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Flat knot 6.486

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,1,1,3,4,2,1,2,2,1,-1,-1,0,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.486']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 14K12 - 8K1K2 + K1 - 2K2*2 + 5K2 + 3*K3 + 8
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.455', '6.486']
Outer characteristic polynomial of the knot is: t^7+68t^5+104t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.486']
2-strand cable arrow polynomial of the knot is: -192K16 - 192K1*4K2*2 + 1376K1*4K2 - 4224K14 + 1056K1*3K2K3 + 96K1*3K3K4 - 992K1*3K3 - 128K12K2*4 + 960K1*2K2*3 + 288K1*2K2*2K4 - 7536K12K2*2 + 256K1*2K2K32 + 64K1*2K2K42 - 928K1*2K2K4 + 10728K1*2K2 - 1984K12K3*2 - 64K1*2K3K5 - 368K1*2K4*2 - 6056K1*2 + 992K1K23K3 + 32K1K2*2K3K4 - 1920K1K22K3 - 384K1K2*2K5 + 192K1K2K33 + 32K1K2K3K42 - 608K1K2K3K4 - 96K1K2K3K6 + 10608K1K2K3 + 2688K1K3K4 + 352K1K4K5 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 288K2*4K4 - 2168K24 + 32K2*3K3K5 - 96K2*3K6 + 64K22K3*2K4 - 1728K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 480K2*2K4*2 + 2744K2*2K4 - 5322K22 - 96K2K32K4 - 32K2K3K4K5 + 1328K2K3K5 + 352K2K4K6 - 160K34 - 80K3*2K4*2 + 152K3*2K6 - 3228K32 + 64K3K4K7 - 8K44 - 1182K4*2 - 260K5*2 - 94K6*2 - 8K7**2 + 5996
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.486']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13887', 'vk6.13984', 'vk6.14131', 'vk6.14354', 'vk6.14962', 'vk6.15085', 'vk6.15583', 'vk6.16053', 'vk6.16306', 'vk6.16329', 'vk6.17410', 'vk6.22617', 'vk6.22648', 'vk6.23922', 'vk6.33706', 'vk6.33783', 'vk6.34138', 'vk6.34266', 'vk6.34597', 'vk6.36197', 'vk6.36222', 'vk6.42292', 'vk6.53861', 'vk6.53904', 'vk6.54092', 'vk6.54406', 'vk6.54587', 'vk6.55570', 'vk6.59030', 'vk6.59059', 'vk6.60064', 'vk6.64546']
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,-1,1,1,1,2,1,1,3,4,2,1,2,2,1,-1,-1,0,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.486']

Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 14*K1**2 - 8*K1*K2 + K1 - 2*K2**2 + 5*K2 + 3*K3 + 8

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.455', '6.486']

Outer characteristic polynomial of the knot is: t^7+68t^5+104t^3+7t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.486']

2-strand cable arrow polynomial of the knot is: -192*K1**6 - 192*K1**4*K2**2 + 1376*K1**4*K2 - 4224*K1**4 + 1056*K1**3*K2*K3 + 96*K1**3*K3*K4 - 992*K1**3*K3 - 128*K1**2*K2**4 + 960*K1**2*K2**3 + 288*K1**2*K2**2*K4 - 7536*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 928*K1**2*K2*K4 + 10728*K1**2*K2 - 1984*K1**2*K3**2 - 64*K1**2*K3*K5 - 368*K1**2*K4**2 - 6056*K1**2 + 992*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1920*K1*K2**2*K3 - 384*K1*K2**2*K5 + 192*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 608*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 10608*K1*K2*K3 + 2688*K1*K3*K4 + 352*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 288*K2**4*K4 - 2168*K2**4 + 32*K2**3*K3*K5 - 96*K2**3*K6 + 64*K2**2*K3**2*K4 - 1728*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 480*K2**2*K4**2 + 2744*K2**2*K4 - 5322*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1328*K2*K3*K5 + 352*K2*K4*K6 - 160*K3**4 - 80*K3**2*K4**2 + 152*K3**2*K6 - 3228*K3**2 + 64*K3*K4*K7 - 8*K4**4 - 1182*K4**2 - 260*K5**2 - 94*K6**2 - 8*K7**2 + 5996

