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Flat knot 6.519

Min(phi) over symmetries of the knot is: [-3,-3,1,1,2,2,0,0,1,3,3,1,2,3,4,0,-1,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.519']
Arrow polynomial of the knot is: 4K12K2 - 4K1K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']
Outer characteristic polynomial of the knot is: t^7+84t^5+194t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.519']
2-strand cable arrow polynomial of the knot is: 480K12K2*3 - 1760K1*2K2*2 - 256K1*2K2K4 + 2520K1*2K2 - 2928K12 + 352K1K23K3 + 224K1K2*2K3K4 - 1312K1K22K3 + 96K1K2*2K4K5 - 544K1K22K5 - 160K1K2K3K4 + 4360K1K2K3 - 64K1K2K4K5 + 864K1K3K4 + 384K1K4K5 + 16K1K5K6 - 32K24K4*2 + 128K2*4K4 - 976K24 + 64K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 1216K22K3*2 - 32K2*2K3K7 - 288K2*2K4*2 - 32K2*2K4K8 + 1696K2*2K4 - 128K22K5*2 - 16K2*2K6*2 - 2768K2*2 - 192K2K32K4 - 64K2K3K4K5 + 1688K2K3K5 + 224K2K4K6 + 72K2K5K7 + 16K2K6K8 + 48K32K6 - 1904K32 - 744K4*2 - 608K5*2 - 48K6*2 - 2K8**2 + 2840
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.519']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73621', 'vk6.73636', 'vk6.73638', 'vk6.74409', 'vk6.74414', 'vk6.75021', 'vk6.75028', 'vk6.75575', 'vk6.75579', 'vk6.75601', 'vk6.75606', 'vk6.76594', 'vk6.76599', 'vk6.76955', 'vk6.78538', 'vk6.78548', 'vk6.78568', 'vk6.78576', 'vk6.79449', 'vk6.79863', 'vk6.79870', 'vk6.80235', 'vk6.80894', 'vk6.80899', 'vk6.83701', 'vk6.84854', 'vk6.84868', 'vk6.85684', 'vk6.85686', 'vk6.87616', 'vk6.88437', 'vk6.89285']
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,-3,1,1,2,2,0,0,1,3,3,1,2,3,4,0,-1,1,0,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.519']

Arrow polynomial of the knot is: 4*K1**2*K2 - 4*K1*K3 + K4

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.111', '6.139', '6.519', '6.566', '6.1228', '6.1254', '6.1259', '6.1912', '6.1936']

Outer characteristic polynomial of the knot is: t^7+84t^5+194t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.519']

2-strand cable arrow polynomial of the knot is: 480*K1**2*K2**3 - 1760*K1**2*K2**2 - 256*K1**2*K2*K4 + 2520*K1**2*K2 - 2928*K1**2 + 352*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 1312*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 544*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4360*K1*K2*K3 - 64*K1*K2*K4*K5 + 864*K1*K3*K4 + 384*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**4*K4**2 + 128*K2**4*K4 - 976*K2**4 + 64*K2**3*K3*K5 + 64*K2**3*K4*K6 - 64*K2**3*K6 - 1216*K2**2*K3**2 - 32*K2**2*K3*K7 - 288*K2**2*K4**2 - 32*K2**2*K4*K8 + 1696*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 2768*K2**2 - 192*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 1688*K2*K3*K5 + 224*K2*K4*K6 + 72*K2*K5*K7 + 16*K2*K6*K8 + 48*K3**2*K6 - 1904*K3**2 - 744*K4**2 - 608*K5**2 - 48*K6**2 - 2*K8**2 + 2840

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.519']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73621', 'vk6.73636', 'vk6.73638', 'vk6.74409', 'vk6.74414', 'vk6.75021', 'vk6.75028', 'vk6.75575', 'vk6.75579', 'vk6.75601', 'vk6.75606', 'vk6.76594', 'vk6.76599', 'vk6.76955', 'vk6.78538', 'vk6.78548', 'vk6.78568', 'vk6.78576', 'vk6.79449', 'vk6.79863', 'vk6.79870', 'vk6.80235', 'vk6.80894', 'vk6.80899', 'vk6.83701', 'vk6.84854', 'vk6.84868', 'vk6.85684', 'vk6.85686', 'vk6.87616', 'vk6.88437', 'vk6.89285']

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	O1O2O3O4O5U2U1U5O6U4U3U6
R3 orbit	{'O1O2O3O4O5U2U1U5O6U4U3U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U3U2O6U1U5U4
Gauss code of K*	O1O2O3U4O5O6O4U2U1U6U5U3
Gauss code of -K*	O1O2O3U1O4O5O6U4U3U2U6U5
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 -3 -3 1 1 2 2],[ 3 0 0 4 3 2 2],[ 3 0 0 3 2 1 2],[-1 -4 -3 0 0 0 2],[-1 -3 -2 0 0 0 1],[-2 -2 -1 0 0 0 0],[-2 -2 -2 -2 -1 0 0]]
Primitive based matrix	[[ 0 2 2 1 1 -3 -3],[-2 0 0 0 0 -1 -2],[-2 0 0 -1 -2 -2 -2],[-1 0 1 0 0 -2 -3],[-1 0 2 0 0 -3 -4],[ 3 1 2 2 3 0 0],[ 3 2 2 3 4 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,-1,3,3,0,0,0,1,2,1,2,2,2,0,2,3,3,4,0]
Phi over symmetry	[-3,-3,1,1,2,2,0,0,1,3,3,1,2,3,4,0,-1,1,0,1,0]
Phi of -K	[-3,-3,1,1,2,2,0,0,1,3,3,1,2,3,4,0,-1,1,0,1,0]
Phi of K*	[-2,-2,-1,-1,3,3,0,-1,0,3,3,1,1,3,4,0,0,1,1,2,0]
Phi of -K*	[-3,-3,1,1,2,2,0,2,3,1,2,3,4,2,2,0,0,1,0,2,0]
Symmetry type of based matrix	c
u-polynomial	2t^3-2t^2-2t
Normalized Jones-Krushkal polynomial	8z^2+25z+19
Enhanced Jones-Krushkal polynomial	-2w^4z^2+10w^3z^2-2w^3z+27w^2z+19w
Inner characteristic polynomial	t^6+56t^4+48t^2
Outer characteristic polynomial	t^7+84t^5+194t^3+8t
Flat arrow polynomial	4K12K2 - 4K1K3 + K4
2-strand cable arrow polynomial	480K12K2*3 - 1760K1*2K2*2 - 256K1*2K2K4 + 2520K1*2K2 - 2928K12 + 352K1K23K3 + 224K1K2*2K3K4 - 1312K1K22K3 + 96K1K2*2K4K5 - 544K1K22K5 - 160K1K2K3K4 + 4360K1K2K3 - 64K1K2K4K5 + 864K1K3K4 + 384K1K4K5 + 16K1K5K6 - 32K24K4*2 + 128K2*4K4 - 976K24 + 64K2*3K3K5 + 64K2*3K4K6 - 64K2*3K6 - 1216K22K3*2 - 32K2*2K3K7 - 288K2*2K4*2 - 32K2*2K4K8 + 1696K2*2K4 - 128K22K5*2 - 16K2*2K6*2 - 2768K2*2 - 192K2K32K4 - 64K2K3K4K5 + 1688K2K3K5 + 224K2K4K6 + 72K2K5K7 + 16K2K6K8 + 48K32K6 - 1904K32 - 744K4*2 - 608K5*2 - 48K6*2 - 2K8**2 + 2840
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}], [{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}, {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}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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