', 'vk6.20299', 'vk6.21632', 'vk6.27591', 'vk6.29143', 'vk6.39013', 'vk6.41261', 'vk6.45777', 'vk6.47454', 'vk6.48758', 'vk6.48959', 'vk6.49557', 'vk6.49771', 'vk6.50768', 'vk6.50972', 'vk6.51245', 'vk6.51452', 'vk6.57150', 'vk6.58334', 'vk6.61772', 'vk6.62891', 'vk6.66771', 'vk6.67647', 'vk6.69415', 'vk6.70137']

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	O1O2O3O4O5U2U3U5O6U4U1U6
R3 orbit	{'O1O2O3O4O5U2U3U5O6U4U1U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U5U2O6U1U3U4
Gauss code of K*	O1O2O3U4O5O6O4U6U1U2U5U3
Gauss code of -K*	O1O2O3U1O4O5O6U4U3U5U6U2
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 -1 -3 -1 1 2 2],[ 1 0 -3 -1 2 2 2],[ 3 3 0 1 3 2 1],[ 1 1 -1 0 2 1 1],[-1 -2 -3 -2 0 0 1],[-2 -2 -2 -1 0 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 2 2 1 -1 -1 -3],[-2 0 0 0 -1 -2 -2],[-2 0 0 -1 -1 -2 -1],[-1 0 1 0 -2 -2 -3],[ 1 1 1 2 0 1 -1],[ 1 2 2 2 -1 0 -3],[ 3 2 1 3 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,1,1,3,0,0,1,2,2,1,1,2,1,2,2,3,-1,1,3]
Phi over symmetry	[-3,-1,-1,1,2,2,-1,1,1,3,4,1,0,1,1,0,2,2,1,0,0]
Phi of -K	[-3,-1,-1,1,2,2,-1,1,1,3,4,1,0,1,1,0,2,2,1,0,0]
Phi of K*	[-2,-2,-1,1,1,3,0,0,1,2,4,1,1,2,3,0,0,1,-1,-1,1]
Phi of -K*	[-3,-1,-1,1,2,2,1,3,3,1,2,1,2,1,1,2,2,2,1,0,0]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	8z^2+29z+27
Enhanced Jones-Krushkal polynomial	8w^3z^2+29w^2z+27w
Inner characteristic polynomial	t^6+44t^4+27t^2+1
Outer characteristic polynomial	t^7+64t^5+87t^3+7t
Flat arrow polynomial	8K13 + 8K1*2K2 - 8K12 - 6K1K2 - 4K1K3 - 3K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial	-512K14K2*2 + 608K1*4K2 - 976K14 + 128K1*3K2*3K3 + 1024K13K2K3 - 576K1*3K3 - 704K12K2*4 + 2752K1*2K2*3 - 448K1*2K2*2K3*2 - 128K1*2K2*2K4*2 + 288K1*2K2*2K4 - 9664K12K2*2 + 128K1*2K2K32 - 864K1*2K2K4 + 8496K1*2K2 - 688K12K3*2 - 96K1*2K4*2 - 5624K1*2 + 2624K1K23K3 + 672K1K2*2K3K4 - 2560K1K22K3 + 192K1K2*2K4K5 - 576K1K22K5 + 64K1K2K33 - 512K1K2K3K4 + 9376K1K2K3 + 1040K1K3K4 + 184K1K4K5 - 64K26 - 256K2*4K3*2 - 192K2*4K4*2 + 832K2*4K4 - 3888K24 + 256K2*3K3K5 + 128K2*3K4K6 - 128K2*3K6 - 1888K22K3*2 - 728K2*2K4*2 + 3008K2*2K4 - 128K22K5*2 - 16K2*2K6*2 - 2858K2*2 - 32K2K32K4 - 32K2K3K4K5 + 944K2K3K5 + 136K2K4K6 + 16K2K5K7 - 2356K3*2 - 660K4*2 - 172K5*2 - 14K6**2 + 4442
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}], [{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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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