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Flat knot 6.359

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,0,1,4,2,4,0,1,0,1,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.359']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 6K12 - 4K1K2 - 2K1K3 - 4K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.169', '6.359']
Outer characteristic polynomial of the knot is: t^7+71t^5+54t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.359']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 448K1*4K2 - 704K14 + 288K1*3K2K3 - 160K1*3K3 - 256K12K2*4 + 832K1*2K2*3 - 128K1*2K2*2K3*2 + 64K1*2K2*2K4 - 3136K12K2*2 + 64K1*2K2K32 + 32K1*2K2K3K5 - 224K12K2K4 + 2992K1*2K2 - 160K12K3*2 - 1764K1*2 + 768K1K23K3 + 128K1K2*2K3K4 - 576K1K22K3 - 224K1K2*2K5 - 64K1K2K3K4 + 2776K1K2K3 + 200K1K3K4 - 64K26 - 64K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 1000K24 + 128K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 544K22K3*2 - 144K2*2K4*2 + 776K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 1076K2*2 + 320K2K3K5 + 40K2K4K6 + 8K3*2K6 - 676K32 - 164K4*2 - 64K5*2 - 12K6**2 + 1490
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.359']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11459', 'vk6.11762', 'vk6.12778', 'vk6.13116', 'vk6.13380', 'vk6.13459', 'vk6.13650', 'vk6.13762', 'vk6.14163', 'vk6.14402', 'vk6.15631', 'vk6.16087', 'vk6.16466', 'vk6.16481', 'vk6.17632', 'vk6.22105', 'vk6.22869', 'vk6.22900', 'vk6.28164', 'vk6.29587', 'vk6.33131', 'vk6.33176', 'vk6.33240', 'vk6.33291', 'vk6.34846', 'vk6.34877', 'vk6.39608', 'vk6.41847', 'vk6.42436', 'vk6.42451', 'vk6.46224', 'vk6.47829', 'vk6.52215', 'vk6.52490', 'vk6.53052', 'vk6.53370', 'vk6.53556', 'vk6.53599', 'vk6.53632', 'vk6.53690']
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,-1,0,1,2,2,0,1,4,2,4,0,1,0,1,1,1,1,1,1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.359']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.169', '6.359']

Outer characteristic polynomial of the knot is: t^7+71t^5+54t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.359']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 704*K1**4 + 288*K1**3*K2*K3 - 160*K1**3*K3 - 256*K1**2*K2**4 + 832*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 3136*K1**2*K2**2 + 64*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 224*K1**2*K2*K4 + 2992*K1**2*K2 - 160*K1**2*K3**2 - 1764*K1**2 + 768*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 576*K1*K2**2*K3 - 224*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 2776*K1*K2*K3 + 200*K1*K3*K4 - 64*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1000*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 544*K2**2*K3**2 - 144*K2**2*K4**2 + 776*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 1076*K2**2 + 320*K2*K3*K5 + 40*K2*K4*K6 + 8*K3**2*K6 - 676*K3**2 - 164*K4**2 - 64*K5**2 - 12*K6**2 + 1490

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.359']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11459', 'vk6.11762', 'vk6.12778', 'vk6.13116', 'vk6.13380', 'vk6.13459', 'vk6.13650', 'vk6.13762', 'vk6.14163', 'vk6.14402', 'vk6.15631', 'vk6.16087', 'vk6.16466', 'vk6.16481', 'vk6.17632', 'vk6.22105', 'vk6.22869', 'vk6.22900', 'vk6.28164', 'vk6.29587', 'vk6.33131', 'vk6.33176', 'vk6.33240', 'vk6.33291', 'vk6.34846', 'vk6.34877', 'vk6.39608', 'vk6.41847', 'vk6.42436', 'vk6.42451', 'vk6.46224', 'vk6.47829', 'vk6.52215', 'vk6.52490', 'vk6.53052', 'vk6.53370', 'vk6.53556', 'vk6.53599', 'vk6.53632', 'vk6.53690']

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	O1O2O3O4O5U3U4O6U2U6U1U5
R3 orbit	{'O1O2O3O4O5U3U4U1O6U2U6U5', 'O1O2O3O4O5U3U4O6U2U6U1U5'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U1U5U6U4O6U2U3
Gauss code of K*	O1O2O3O4U3U1U5U6U4O5O6U2
Gauss code of -K*	O1O2O3O4U3O5O6U1U5U6U4U2
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 -2 -2 0 4 1],[ 1 0 -1 -1 1 4 1],[ 2 1 0 -1 1 4 1],[ 2 1 1 0 1 2 0],[ 0 -1 -1 -1 0 1 0],[-4 -4 -4 -2 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix	[[ 0 4 1 0 -1 -2 -2],[-4 0 0 -1 -4 -2 -4],[-1 0 0 0 -1 0 -1],[ 0 1 0 0 -1 -1 -1],[ 1 4 1 1 0 -1 -1],[ 2 2 0 1 1 0 1],[ 2 4 1 1 1 -1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-1,0,1,2,2,0,1,4,2,4,0,1,0,1,1,1,1,1,1,-1]
Phi over symmetry	[-4,-1,0,1,2,2,0,1,4,2,4,0,1,0,1,1,1,1,1,1,-1]
Phi of -K	[-2,-2,-1,0,1,4,-1,0,1,3,4,0,1,2,2,0,1,1,1,3,3]
Phi of K*	[-4,-1,0,1,2,2,3,3,1,2,4,1,1,2,3,0,1,1,0,0,-1]
Phi of -K*	[-2,-2,-1,0,1,4,-1,1,1,1,4,1,1,0,2,1,1,4,0,1,0]
Symmetry type of based matrix	c
u-polynomial	-t^4+2t^2
Normalized Jones-Krushkal polynomial	3z^2+16z+21
Enhanced Jones-Krushkal polynomial	3w^3z^2+16w^2z+21w
Inner characteristic polynomial	t^6+45t^4+20t^2
Outer characteristic polynomial	t^7+71t^5+54t^3+3t
Flat arrow polynomial	8K13 + 4K1*2K2 - 6K12 - 4K1K2 - 2K1K3 - 4K1 + 2*K2 + 3
2-strand cable arrow polynomial	-256K14K2*2 + 448K1*4K2 - 704K14 + 288K1*3K2K3 - 160K1*3K3 - 256K12K2*4 + 832K1*2K2*3 - 128K1*2K2*2K3*2 + 64K1*2K2*2K4 - 3136K12K2*2 + 64K1*2K2K32 + 32K1*2K2K3K5 - 224K12K2K4 + 2992K1*2K2 - 160K12K3*2 - 1764K1*2 + 768K1K23K3 + 128K1K2*2K3K4 - 576K1K22K3 - 224K1K2*2K5 - 64K1K2K3K4 + 2776K1K2K3 + 200K1K3K4 - 64K26 - 64K2*4K3*2 - 32K2*4K4*2 + 192K2*4K4 - 1000K24 + 128K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 544K22K3*2 - 144K2*2K4*2 + 776K2*2K4 - 48K22K5*2 - 8K2*2K6*2 - 1076K2*2 + 320K2K3K5 + 40K2K4K6 + 8K3*2K6 - 676K32 - 164K4*2 - 64K5*2 - 12K6**2 + 1490
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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