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Flat knot 6.58

Min(phi) over symmetries of the knot is: [-4,-4,-1,2,3,4,0,1,2,4,3,2,3,5,4,1,3,2,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.58']
Arrow polynomial of the knot is: -2K1K2 - 2K1K3 + K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511']
Outer characteristic polynomial of the knot is: t^7+163t^5+307t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.58']
2-strand cable arrow polynomial of the knot is: -416K13K3 - 128K12K2*2 + 128K1*2K2K32 + 1776K1*2K2 - 1280K12K3*2 - 3684K1*2 + 64K1K23K3 - 448K1K2*2K3 + 480K1K2K33 - 160K1K2K3K4 - 224K1K2K3K6 + 5896K1K2K3 - 32K1K3*2K5 + 1928K1K3K4 + 16K1K4K5 + 8K1K5K6 + 8K1K6K7 - 32K24 - 256K2*2K3*4 + 224K2*2K3*2K6 - 2608K22K3*2 - 8K2*2K4*2 + 280K2*2K4 - 72K22K6*2 - 2762K2*2 + 32K2K33K5 - 224K2K3*2K4 + 1656K2K3K5 + 144K2K4K6 + 8K2K6K8 - 192K3*4 - 64K3*2K4*2 + 176K3*2K6 - 2552K32 + 72K3K4K7 - 686K42 - 172K5*2 - 54K6*2 - 24K7*2 - 2K8**2 + 3182
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.58']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81841', 'vk6.81895', 'vk6.82061', 'vk6.82087', 'vk6.82557', 'vk6.82615', 'vk6.82774', 'vk6.82787', 'vk6.82826', 'vk6.82838', 'vk6.82951', 'vk6.83053', 'vk6.83067', 'vk6.83269', 'vk6.83330', 'vk6.83370', 'vk6.83524', 'vk6.84537', 'vk6.84644', 'vk6.84905', 'vk6.84960', 'vk6.85823', 'vk6.86094', 'vk6.86114', 'vk6.86160', 'vk6.86836', 'vk6.88454', 'vk6.88889', 'vk6.89036', 'vk6.89689', 'vk6.89925', 'vk6.90017']
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,-4,-1,2,3,4,0,1,2,4,3,2,3,5,4,1,3,2,1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.58']

Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511']

Outer characteristic polynomial of the knot is: t^7+163t^5+307t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.58']

2-strand cable arrow polynomial of the knot is: -416*K1**3*K3 - 128*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 1776*K1**2*K2 - 1280*K1**2*K3**2 - 3684*K1**2 + 64*K1*K2**3*K3 - 448*K1*K2**2*K3 + 480*K1*K2*K3**3 - 160*K1*K2*K3*K4 - 224*K1*K2*K3*K6 + 5896*K1*K2*K3 - 32*K1*K3**2*K5 + 1928*K1*K3*K4 + 16*K1*K4*K5 + 8*K1*K5*K6 + 8*K1*K6*K7 - 32*K2**4 - 256*K2**2*K3**4 + 224*K2**2*K3**2*K6 - 2608*K2**2*K3**2 - 8*K2**2*K4**2 + 280*K2**2*K4 - 72*K2**2*K6**2 - 2762*K2**2 + 32*K2*K3**3*K5 - 224*K2*K3**2*K4 + 1656*K2*K3*K5 + 144*K2*K4*K6 + 8*K2*K6*K8 - 192*K3**4 - 64*K3**2*K4**2 + 176*K3**2*K6 - 2552*K3**2 + 72*K3*K4*K7 - 686*K4**2 - 172*K5**2 - 54*K6**2 - 24*K7**2 - 2*K8**2 + 3182

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.58']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81841', 'vk6.81895', 'vk6.82061', 'vk6.82087', 'vk6.82557', 'vk6.82615', 'vk6.82774', 'vk6.82787', 'vk6.82826', 'vk6.82838', 'vk6.82951', 'vk6.83053', 'vk6.83067', 'vk6.83269', 'vk6.83330', 'vk6.83370', 'vk6.83524', 'vk6.84537', 'vk6.84644', 'vk6.84905', 'vk6.84960', 'vk6.85823', 'vk6.86094', 'vk6.86114', 'vk6.86160', 'vk6.86836', 'vk6.88454', 'vk6.88889', 'vk6.89036', 'vk6.89689', 'vk6.89925', 'vk6.90017']

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	O1O2O3O4O5O6U2U1U3U5U6U4
R3 orbit	{'O1O2O3O4O5O6U2U1U3U5U6U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U3U1U2U4U6U5
Gauss code of K*	O1O2O3O4O5O6U2U1U3U6U4U5
Gauss code of -K*	O1O2O3O4O5O6U2U3U1U4U6U5
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 -4 -1 3 2 4],[ 4 0 0 2 5 3 4],[ 4 0 0 1 4 2 3],[ 1 -2 -1 0 3 1 2],[-3 -5 -4 -3 0 -1 1],[-2 -3 -2 -1 1 0 1],[-4 -4 -3 -2 -1 -1 0]]
Primitive based matrix	[[ 0 4 3 2 -1 -4 -4],[-4 0 -1 -1 -2 -3 -4],[-3 1 0 -1 -3 -4 -5],[-2 1 1 0 -1 -2 -3],[ 1 2 3 1 0 -1 -2],[ 4 3 4 2 1 0 0],[ 4 4 5 3 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-3,-2,1,4,4,1,1,2,3,4,1,3,4,5,1,2,3,1,2,0]
Phi over symmetry	[-4,-4,-1,2,3,4,0,1,2,4,3,2,3,5,4,1,3,2,1,1,1]
Phi of -K	[-4,-4,-1,2,3,4,0,1,3,2,4,2,4,3,5,2,1,3,0,1,0]
Phi of K*	[-4,-3,-2,1,4,4,0,1,3,4,5,0,1,2,3,2,3,4,1,2,0]
Phi of -K*	[-4,-4,-1,2,3,4,0,1,2,4,3,2,3,5,4,1,3,2,1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial	7z^2+26z+25
Enhanced Jones-Krushkal polynomial	-2w^4z^2+9w^3z^2+26w^2z+25w
Inner characteristic polynomial	t^6+101t^4+50t^2
Outer characteristic polynomial	t^7+163t^5+307t^3+9t
Flat arrow polynomial	-2K1K2 - 2K1K3 + K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial	-416K13K3 - 128K12K2*2 + 128K1*2K2K32 + 1776K1*2K2 - 1280K12K3*2 - 3684K1*2 + 64K1K23K3 - 448K1K2*2K3 + 480K1K2K33 - 160K1K2K3K4 - 224K1K2K3K6 + 5896K1K2K3 - 32K1K3*2K5 + 1928K1K3K4 + 16K1K4K5 + 8K1K5K6 + 8K1K6K7 - 32K24 - 256K2*2K3*4 + 224K2*2K3*2K6 - 2608K22K3*2 - 8K2*2K4*2 + 280K2*2K4 - 72K22K6*2 - 2762K2*2 + 32K2K33K5 - 224K2K3*2K4 + 1656K2K3K5 + 144K2K4K6 + 8K2K6K8 - 192K3*4 - 64K3*2K4*2 + 176K3*2K6 - 2552K32 + 72K3K4K7 - 686K42 - 172K5*2 - 54K6*2 - 24K7*2 - 2K8**2 + 3182
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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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