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Flat knot 6.431

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,-1,1,3,2,4,1,1,1,1,1,1,1,1,1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.431']
Arrow polynomial of the knot is: 8K13 - 8K1*2 - 6K1K2 - 3K1 + 4*K2 + K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']
Outer characteristic polynomial of the knot is: t^7+55t^5+103t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.431']
2-strand cable arrow polynomial of the knot is: -640K14K2*2 + 1152K1*4K2 - 1792K14 + 256K1*3K2K3 - 192K1*3K3 + 640K12K2*5 - 3136K1*2K2*4 + 5888K1*2K2*3 + 128K1*2K2*2K4 - 11008K12K2*2 - 384K1*2K2K4 + 8168K1*2K2 - 128K12K3*2 - 3536K1*2 - 512K1K24K3 + 2752K1K2*3K3 - 2208K1K2*2K3 - 256K1K2*2K5 - 64K1K2K3K4 + 5936K1K2K3 + 216K1K3K4 + 8K1K4K5 - 704K2*6 + 608K2*4K4 - 4032K24 - 32K2*3K6 - 704K22K3*2 - 88K2*2K4*2 + 2328K2*2K4 - 582K22 + 200K2K3K5 + 24K2K4K6 - 764K3*2 - 196K4*2 - 20K5*2 - 2K6**2 + 2674
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.431']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20116', 'vk6.20120', 'vk6.21394', 'vk6.21402', 'vk6.27204', 'vk6.27212', 'vk6.28884', 'vk6.28888', 'vk6.38616', 'vk6.38624', 'vk6.40804', 'vk6.40820', 'vk6.45490', 'vk6.45506', 'vk6.47222', 'vk6.47230', 'vk6.56933', 'vk6.56941', 'vk6.58071', 'vk6.58087', 'vk6.61487', 'vk6.61503', 'vk6.62628', 'vk6.62636', 'vk6.66643', 'vk6.66647', 'vk6.67430', 'vk6.67438', 'vk6.69281', 'vk6.69289', 'vk6.70016', 'vk6.70020']
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,-1,0,0,2,2,-1,1,3,2,4,1,1,1,1,1,1,1,1,1,-1]

Flat knots (up to 7 crossings) with same phi are :['6.431']

Arrow polynomial of the knot is: 8*K1**3 - 8*K1**2 - 6*K1*K2 - 3*K1 + 4*K2 + K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.238', '6.431', '6.945', '6.977', '6.981', '6.997', '6.1050', '6.1070', '6.1098', '6.1376']

Outer characteristic polynomial of the knot is: t^7+55t^5+103t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.431']

2-strand cable arrow polynomial of the knot is: -640*K1**4*K2**2 + 1152*K1**4*K2 - 1792*K1**4 + 256*K1**3*K2*K3 - 192*K1**3*K3 + 640*K1**2*K2**5 - 3136*K1**2*K2**4 + 5888*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 11008*K1**2*K2**2 - 384*K1**2*K2*K4 + 8168*K1**2*K2 - 128*K1**2*K3**2 - 3536*K1**2 - 512*K1*K2**4*K3 + 2752*K1*K2**3*K3 - 2208*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 5936*K1*K2*K3 + 216*K1*K3*K4 + 8*K1*K4*K5 - 704*K2**6 + 608*K2**4*K4 - 4032*K2**4 - 32*K2**3*K6 - 704*K2**2*K3**2 - 88*K2**2*K4**2 + 2328*K2**2*K4 - 582*K2**2 + 200*K2*K3*K5 + 24*K2*K4*K6 - 764*K3**2 - 196*K4**2 - 20*K5**2 - 2*K6**2 + 2674

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.431']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20116', 'vk6.20120', 'vk6.21394', 'vk6.21402', 'vk6.27204', 'vk6.27212', 'vk6.28884', 'vk6.28888', 'vk6.38616', 'vk6.38624', 'vk6.40804', 'vk6.40820', 'vk6.45490', 'vk6.45506', 'vk6.47222', 'vk6.47230', 'vk6.56933', 'vk6.56941', 'vk6.58071', 'vk6.58087', 'vk6.61487', 'vk6.61503', 'vk6.62628', 'vk6.62636', 'vk6.66643', 'vk6.66647', 'vk6.67430', 'vk6.67438', 'vk6.69281', 'vk6.69289', 'vk6.70016', 'vk6.70020']

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	O1O2O3O4O5U6U1O6U4U5U2U3
R3 orbit	{'O1O2O3O4O5U6U1O6U4U5U2U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U4U1U2O6U5U6
Gauss code of K*	O1O2O3O4U5U3U4U1U2O6O5U6
Gauss code of -K*	O1O2O3O4U5O6O5U3U4U1U2U6
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 -3 0 2 0 2 -1],[ 3 0 2 3 0 1 3],[ 0 -2 0 1 -1 1 0],[-2 -3 -1 0 -1 1 -2],[ 0 0 1 1 0 1 0],[-2 -1 -1 -1 -1 0 -2],[ 1 -3 0 2 0 2 0]]
Primitive based matrix	[[ 0 2 2 0 0 -1 -3],[-2 0 1 -1 -1 -2 -3],[-2 -1 0 -1 -1 -2 -1],[ 0 1 1 0 1 0 0],[ 0 1 1 -1 0 0 -2],[ 1 2 2 0 0 0 -3],[ 3 3 1 0 2 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,0,1,3,-1,1,1,2,3,1,1,2,1,-1,0,0,0,2,3]
Phi over symmetry	[-3,-1,0,0,2,2,-1,1,3,2,4,1,1,1,1,1,1,1,1,1,-1]
Phi of -K	[-3,-1,0,0,2,2,-1,1,3,2,4,1,1,1,1,1,1,1,1,1,-1]
Phi of K*	[-2,-2,0,0,1,3,-1,1,1,1,4,1,1,1,2,-1,1,1,1,3,-1]
Phi of -K*	[-3,-1,0,0,2,2,3,0,2,1,3,0,0,2,2,1,1,1,1,1,-1]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	5z^2+22z+25
Enhanced Jones-Krushkal polynomial	-2w^4z^2+7w^3z^2-2w^3z+24w^2z+25w
Inner characteristic polynomial	t^6+37t^4+45t^2+1
Outer characteristic polynomial	t^7+55t^5+103t^3+9t
Flat arrow polynomial	8K13 - 8K1*2 - 6K1K2 - 3K1 + 4*K2 + K3 + 5
2-strand cable arrow polynomial	-640K14K2*2 + 1152K1*4K2 - 1792K14 + 256K1*3K2K3 - 192K1*3K3 + 640K12K2*5 - 3136K1*2K2*4 + 5888K1*2K2*3 + 128K1*2K2*2K4 - 11008K12K2*2 - 384K1*2K2K4 + 8168K1*2K2 - 128K12K3*2 - 3536K1*2 - 512K1K24K3 + 2752K1K2*3K3 - 2208K1K2*2K3 - 256K1K2*2K5 - 64K1K2K3K4 + 5936K1K2K3 + 216K1K3K4 + 8K1K4K5 - 704K2*6 + 608K2*4K4 - 4032K24 - 32K2*3K6 - 704K22K3*2 - 88K2*2K4*2 + 2328K2*2K4 - 582K22 + 200K2K3K5 + 24K2K4K6 - 764K3*2 - 196K4*2 - 20K5*2 - 2K6**2 + 2674
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}], [{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}], [{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}, {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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