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Flat knot 6.482

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,1,4,2,2,3,2,1,1,2,0,2,2,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.482']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 6K12 - 4K1K2 - 2K1K3 - K1 + 2K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.94', '6.482', '6.492']
Outer characteristic polynomial of the knot is: t^7+81t^5+96t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.482']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 544K1*4K2 - 624K14 + 128K1*3K2*3K3 - 384K13K2*2K3 + 544K13K2K3 - 512K1*3K3 - 256K12K2*4 - 128K1*2K2*3K4 + 2176K12K2*3 - 192K1*2K2*2K3*2 + 128K1*2K2*2K4 - 7088K12K2*2 + 192K1*2K2K32 - 576K1*2K2K4 + 6512K1*2K2 - 272K12K3*2 - 4936K1*2 - 128K1K24K3 + 2336K1K2*3K3 + 448K1K2*2K3K4 - 2048K1K22K3 + 128K1K2*2K4K5 - 416K1K22K5 - 416K1K2K3K4 + 7400K1K2K3 - 96K1K2K4K5 + 856K1K3K4 + 88K1K4K5 + 24K1K5K6 - 32K26 - 128K2*4K3*2 - 32K2*4K4*2 + 352K2*4K4 - 3224K24 + 256K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 - 1872K22K3*2 - 32K2*2K3K7 - 416K2*2K4*2 + 2384K2*2K4 - 160K22K5*2 - 8K2*2K6*2 - 2330K2*2 + 960K2K3K5 + 128K2K4K6 + 16K2K5K7 - 2124K32 - 544K4*2 - 156K5*2 - 22K6**2 + 3774
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.482']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72569', 'vk6.72572', 'vk6.72672', 'vk6.72677', 'vk6.72985', 'vk6.72993', 'vk6.73144', 'vk6.73149', 'vk6.74209', 'vk6.74214', 'vk6.74841', 'vk6.74845', 'vk6.76400', 'vk6.76407', 'vk6.76891', 'vk6.77856', 'vk6.77887', 'vk6.77895', 'vk6.78000', 'vk6.79251', 'vk6.79256', 'vk6.79736', 'vk6.80746', 'vk6.80748', 'vk6.81149', 'vk6.81152', 'vk6.82310', 'vk6.83984', 'vk6.86359', 'vk6.87274', 'vk6.88229', 'vk6.88230']
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,-1,0,1,2,2,1,4,2,2,3,2,1,1,2,0,2,2,1,1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.482']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.94', '6.482', '6.492']

Outer characteristic polynomial of the knot is: t^7+81t^5+96t^3+13t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.482']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 544*K1**4*K2 - 624*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 544*K1**3*K2*K3 - 512*K1**3*K3 - 256*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2176*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 7088*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 576*K1**2*K2*K4 + 6512*K1**2*K2 - 272*K1**2*K3**2 - 4936*K1**2 - 128*K1*K2**4*K3 + 2336*K1*K2**3*K3 + 448*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 - 416*K1*K2*K3*K4 + 7400*K1*K2*K3 - 96*K1*K2*K4*K5 + 856*K1*K3*K4 + 88*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 352*K2**4*K4 - 3224*K2**4 + 256*K2**3*K3*K5 + 32*K2**3*K4*K6 - 64*K2**3*K6 - 1872*K2**2*K3**2 - 32*K2**2*K3*K7 - 416*K2**2*K4**2 + 2384*K2**2*K4 - 160*K2**2*K5**2 - 8*K2**2*K6**2 - 2330*K2**2 + 960*K2*K3*K5 + 128*K2*K4*K6 + 16*K2*K5*K7 - 2124*K3**2 - 544*K4**2 - 156*K5**2 - 22*K6**2 + 3774

