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Flat knot 6.182

Min(phi) over symmetries of the knot is: [-4,0,0,1,1,2,1,2,1,4,3,1,1,1,1,1,1,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.182']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 12K12 - 4K1K2 - 2K1K3 - 4K1 + 5*K2 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.182']
Outer characteristic polynomial of the knot is: t^7+63t^5+96t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.182']
2-strand cable arrow polynomial of the knot is: -512K16 - 832K1*4K2*2 + 1472K1*4K2 - 4352K14 + 1024K1*3K2K3 - 960K1*3K3 - 512K12K2*4 + 1376K1*2K2*3 - 256K1*2K2*2K3*2 + 64K1*2K2*2K4 - 10032K12K2*2 + 64K1*2K2K32 - 1120K1*2K2K4 + 12496K1*2K2 - 512K12K3*2 - 32K1*2K3K5 - 64K1*2K4*2 - 5816K1*2 + 3040K1K23K3 + 480K1K2*2K3K4 - 1568K1K22K3 - 576K1K2*2K5 - 320K1K2K3K4 - 32K1K2K3K6 + 9736K1K2K3 - 32K1K2K4K5 + 1216K1K3K4 + 88K1K4K5 - 192K2*6 - 192K2*4K3*2 - 32K2*4K4*2 + 224K2*4K4 - 2464K24 + 128K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 1728K22K3*2 - 304K2*2K4*2 + 1976K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 3780K2*2 + 600K2K3K5 + 88K2K4K6 - 2288K3*2 - 502K4*2 - 32K5*2 - 4K6**2 + 4996
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.182']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4690', 'vk6.4993', 'vk6.6172', 'vk6.6643', 'vk6.8165', 'vk6.8583', 'vk6.9555', 'vk6.9894', 'vk6.17396', 'vk6.20916', 'vk6.20987', 'vk6.22326', 'vk6.22411', 'vk6.23563', 'vk6.23900', 'vk6.28392', 'vk6.36164', 'vk6.40046', 'vk6.40188', 'vk6.42097', 'vk6.43075', 'vk6.43379', 'vk6.46574', 'vk6.46693', 'vk6.48730', 'vk6.49522', 'vk6.49725', 'vk6.51426', 'vk6.55554', 'vk6.58906', 'vk6.65292', 'vk6.69760']
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: [-4,0,0,1,1,2,1,2,1,4,3,1,1,1,1,1,1,1,-1,-1,1]

Flat knots (up to 7 crossings) with same phi are :['6.182']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 12*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + 5*K2 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.182']

Outer characteristic polynomial of the knot is: t^7+63t^5+96t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.182']

2-strand cable arrow polynomial of the knot is: -512*K1**6 - 832*K1**4*K2**2 + 1472*K1**4*K2 - 4352*K1**4 + 1024*K1**3*K2*K3 - 960*K1**3*K3 - 512*K1**2*K2**4 + 1376*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 10032*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 1120*K1**2*K2*K4 + 12496*K1**2*K2 - 512*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 5816*K1**2 + 3040*K1*K2**3*K3 + 480*K1*K2**2*K3*K4 - 1568*K1*K2**2*K3 - 576*K1*K2**2*K5 - 320*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 9736*K1*K2*K3 - 32*K1*K2*K4*K5 + 1216*K1*K3*K4 + 88*K1*K4*K5 - 192*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 224*K2**4*K4 - 2464*K2**4 + 128*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1728*K2**2*K3**2 - 304*K2**2*K4**2 + 1976*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3780*K2**2 + 600*K2*K3*K5 + 88*K2*K4*K6 - 2288*K3**2 - 502*K4**2 - 32*K5**2 - 4*K6**2 + 4996

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.182']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4690', 'vk6.4993', 'vk6.6172', 'vk6.6643', 'vk6.8165', 'vk6.8583', 'vk6.9555', 'vk6.9894', 'vk6.17396', 'vk6.20916', 'vk6.20987', 'vk6.22326', 'vk6.22411', 'vk6.23563', 'vk6.23900', 'vk6.28392', 'vk6.36164', 'vk6.40046', 'vk6.40188', 'vk6.42097', 'vk6.43075', 'vk6.43379', 'vk6.46574', 'vk6.46693', 'vk6.48730', 'vk6.49522', 'vk6.49725', 'vk6.51426', 'vk6.55554', 'vk6.58906', 'vk6.65292', 'vk6.69760']

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	O1O2O3O4O5U1O6U5U6U3U4U2
R3 orbit	{'O1O2O3O4O5U1O6U5U6U3U4U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U2U3U6U1O6U5
Gauss code of K*	O1O2O3O4O5U6U5U3U4U1O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U5U2U3U1U6
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 1 0 2 0 1],[ 4 0 4 2 3 1 1],[-1 -4 0 -1 1 -1 1],[ 0 -2 1 0 1 -1 1],[-2 -3 -1 -1 0 -1 1],[ 0 -1 1 1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 0 -4],[-2 0 1 -1 -1 -1 -3],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 -1 -1 -4],[ 0 1 1 1 0 1 -1],[ 0 1 1 1 -1 0 -2],[ 4 3 1 4 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,0,4,-1,1,1,1,3,1,1,1,1,1,1,4,-1,1,2]
Phi over symmetry	[-4,0,0,1,1,2,1,2,1,4,3,1,1,1,1,1,1,1,-1,-1,1]
Phi of -K	[-4,0,0,1,1,2,2,3,1,4,3,1,0,0,1,0,0,1,-1,0,2]
Phi of K*	[-2,-1,-1,0,0,4,0,2,1,1,3,1,0,0,1,0,0,4,-1,2,3]
Phi of -K*	[-4,0,0,1,1,2,1,2,1,4,3,1,1,1,1,1,1,1,-1,-1,1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^2-2t
Normalized Jones-Krushkal polynomial	5z^2+26z+33
Enhanced Jones-Krushkal polynomial	5w^3z^2+26w^2z+33w
Inner characteristic polynomial	t^6+41t^4+29t^2+1
Outer characteristic polynomial	t^7+63t^5+96t^3+8t
Flat arrow polynomial	8K13 + 4K1*2K2 - 12K12 - 4K1K2 - 2K1K3 - 4K1 + 5*K2 + 6
2-strand cable arrow polynomial	-512K16 - 832K1*4K2*2 + 1472K1*4K2 - 4352K14 + 1024K1*3K2K3 - 960K1*3K3 - 512K12K2*4 + 1376K1*2K2*3 - 256K1*2K2*2K3*2 + 64K1*2K2*2K4 - 10032K12K2*2 + 64K1*2K2K32 - 1120K1*2K2K4 + 12496K1*2K2 - 512K12K3*2 - 32K1*2K3K5 - 64K1*2K4*2 - 5816K1*2 + 3040K1K23K3 + 480K1K2*2K3K4 - 1568K1K22K3 - 576K1K2*2K5 - 320K1K2K3K4 - 32K1K2K3K6 + 9736K1K2K3 - 32K1K2K4K5 + 1216K1K3K4 + 88K1K4K5 - 192K2*6 - 192K2*4K3*2 - 32K2*4K4*2 + 224K2*4K4 - 2464K24 + 128K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 1728K22K3*2 - 304K2*2K4*2 + 1976K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 3780K2*2 + 600K2K3K5 + 88K2K4K6 - 2288K3*2 - 502K4*2 - 32K5*2 - 4K6**2 + 4996
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}], [{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}, {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}, {3, 4}, {2}, {1}]]
If K is slice	False

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