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.486']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13887', 'vk6.13984', 'vk6.14131', 'vk6.14354', 'vk6.14962', 'vk6.15085', 'vk6.15583', 'vk6.16053', 'vk6.16306', 'vk6.16329', 'vk6.17410', 'vk6.22617', 'vk6.22648', 'vk6.23922', 'vk6.33706', 'vk6.33783', 'vk6.34138', 'vk6.34266', 'vk6.34597', 'vk6.36197', 'vk6.36222', 'vk6.42292', 'vk6.53861', 'vk6.53904', 'vk6.54092', 'vk6.54406', 'vk6.54587', 'vk6.55570', 'vk6.59030', 'vk6.59059', 'vk6.60064', 'vk6.64546']

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	O1O2O3O4O5U1U3U5O6U4U6U2
R3 orbit	{'O1O2O3O4O5U1U3U5O6U4U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U6U2O6U1U3U5
Gauss code of K*	O1O2O3U4O5O4O6U1U6U2U5U3
Gauss code of -K*	O1O2O3U2O4O5O6U4U3U5U1U6
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 1 -1 1 2 1],[ 4 0 4 1 3 2 1],[-1 -4 0 -2 0 1 1],[ 1 -1 2 0 2 1 1],[-1 -3 0 -2 0 0 1],[-2 -2 -1 -1 0 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 2 1 1 1 -1 -4],[-2 0 0 0 -1 -1 -2],[-1 0 0 1 0 -2 -3],[-1 0 -1 0 -1 -1 -1],[-1 1 0 1 0 -2 -4],[ 1 1 2 1 2 0 -1],[ 4 2 3 1 4 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,-1,1,4,0,0,1,1,2,-1,0,2,3,1,1,1,2,4,1]
Phi over symmetry	[-4,-1,1,1,1,2,1,1,3,4,2,1,2,2,1,-1,-1,0,0,0,1]
Phi of -K	[-4,-1,1,1,1,2,2,1,2,4,4,0,0,1,2,0,-1,0,-1,1,1]
Phi of K*	[-2,-1,-1,-1,1,4,0,1,1,2,4,0,1,0,1,1,0,2,1,4,2]
Phi of -K*	[-4,-1,1,1,1,2,1,1,3,4,2,1,2,2,1,-1,-1,0,0,0,1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^2-2t
Normalized Jones-Krushkal polynomial	4z^2+25z+35
Enhanced Jones-Krushkal polynomial	4w^3z^2+25w^2z+35w
Inner characteristic polynomial	t^6+44t^4+29t^2+1
Outer characteristic polynomial	t^7+68t^5+104t^3+7t
Flat arrow polynomial	4K13 + 4K1*2K2 - 14K12 - 8K1K2 + K1 - 2K2*2 + 5K2 + 3*K3 + 8
2-strand cable arrow polynomial	-192K16 - 192K1*4K2*2 + 1376K1*4K2 - 4224K14 + 1056K1*3K2K3 + 96K1*3K3K4 - 992K1*3K3 - 128K12K2*4 + 960K1*2K2*3 + 288K1*2K2*2K4 - 7536K12K2*2 + 256K1*2K2K32 + 64K1*2K2K42 - 928K1*2K2K4 + 10728K1*2K2 - 1984K12K3*2 - 64K1*2K3K5 - 368K1*2K4*2 - 6056K1*2 + 992K1K23K3 + 32K1K2*2K3K4 - 1920K1K22K3 - 384K1K2*2K5 + 192K1K2K33 + 32K1K2K3K42 - 608K1K2K3K4 - 96K1K2K3K6 + 10608K1K2K3 + 2688K1K3K4 + 352K1K4K5 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 288K2*4K4 - 2168K24 + 32K2*3K3K5 - 96K2*3K6 + 64K22K3*2K4 - 1728K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 480K2*2K4*2 + 2744K2*2K4 - 5322K22 - 96K2K32K4 - 32K2K3K4K5 + 1328K2K3K5 + 352K2K4K6 - 160K34 - 80K3*2K4*2 + 152K3*2K6 - 3228K32 + 64K3K4K7 - 8K44 - 1182K4*2 - 260K5*2 - 94K6*2 - 8K7**2 + 5996
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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {1, 5}, {4}, {3}, {2}], [{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}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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