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Flat knot 6.232

Min(phi) over symmetries of the knot is: [-4,-2,-1,2,2,3,1,0,2,4,4,0,1,3,3,0,1,1,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.232']
Arrow polynomial of the knot is: 8K13 - 6K1K2 - 2K1K3 - 3K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.232']
Outer characteristic polynomial of the knot is: t^7+97t^5+101t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.232']
2-strand cable arrow polynomial of the knot is: -256K14K2*2 + 96K1*4K2 - 176K14 + 128K1*3K2*3K3 + 416K13K2K3 - 576K1*2K2*4 + 352K1*2K2*3 - 128K1*2K2*2K3*2 - 2032K1*2K2*2 + 1048K1*2K2 - 272K12K3*2 - 584K1*2 + 704K1K23K3 + 32K1K2K33 + 1752K1K2K3 + 256K1K3K4 + 56K1K4K5 + 8K1K5K6 - 192K2*6 + 320K2*4K4 - 944K24 + 64K2*3K3K5 - 400K2*2K3*2 - 184K2*2K4*2 + 664K2*2K4 - 80K22K5*2 - 8K2*2K6*2 - 314K2*2 + 288K2K3K5 + 48K2K4K6 + 32K2K5K7 + 8K2K6K8 + 8K3*2K6 - 604K32 - 310K4*2 - 128K5*2 - 14K6*2 - 4K7*2 - 2K8**2 + 926
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.232']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16916', 'vk6.17160', 'vk6.17497', 'vk6.17508', 'vk6.17552', 'vk6.17565', 'vk6.21892', 'vk6.24021', 'vk6.24036', 'vk6.24105', 'vk6.27946', 'vk6.29427', 'vk6.35330', 'vk6.35764', 'vk6.36275', 'vk6.36290', 'vk6.36357', 'vk6.39362', 'vk6.41540', 'vk6.43430', 'vk6.43441', 'vk6.43470', 'vk6.45925', 'vk6.47614', 'vk6.55067', 'vk6.55318', 'vk6.55619', 'vk6.55622', 'vk6.55651', 'vk6.58543', 'vk6.60133', 'vk6.60136', 'vk6.60168', 'vk6.63029', 'vk6.64904', 'vk6.65119', 'vk6.65322', 'vk6.65357', 'vk6.68494', 'vk6.68518']
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,-2,-1,2,2,3,1,0,2,4,4,0,1,3,3,0,1,1,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.232']

Arrow polynomial of the knot is: 8*K1**3 - 6*K1*K2 - 2*K1*K3 - 3*K1 + K2 + K3 + K4 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.232']

Outer characteristic polynomial of the knot is: t^7+97t^5+101t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.232']

2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 96*K1**4*K2 - 176*K1**4 + 128*K1**3*K2**3*K3 + 416*K1**3*K2*K3 - 576*K1**2*K2**4 + 352*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 2032*K1**2*K2**2 + 1048*K1**2*K2 - 272*K1**2*K3**2 - 584*K1**2 + 704*K1*K2**3*K3 + 32*K1*K2*K3**3 + 1752*K1*K2*K3 + 256*K1*K3*K4 + 56*K1*K4*K5 + 8*K1*K5*K6 - 192*K2**6 + 320*K2**4*K4 - 944*K2**4 + 64*K2**3*K3*K5 - 400*K2**2*K3**2 - 184*K2**2*K4**2 + 664*K2**2*K4 - 80*K2**2*K5**2 - 8*K2**2*K6**2 - 314*K2**2 + 288*K2*K3*K5 + 48*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 604*K3**2 - 310*K4**2 - 128*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 926