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.482']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.72569', 'vk6.72572', 'vk6.72672', 'vk6.72677', 'vk6.72985', 'vk6.72993', 'vk6.73144', 'vk6.73149', 'vk6.74209', 'vk6.74214', 'vk6.74841', 'vk6.74845', 'vk6.76400', 'vk6.76407', 'vk6.76891', 'vk6.77856', 'vk6.77887', 'vk6.77895', 'vk6.78000', 'vk6.79251', 'vk6.79256', 'vk6.79736', 'vk6.80746', 'vk6.80748', 'vk6.81149', 'vk6.81152', 'vk6.82310', 'vk6.83984', 'vk6.86359', 'vk6.87274', 'vk6.88229', 'vk6.88230']

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	O1O2O3O4O5U1U3U4O6U5U2U6
R3 orbit	{'O1O2O3O4O5U1U3U4O6U5U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U4U1O6U2U3U5
Gauss code of K*	O1O2O3U4O5O6O4U1U6U2U3U5
Gauss code of -K*	O1O2O3U1O4O5O6U3U4U5U2U6
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 1 2 2],[ 4 0 4 1 2 3 2],[ 0 -4 0 -2 0 2 2],[ 1 -1 2 0 1 2 1],[-1 -2 0 -1 0 1 1],[-2 -3 -2 -2 -1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 2 1 0 -1 -4],[-2 0 1 -1 -2 -2 -3],[-2 -1 0 -1 -2 -1 -2],[-1 1 1 0 0 -1 -2],[ 0 2 2 0 0 -2 -4],[ 1 2 1 1 2 0 -1],[ 4 3 2 2 4 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,1,4,-1,1,2,2,3,1,2,1,2,0,1,2,2,4,1]
Phi over symmetry	[-4,-1,0,1,2,2,1,4,2,2,3,2,1,1,2,0,2,2,1,1,-1]
Phi of -K	[-4,-1,0,1,2,2,2,0,3,3,4,-1,1,1,2,1,0,0,0,0,-1]
Phi of K*	[-2,-2,-1,0,1,4,-1,0,0,2,4,0,0,1,3,1,1,3,-1,0,2]
Phi of -K*	[-4,-1,0,1,2,2,1,4,2,2,3,2,1,1,2,0,2,2,1,1,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^2
Normalized Jones-Krushkal polynomial	8z^2+29z+27
Enhanced Jones-Krushkal polynomial	8w^3z^2+29w^2z+27w
Inner characteristic polynomial	t^6+55t^4+14t^2
Outer characteristic polynomial	t^7+81t^5+96t^3+13t
Flat arrow polynomial	4K13 + 4K1*2K2 - 6K12 - 4K1K2 - 2K1K3 - K1 + 2K2 + K3 + 3
2-strand cable arrow polynomial	-192K14K2*2 + 544K1*4K2 - 624K14 + 128K1*3K2*3K3 - 384K13K2*2K3 + 544K13K2K3 - 512K1*3K3 - 256K12K2*4 - 128K1*2K2*3K4 + 2176K12K2*3 - 192K1*2K2*2K3*2 + 128K1*2K2*2K4 - 7088K12K2*2 + 192K1*2K2K32 - 576K1*2K2K4 + 6512K1*2K2 - 272K12K3*2 - 4936K1*2 - 128K1K24K3 + 2336K1K2*3K3 + 448K1K2*2K3K4 - 2048K1K22K3 + 128K1K2*2K4K5 - 416K1K22K5 - 416K1K2K3K4 + 7400K1K2K3 - 96K1K2K4K5 + 856K1K3K4 + 88K1K4K5 + 24K1K5K6 - 32K26 - 128K2*4K3*2 - 32K2*4K4*2 + 352K2*4K4 - 3224K24 + 256K2*3K3K5 + 32K2*3K4K6 - 64K2*3K6 - 1872K22K3*2 - 32K2*2K3K7 - 416K2*2K4*2 + 2384K2*2K4 - 160K22K5*2 - 8K2*2K6*2 - 2330K2*2 + 960K2K3K5 + 128K2K4K6 + 16K2K5K7 - 2124K32 - 544K4*2 - 156K5*2 - 22K6**2 + 3774
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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