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Flat knot 6.130

Min(phi) over symmetries of the knot is: [-3,-2,-1,2,2,2,0,2,2,3,3,2,1,2,2,2,2,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.130']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 2K12 - 2K1K2 - 2K1K3 - 2K1 - 2K2*2 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.130', '6.560']
Outer characteristic polynomial of the knot is: t^7+79t^5+72t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.130']
2-strand cable arrow polynomial of the knot is: -64K13K3 - 192K12K2*4 + 800K1*2K2*3 + 160K1*2K2*2K4 - 4304K12K2*2 + 96K1*2K2K32 - 864K1*2K2K4 + 5184K1*2K2 - 128K12K3*2 - 128K1*2K4*2 - 5096K1*2 + 352K1K23K3 + 160K1K2*2K3K4 - 1056K1K22K3 - 256K1K2*2K5 - 512K1K2K3K4 - 64K1K2K3K6 + 6272K1K2K3 - 64K1K2K4K5 - 32K1K2K4K7 + 1552K1K3K4 + 448K1K4K5 + 112K1K5K6 + 56K1K6K7 - 32K2*6 - 32K2*4K4*2 + 160K2*4K4 - 1176K24 + 64K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 656K22K3*2 + 32K2*2K4*3 - 424K2*2K4*2 + 2416K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 4148K2*2 - 96K2K3K4K5 + 976K2K3K5 - 32K2K4*2K6 + 408K2K4K6 + 96K2K5K7 + 8K2K6K8 + 8K3*2K6 - 2336K32 + 104K3K4K7 - 8K44 + 8K4*2K8 - 1400K42 - 440K5*2 - 180K6*2 - 88K7*2 - 2K8**2 + 4408
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.130']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4734', 'vk6.5058', 'vk6.6264', 'vk6.6709', 'vk6.8231', 'vk6.8676', 'vk6.9616', 'vk6.9938', 'vk6.20660', 'vk6.22091', 'vk6.28146', 'vk6.29575', 'vk6.39588', 'vk6.41819', 'vk6.46203', 'vk6.47821', 'vk6.48766', 'vk6.48972', 'vk6.49568', 'vk6.49779', 'vk6.50776', 'vk6.50985', 'vk6.51256', 'vk6.51460', 'vk6.57576', 'vk6.58742', 'vk6.62246', 'vk6.63192', 'vk6.67046', 'vk6.67919', 'vk6.69671', 'vk6.70352']
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,-2,-1,2,2,2,0,2,2,3,3,2,1,2,2,2,2,2,-1,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.130']

Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1*K3 - 2*K1 - 2*K2**2 + K4 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.130', '6.560']

Outer characteristic polynomial of the knot is: t^7+79t^5+72t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.130']

2-strand cable arrow polynomial of the knot is: -64*K1**3*K3 - 192*K1**2*K2**4 + 800*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 4304*K1**2*K2**2 + 96*K1**2*K2*K3**2 - 864*K1**2*K2*K4 + 5184*K1**2*K2 - 128*K1**2*K3**2 - 128*K1**2*K4**2 - 5096*K1**2 + 352*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1056*K1*K2**2*K3 - 256*K1*K2**2*K5 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 6272*K1*K2*K3 - 64*K1*K2*K4*K5 - 32*K1*K2*K4*K7 + 1552*K1*K3*K4 + 448*K1*K4*K5 + 112*K1*K5*K6 + 56*K1*K6*K7 - 32*K2**6 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1176*K2**4 + 64*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 656*K2**2*K3**2 + 32*K2**2*K4**3 - 424*K2**2*K4**2 + 2416*K2**2*K4 - 64*K2**2*K5**2 - 8*K2**2*K6**2 - 4148*K2**2 - 96*K2*K3*K4*K5 + 976*K2*K3*K5 - 32*K2*K4**2*K6 + 408*K2*K4*K6 + 96*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 2336*K3**2 + 104*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1400*K4**2 - 440*K5**2 - 180*K6**2 - 88*K7**2 - 2*K8**2 + 4408