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.232']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16916', 'vk6.17160', 'vk6.17497', 'vk6.17508', 'vk6.17552', 'vk6.17565', 'vk6.21892', 'vk6.24021', 'vk6.24036', 'vk6.24105', 'vk6.27946', 'vk6.29427', 'vk6.35330', 'vk6.35764', 'vk6.36275', 'vk6.36290', 'vk6.36357', 'vk6.39362', 'vk6.41540', 'vk6.43430', 'vk6.43441', 'vk6.43470', 'vk6.45925', 'vk6.47614', 'vk6.55067', 'vk6.55318', 'vk6.55619', 'vk6.55622', 'vk6.55651', 'vk6.58543', 'vk6.60133', 'vk6.60136', 'vk6.60168', 'vk6.63029', 'vk6.64904', 'vk6.65119', 'vk6.65322', 'vk6.65357', 'vk6.68494', 'vk6.68518']

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	O1O2O3O4O5U3O6U2U6U1U4U5
R3 orbit	{'O1O2O3O4O5U3U1O6U2U6U4U5', 'O1O2O3O4O5U3O6U2U6U1U4U5'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U1U2U5U6U4O6U3
Gauss code of K*	O1O2O3O4O5U3U1U6U4U5O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U1U2U6U5U3
Diagrammatic symmetry type	c
Flat genus of the diagram	2
If K is checkerboard colorable	False
If K is almost classical	False
Based matrix from Gauss code	[[ 0 -2 -3 -2 2 4 1],[ 2 0 -1 0 3 4 1],[ 3 1 0 0 3 4 1],[ 2 0 0 0 1 2 0],[-2 -3 -3 -1 0 1 0],[-4 -4 -4 -2 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix	[[ 0 4 2 1 -2 -2 -3],[-4 0 -1 0 -2 -4 -4],[-2 1 0 0 -1 -3 -3],[-1 0 0 0 0 -1 -1],[ 2 2 1 0 0 0 0],[ 2 4 3 1 0 0 -1],[ 3 4 3 1 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,-1,2,2,3,1,0,2,4,4,0,1,3,3,0,1,1,0,0,1]
Phi over symmetry	[-4,-2,-1,2,2,3,1,0,2,4,4,0,1,3,3,0,1,1,0,0,1]
Phi of -K	[-3,-2,-2,1,2,4,0,1,3,2,3,0,2,1,2,3,3,4,1,3,1]
Phi of K*	[-4,-2,-1,2,2,3,1,3,2,4,3,1,1,3,2,2,3,3,0,0,1]
Phi of -K*	[-3,-2,-2,1,2,4,0,1,1,3,4,0,0,1,2,1,3,4,0,0,1]
Symmetry type of based matrix	c
u-polynomial	-t^4+t^3+t^2-t
Normalized Jones-Krushkal polynomial	3z+7
Enhanced Jones-Krushkal polynomial	-10w^3z+13w^2z+7w
Inner characteristic polynomial	t^6+59t^4+26t^2
Outer characteristic polynomial	t^7+97t^5+101t^3
Flat arrow polynomial	8K13 - 6K1K2 - 2K1K3 - 3K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial	-256K14K2*2 + 96K1*4K2 - 176K14 + 128K1*3K2*3K3 + 416K13K2K3 - 576K1*2K2*4 + 352K1*2K2*3 - 128K1*2K2*2K3*2 - 2032K1*2K2*2 + 1048K1*2K2 - 272K12K3*2 - 584K1*2 + 704K1K23K3 + 32K1K2K33 + 1752K1K2K3 + 256K1K3K4 + 56K1K4K5 + 8K1K5K6 - 192K2*6 + 320K2*4K4 - 944K24 + 64K2*3K3K5 - 400K2*2K3*2 - 184K2*2K4*2 + 664K2*2K4 - 80K22K5*2 - 8K2*2K6*2 - 314K2*2 + 288K2K3K5 + 48K2K4K6 + 32K2K5K7 + 8K2K6K8 + 8K3*2K6 - 604K32 - 310K4*2 - 128K5*2 - 14K6*2 - 4K7*2 - 2K8**2 + 926
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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}], [{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}, {1, 5}, {4}, {3}, {2}], [{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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}], [{6}, {5}, {4}, {2, 3}, {1}], [{6}, {5}, {4}, {3}, {1, 2}], [{6}, {5}, {4}, {3}, {2}, {1}]]
If K is slice	False

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