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Flat knot 6.96

Min(phi) over symmetries of the knot is: [-4,-3,1,1,2,3,0,3,4,2,4,2,3,1,3,0,0,1,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.96']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 8K12 - 4K1K2 - 2K1K3 - K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.96', '6.149', '6.269', '6.441', '6.457']
Outer characteristic polynomial of the knot is: t^7+104t^5+72t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.96']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 320K1*4K2 - 1120K14 + 160K1*3K2K3 - 96K1*3K3 - 256K12K2*4 + 672K1*2K2*3 + 64K1*2K2*2K4 - 3152K12K2*2 - 288K1*2K2K4 + 3528K1*2K2 - 256K12K3*2 - 32K1*2K3K5 - 1868K1*2 + 576K1K23K3 + 256K1K2*2K3K4 - 672K1K22K3 + 32K1K2*2K4K5 - 160K1K22K5 - 224K1K2K3K4 - 32K1K2K3K6 + 3280K1K2K3 + 592K1K3K4 + 88K1K4K5 + 8K1K5K6 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 864K24 + 64K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 592K22K3*2 - 304K2*2K4*2 + 1016K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 1410K2*2 - 32K2K32K4 + 392K2K3K5 + 120K2K4K6 + 8K2K5K7 - 880K3*2 - 354K4*2 - 76K5*2 - 14K6**2 + 1752
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.96']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17085', 'vk6.17326', 'vk6.20058', 'vk6.20255', 'vk6.21185', 'vk6.21561', 'vk6.23467', 'vk6.26901', 'vk6.27119', 'vk6.27488', 'vk6.28655', 'vk6.29084', 'vk6.35604', 'vk6.38322', 'vk6.38516', 'vk6.38904', 'vk6.40461', 'vk6.41103', 'vk6.42977', 'vk6.45198', 'vk6.45412', 'vk6.45657', 'vk6.47022', 'vk6.47389', 'vk6.55218', 'vk6.56736', 'vk6.56864', 'vk6.57837', 'vk6.59616', 'vk6.61165', 'vk6.61389', 'vk6.61624', 'vk6.62405', 'vk6.62803', 'vk6.65021', 'vk6.66571', 'vk6.68291', 'vk6.69084', 'vk6.69219', 'vk6.69866']
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,-3,1,1,2,3,0,3,4,2,4,2,3,1,3,0,0,1,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.96']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.96', '6.149', '6.269', '6.441', '6.457']

Outer characteristic polynomial of the knot is: t^7+104t^5+72t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.96']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 320*K1**4*K2 - 1120*K1**4 + 160*K1**3*K2*K3 - 96*K1**3*K3 - 256*K1**2*K2**4 + 672*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 3152*K1**2*K2**2 - 288*K1**2*K2*K4 + 3528*K1**2*K2 - 256*K1**2*K3**2 - 32*K1**2*K3*K5 - 1868*K1**2 + 576*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 160*K1*K2**2*K5 - 224*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3280*K1*K2*K3 + 592*K1*K3*K4 + 88*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 864*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 592*K2**2*K3**2 - 304*K2**2*K4**2 + 1016*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 1410*K2**2 - 32*K2*K3**2*K4 + 392*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 - 880*K3**2 - 354*K4**2 - 76*K5**2 - 14*K6**2 + 1752

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.96']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17085', 'vk6.17326', 'vk6.20058', 'vk6.20255', 'vk6.21185', 'vk6.21561', 'vk6.23467', 'vk6.26901', 'vk6.27119', 'vk6.27488', 'vk6.28655', 'vk6.29084', 'vk6.35604', 'vk6.38322', 'vk6.38516', 'vk6.38904', 'vk6.40461', 'vk6.41103', 'vk6.42977', 'vk6.45198', 'vk6.45412', 'vk6.45657', 'vk6.47022', 'vk6.47389', 'vk6.55218', 'vk6.56736', 'vk6.56864', 'vk6.57837', 'vk6.59616', 'vk6.61165', 'vk6.61389', 'vk6.61624', 'vk6.62405', 'vk6.62803', 'vk6.65021', 'vk6.66571', 'vk6.68291', 'vk6.69084', 'vk6.69219', 'vk6.69866']

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	O1O2O3O4O5O6U2U6U1U4U5U3
R3 orbit	{'O1O2O3O4O5U1O6U2U6U4U5U3', 'O1O2O3O4O5O6U2U6U1U4U5U3'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5O6U4U2U3U6U1U5
Gauss code of K*	O1O2O3O4O5O6U3U1U6U4U5U2
Gauss code of -K*	O1O2O3O4O5O6U5U2U3U1U6U4
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 -4 2 1 3 1],[ 3 0 -1 4 2 3 1],[ 4 1 0 4 2 3 1],[-2 -4 -4 0 -1 1 0],[-1 -2 -2 1 0 1 0],[-3 -3 -3 -1 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix	[[ 0 3 2 1 1 -3 -4],[-3 0 -1 0 -1 -3 -3],[-2 1 0 0 -1 -4 -4],[-1 0 0 0 0 -1 -1],[-1 1 1 0 0 -2 -2],[ 3 3 4 1 2 0 -1],[ 4 3 4 1 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,-1,3,4,1,0,1,3,3,0,1,4,4,0,1,1,2,2,1]
Phi over symmetry	[-4,-3,1,1,2,3,0,3,4,2,4,2,3,1,3,0,0,1,1,2,0]
Phi of -K	[-4,-3,1,1,2,3,0,3,4,2,4,2,3,1,3,0,0,1,1,2,0]
Phi of K*	[-3,-2,-1,-1,3,4,0,1,2,3,4,0,1,1,2,0,2,3,3,4,0]
Phi of -K*	[-4,-3,1,1,2,3,1,1,2,4,3,1,2,4,3,0,0,0,1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^2-2t
Normalized Jones-Krushkal polynomial	3z^2+16z+21
Enhanced Jones-Krushkal polynomial	3w^3z^2+16w^2z+21w
Inner characteristic polynomial	t^6+64t^4+11t^2
Outer characteristic polynomial	t^7+104t^5+72t^3+3t
Flat arrow polynomial	4K13 + 4K1*2K2 - 8K12 - 4K1K2 - 2K1K3 - K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-192K14K2*2 + 320K1*4K2 - 1120K14 + 160K1*3K2K3 - 96K1*3K3 - 256K12K2*4 + 672K1*2K2*3 + 64K1*2K2*2K4 - 3152K12K2*2 - 288K1*2K2K4 + 3528K1*2K2 - 256K12K3*2 - 32K1*2K3K5 - 1868K1*2 + 576K1K23K3 + 256K1K2*2K3K4 - 672K1K22K3 + 32K1K2*2K4K5 - 160K1K22K5 - 224K1K2K3K4 - 32K1K2K3K6 + 3280K1K2K3 + 592K1K3K4 + 88K1K4K5 + 8K1K5K6 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 864K24 + 64K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 592K22K3*2 - 304K2*2K4*2 + 1016K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 1410K2*2 - 32K2K32K4 + 392K2K3K5 + 120K2K4K6 + 8K2K5K7 - 880K3*2 - 354K4*2 - 76K5*2 - 14K6**2 + 1752
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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

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