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.130']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4734', 'vk6.5058', 'vk6.6264', 'vk6.6709', 'vk6.8231', 'vk6.8676', 'vk6.9616', 'vk6.9938', 'vk6.20660', 'vk6.22091', 'vk6.28146', 'vk6.29575', 'vk6.39588', 'vk6.41819', 'vk6.46203', 'vk6.47821', 'vk6.48766', 'vk6.48972', 'vk6.49568', 'vk6.49779', 'vk6.50776', 'vk6.50985', 'vk6.51256', 'vk6.51460', 'vk6.57576', 'vk6.58742', 'vk6.62246', 'vk6.63192', 'vk6.67046', 'vk6.67919', 'vk6.69671', 'vk6.70352']

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	O1O2O3O4O5O6U4U2U6U5U1U3
R3 orbit	{'O1O2O3O4O5O6U4U2U6U5U1U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U4U6U2U1U5U3
Gauss code of K*	O1O2O3O4O5O6U5U2U6U1U4U3
Gauss code of -K*	O1O2O3O4O5O6U4U3U6U1U5U2
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 -3 2 -2 2 2],[ 1 0 -2 2 -2 2 2],[ 3 2 0 3 0 3 2],[-2 -2 -3 0 -2 1 1],[ 2 2 0 2 0 2 1],[-2 -2 -3 -1 -2 0 0],[-2 -2 -2 -1 -1 0 0]]
Primitive based matrix	[[ 0 2 2 2 -1 -2 -3],[-2 0 1 1 -2 -2 -3],[-2 -1 0 0 -2 -1 -2],[-2 -1 0 0 -2 -2 -3],[ 1 2 2 2 0 -2 -2],[ 2 2 1 2 2 0 0],[ 3 3 2 3 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-2,1,2,3,-1,-1,2,2,3,0,2,1,2,2,2,3,2,2,0]
Phi over symmetry	[-3,-2,-1,2,2,2,0,2,2,3,3,2,1,2,2,2,2,2,-1,0,1]
Phi of -K	[-3,-2,-1,2,2,2,1,0,2,2,3,-1,2,2,3,1,1,1,-1,-1,0]
Phi of K*	[-2,-2,-2,1,2,3,-1,0,1,2,2,1,1,2,2,1,3,3,-1,0,1]
Phi of -K*	[-3,-2,-1,2,2,2,0,2,2,3,3,2,1,2,2,2,2,2,-1,0,1]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	8z^2+27z+23
Enhanced Jones-Krushkal polynomial	8w^3z^2-2w^3z+29w^2z+23w
Inner characteristic polynomial	t^6+53t^4+32t^2
Outer characteristic polynomial	t^7+79t^5+72t^3+9t
Flat arrow polynomial	4K13 + 4K1*2K2 - 2K12 - 2K1K2 - 2K1K3 - 2K1 - 2K2*2 + K4 + 2
2-strand cable arrow polynomial	-64K13K3 - 192K12K2*4 + 800K1*2K2*3 + 160K1*2K2*2K4 - 4304K12K2*2 + 96K1*2K2K32 - 864K1*2K2K4 + 5184K1*2K2 - 128K12K3*2 - 128K1*2K4*2 - 5096K1*2 + 352K1K23K3 + 160K1K2*2K3K4 - 1056K1K22K3 - 256K1K2*2K5 - 512K1K2K3K4 - 64K1K2K3K6 + 6272K1K2K3 - 64K1K2K4K5 - 32K1K2K4K7 + 1552K1K3K4 + 448K1K4K5 + 112K1K5K6 + 56K1K6K7 - 32K2*6 - 32K2*4K4*2 + 160K2*4K4 - 1176K24 + 64K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 656K22K3*2 + 32K2*2K4*3 - 424K2*2K4*2 + 2416K2*2K4 - 64K22K5*2 - 8K2*2K6*2 - 4148K2*2 - 96K2K3K4K5 + 976K2K3K5 - 32K2K4*2K6 + 408K2K4K6 + 96K2K5K7 + 8K2K6K8 + 8K3*2K6 - 2336K32 + 104K3K4K7 - 8K44 + 8K4*2K8 - 1400K42 - 440K5*2 - 180K6*2 - 88K7*2 - 2K8**2 + 4408
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}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]]
If K is slice	False

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