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Flat knot 6.193

Min(phi) over symmetries of the knot is: [-4,-3,0,1,2,4,0,3,1,4,5,1,0,2,3,0,1,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.193']
Arrow polynomial of the knot is: -16K14 + 4K1*3 + 8K1*2K2 + 6K12 - 2K1K2 - 2K1 + K2 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.193', '6.868']
Outer characteristic polynomial of the knot is: t^7+117t^5+167t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.193']
2-strand cable arrow polynomial of the knot is: -704K14 - 32K1*3K3 - 768K12K2*6 + 2048K1*2K2*5 - 4672K1*2K2*4 - 128K1*2K2*3K4 + 3488K12K2*3 - 4640K1*2K2*2 - 128K1*2K2K4 + 4048K1*2K2 - 64K12K3*2 - 2580K1*2 + 1152K1K25K3 + 128K1K2*4K3K4 - 1280K1K24K3 - 128K1K2*4K5 + 3296K1K2*3K3 + 32K1K2*2K3K4 - 704K1K22K3 - 96K1K2*2K5 - 32K1K2K3K4 + 3296K1K2K3 + 336K1K3K4 + 48K1K4K5 - 256K2*8 + 256K2*6K4 - 2208K26 - 576K2*4K3*2 - 192K2*4K4*2 + 1664K2*4K4 - 1176K24 + 128K2*3K3K5 + 32K2*3K4K6 - 752K2*2K3*2 - 264K2*2K4*2 + 984K2*2K4 - 600K22 + 176K2K3K5 + 32K2K4K6 - 856K3*2 - 278K4*2 - 36K5**2 + 2004
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.193']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16902', 'vk6.17146', 'vk6.20201', 'vk6.21479', 'vk6.23290', 'vk6.23591', 'vk6.27367', 'vk6.29005', 'vk6.35300', 'vk6.35740', 'vk6.38798', 'vk6.40965', 'vk6.42807', 'vk6.43091', 'vk6.45549', 'vk6.47338', 'vk6.55047', 'vk6.55292', 'vk6.57044', 'vk6.58140', 'vk6.59435', 'vk6.59724', 'vk6.61537', 'vk6.62719', 'vk6.64890', 'vk6.65105', 'vk6.66658', 'vk6.67481', 'vk6.68199', 'vk6.68345', 'vk6.69303', 'vk6.70067']
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,-3,0,1,2,4,0,3,1,4,5,1,0,2,3,0,1,2,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.193']

Arrow polynomial of the knot is: -16*K1**4 + 4*K1**3 + 8*K1**2*K2 + 6*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.193', '6.868']

Outer characteristic polynomial of the knot is: t^7+117t^5+167t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.193']

2-strand cable arrow polynomial of the knot is: -704*K1**4 - 32*K1**3*K3 - 768*K1**2*K2**6 + 2048*K1**2*K2**5 - 4672*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 3488*K1**2*K2**3 - 4640*K1**2*K2**2 - 128*K1**2*K2*K4 + 4048*K1**2*K2 - 64*K1**2*K3**2 - 2580*K1**2 + 1152*K1*K2**5*K3 + 128*K1*K2**4*K3*K4 - 1280*K1*K2**4*K3 - 128*K1*K2**4*K5 + 3296*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 704*K1*K2**2*K3 - 96*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3296*K1*K2*K3 + 336*K1*K3*K4 + 48*K1*K4*K5 - 256*K2**8 + 256*K2**6*K4 - 2208*K2**6 - 576*K2**4*K3**2 - 192*K2**4*K4**2 + 1664*K2**4*K4 - 1176*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 752*K2**2*K3**2 - 264*K2**2*K4**2 + 984*K2**2*K4 - 600*K2**2 + 176*K2*K3*K5 + 32*K2*K4*K6 - 856*K3**2 - 278*K4**2 - 36*K5**2 + 2004

