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Flat knot 6.775

Min(phi) over symmetries of the knot is: [-3,-2,1,1,1,2,0,1,3,3,2,1,2,3,2,0,-1,0,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.775']
Arrow polynomial of the knot is: 8K13 - 4K1*2 - 6K1K2 - 3K1 + 2*K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.205', '6.660', '6.775', '6.820']
Outer characteristic polynomial of the knot is: t^7+64t^5+63t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.775']
2-strand cable arrow polynomial of the knot is: 2112K14K2 - 3920K14 - 512K1*3K2*2K3 + 1536K13K2K3 + 32K1*3K3K4 - 1792K1*3K3 - 384K12K2*4 - 128K1*2K2*3K4 + 2240K12K2*3 + 352K1*2K2*2K4 - 8816K12K2*2 + 256K1*2K2K32 + 64K1*2K2K42 - 1280K1*2K2K4 + 9552K1*2K2 - 1040K12K3*2 - 160K1*2K4*2 - 4808K1*2 + 1408K1K23K3 - 1664K1K2*2K3 - 288K1K2*2K5 - 288K1K2K3K4 + 8496K1K2K3 + 1456K1K3K4 + 128K1K4K5 - 192K2*6 + 288K2*4K4 - 1904K24 - 624K2*2K3*2 - 120K2*2K4*2 + 1752K2*2K4 - 3366K22 + 176K2K3K5 + 16K2K4K6 - 2092K3*2 - 644K4*2 - 12K5*2 - 2K6**2 + 4170
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.775']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13918', 'vk6.14013', 'vk6.14167', 'vk6.14406', 'vk6.14987', 'vk6.15108', 'vk6.15635', 'vk6.16089', 'vk6.16720', 'vk6.16751', 'vk6.16836', 'vk6.18802', 'vk6.19290', 'vk6.19582', 'vk6.23158', 'vk6.23217', 'vk6.25396', 'vk6.26479', 'vk6.33729', 'vk6.33804', 'vk6.34283', 'vk6.35154', 'vk6.37521', 'vk6.42716', 'vk6.44703', 'vk6.54118', 'vk6.54925', 'vk6.54954', 'vk6.56396', 'vk6.56613', 'vk6.59349', 'vk6.64597']
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,-2,1,1,1,2,0,1,3,3,2,1,2,3,2,0,-1,0,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.775']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.205', '6.660', '6.775', '6.820']

Outer characteristic polynomial of the knot is: t^7+64t^5+63t^3+10t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.775']

2-strand cable arrow polynomial of the knot is: 2112*K1**4*K2 - 3920*K1**4 - 512*K1**3*K2**2*K3 + 1536*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1792*K1**3*K3 - 384*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2240*K1**2*K2**3 + 352*K1**2*K2**2*K4 - 8816*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1280*K1**2*K2*K4 + 9552*K1**2*K2 - 1040*K1**2*K3**2 - 160*K1**2*K4**2 - 4808*K1**2 + 1408*K1*K2**3*K3 - 1664*K1*K2**2*K3 - 288*K1*K2**2*K5 - 288*K1*K2*K3*K4 + 8496*K1*K2*K3 + 1456*K1*K3*K4 + 128*K1*K4*K5 - 192*K2**6 + 288*K2**4*K4 - 1904*K2**4 - 624*K2**2*K3**2 - 120*K2**2*K4**2 + 1752*K2**2*K4 - 3366*K2**2 + 176*K2*K3*K5 + 16*K2*K4*K6 - 2092*K3**2 - 644*K4**2 - 12*K5**2 - 2*K6**2 + 4170

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.775']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13918', 'vk6.14013', 'vk6.14167', 'vk6.14406', 'vk6.14987', 'vk6.15108', 'vk6.15635', 'vk6.16089', 'vk6.16720', 'vk6.16751', 'vk6.16836', 'vk6.18802', 'vk6.19290', 'vk6.19582', 'vk6.23158', 'vk6.23217', 'vk6.25396', 'vk6.26479', 'vk6.33729', 'vk6.33804', 'vk6.34283', 'vk6.35154', 'vk6.37521', 'vk6.42716', 'vk6.44703', 'vk6.54118', 'vk6.54925', 'vk6.54954', 'vk6.56396', 'vk6.56613', 'vk6.59349', 'vk6.64597']

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	O1O2O3O4U3O5U2U6U4O6U1U5
R3 orbit	{'O1O2O3O4U3O5U2U6U4O6U1U5'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U4O6U1U6U3O5U2
Gauss code of K*	O1O2O3U2O4O5U4U1U6U3O6U5
Gauss code of -K*	O1O2O3U4O5U1U5U3U6O4O6U2
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 -2 -1 2 3 -1],[ 1 0 -1 -1 3 3 0],[ 2 1 0 0 2 2 1],[ 1 1 0 0 1 1 0],[-2 -3 -2 -1 0 0 -2],[-3 -3 -2 -1 0 0 -3],[ 1 0 -1 0 2 3 0]]
Primitive based matrix	[[ 0 3 2 -1 -1 -1 -2],[-3 0 0 -1 -3 -3 -2],[-2 0 0 -1 -2 -3 -2],[ 1 1 1 0 0 1 0],[ 1 3 2 0 0 0 -1],[ 1 3 3 -1 0 0 -1],[ 2 2 2 0 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,1,1,1,2,0,1,3,3,2,1,2,3,2,0,-1,0,0,1,1]
Phi over symmetry	[-3,-2,1,1,1,2,0,1,3,3,2,1,2,3,2,0,-1,0,0,1,1]
Phi of -K	[-2,-1,-1,-1,2,3,0,0,1,2,3,0,0,1,1,1,0,1,2,3,1]
Phi of K*	[-3,-2,1,1,1,2,1,1,1,3,3,0,1,2,2,0,-1,0,0,0,1]
Phi of -K*	[-2,-1,-1,-1,2,3,0,1,1,2,2,0,1,1,1,0,2,3,3,3,0]
Symmetry type of based matrix	c
u-polynomial	-t^3+3t
Normalized Jones-Krushkal polynomial	6z^2+25z+27
Enhanced Jones-Krushkal polynomial	6w^3z^2-2w^3z+27w^2z+27w
Inner characteristic polynomial	t^6+44t^4+33t^2+1
Outer characteristic polynomial	t^7+64t^5+63t^3+10t
Flat arrow polynomial	8K13 - 4K1*2 - 6K1K2 - 3K1 + 2*K2 + K3 + 3
2-strand cable arrow polynomial	2112K14K2 - 3920K14 - 512K1*3K2*2K3 + 1536K13K2K3 + 32K1*3K3K4 - 1792K1*3K3 - 384K12K2*4 - 128K1*2K2*3K4 + 2240K12K2*3 + 352K1*2K2*2K4 - 8816K12K2*2 + 256K1*2K2K32 + 64K1*2K2K42 - 1280K1*2K2K4 + 9552K1*2K2 - 1040K12K3*2 - 160K1*2K4*2 - 4808K1*2 + 1408K1K23K3 - 1664K1K2*2K3 - 288K1K2*2K5 - 288K1K2K3K4 + 8496K1K2K3 + 1456K1K3K4 + 128K1K4K5 - 192K2*6 + 288K2*4K4 - 1904K24 - 624K2*2K3*2 - 120K2*2K4*2 + 1752K2*2K4 - 3366K22 + 176K2K3K5 + 16K2K4K6 - 2092K3*2 - 644K4*2 - 12K5*2 - 2K6**2 + 4170
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}], [{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}], [{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}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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