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.193']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16902', 'vk6.17146', 'vk6.20201', 'vk6.21479', 'vk6.23290', 'vk6.23591', 'vk6.27367', 'vk6.29005', 'vk6.35300', 'vk6.35740', 'vk6.38798', 'vk6.40965', 'vk6.42807', 'vk6.43091', 'vk6.45549', 'vk6.47338', 'vk6.55047', 'vk6.55292', 'vk6.57044', 'vk6.58140', 'vk6.59435', 'vk6.59724', 'vk6.61537', 'vk6.62719', 'vk6.64890', 'vk6.65105', 'vk6.66658', 'vk6.67481', 'vk6.68199', 'vk6.68345', 'vk6.69303', 'vk6.70067']

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	O1O2O3O4O5U2O6U1U6U3U4U5
R3 orbit	{'O1O2O3O4O5U2O6U1U6U3U4U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U1U2U3U6U5O6U4
Gauss code of K*	O1O2O3O4O5U1U6U3U4U5O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U1U2U3U6U5
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 -3 0 2 4 1],[ 4 0 0 3 4 5 1],[ 3 0 0 1 2 3 0],[ 0 -3 -1 0 1 2 0],[-2 -4 -2 -1 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix	[[ 0 4 2 1 0 -3 -4],[-4 0 -1 0 -2 -3 -5],[-2 1 0 0 -1 -2 -4],[-1 0 0 0 0 0 -1],[ 0 2 1 0 0 -1 -3],[ 3 3 2 0 1 0 0],[ 4 5 4 1 3 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-2,-1,0,3,4,1,0,2,3,5,0,1,2,4,0,0,1,1,3,0]
Phi over symmetry	[-4,-3,0,1,2,4,0,3,1,4,5,1,0,2,3,0,1,2,0,0,1]
Phi of -K	[-4,-3,0,1,2,4,1,1,4,2,3,2,4,3,4,1,1,2,1,3,1]
Phi of K*	[-4,-2,-1,0,3,4,1,3,2,4,3,1,1,3,2,1,4,4,2,1,1]
Phi of -K*	[-4,-3,0,1,2,4,0,3,1,4,5,1,0,2,3,0,1,2,0,0,1]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	5z+11
Enhanced Jones-Krushkal polynomial	-6w^4z^2+6w^3z^2-10w^3z+15w^2z+11w
Inner characteristic polynomial	t^6+71t^4+44t^2
Outer characteristic polynomial	t^7+117t^5+167t^3+6t
Flat arrow polynomial	-16K14 + 4K1*3 + 8K1*2K2 + 6K12 - 2K1K2 - 2K1 + K2 + 2
2-strand cable arrow polynomial	-704K14 - 32K1*3K3 - 768K12K2*6 + 2048K1*2K2*5 - 4672K1*2K2*4 - 128K1*2K2*3K4 + 3488K12K2*3 - 4640K1*2K2*2 - 128K1*2K2K4 + 4048K1*2K2 - 64K12K3*2 - 2580K1*2 + 1152K1K25K3 + 128K1K2*4K3K4 - 1280K1K24K3 - 128K1K2*4K5 + 3296K1K2*3K3 + 32K1K2*2K3K4 - 704K1K22K3 - 96K1K2*2K5 - 32K1K2K3K4 + 3296K1K2K3 + 336K1K3K4 + 48K1K4K5 - 256K2*8 + 256K2*6K4 - 2208K26 - 576K2*4K3*2 - 192K2*4K4*2 + 1664K2*4K4 - 1176K24 + 128K2*3K3K5 + 32K2*3K4K6 - 752K2*2K3*2 - 264K2*2K4*2 + 984K2*2K4 - 600K22 + 176K2K3K5 + 32K2K4K6 - 856K3*2 - 278K4*2 - 36K5**2 + 2004
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}], [{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}, {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}, {3}, {1, 2}]]
If K is slice	False

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