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Flat knot 6.152

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,1,2,1,3,4,1,0,2,2,0,1,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.152']
Arrow polynomial of the knot is: -8K14 + 4K1*3 + 4K1*2K2 - 2K1K2 - 2K1 + 2K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.152', '6.164', '6.256', '6.433']
Outer characteristic polynomial of the knot is: t^7+113t^5+71t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.152']
2-strand cable arrow polynomial of the knot is: -544K14 + 128K1*2K2*5 - 896K1*2K2*4 + 672K1*2K2*3 - 1584K1*2K2*2 + 1656K1*2K2 - 32K12K3*2 - 768K1*2 + 128K1K25K3 + 576K1K2*3K3 + 1064K1K2K3 + 72K1K3K4 - 128K28 + 128K2*6K4 - 672K26 - 128K2*4K3*2 - 32K2*4K4*2 + 352K2*4K4 - 352K24 + 32K2*3K3K5 - 176K2*2K3*2 - 40K2*2K4*2 + 296K2*2K4 - 176K22 + 64K2K3K5 - 252K32 - 64K4*2 - 12K5**2 + 686
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.152']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16903', 'vk6.17145', 'vk6.20203', 'vk6.21483', 'vk6.23291', 'vk6.23590', 'vk6.27375', 'vk6.29009', 'vk6.35301', 'vk6.35739', 'vk6.38802', 'vk6.40973', 'vk6.42808', 'vk6.43090', 'vk6.45553', 'vk6.47340', 'vk6.55048', 'vk6.55291', 'vk6.57042', 'vk6.58136', 'vk6.59436', 'vk6.59723', 'vk6.61530', 'vk6.62715', 'vk6.64891', 'vk6.65104', 'vk6.66654', 'vk6.67474', 'vk6.68200', 'vk6.68344', 'vk6.69300', 'vk6.70065', 'vk6.81966', 'vk6.82701', 'vk6.85843', 'vk6.85844', 'vk6.88479', 'vk6.89100', 'vk6.89645', 'vk6.90064']
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,-2,0,1,2,3,1,2,1,3,4,1,0,2,2,0,1,1,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.152']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.152', '6.164', '6.256', '6.433']

Outer characteristic polynomial of the knot is: t^7+113t^5+71t^3

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.152']

2-strand cable arrow polynomial of the knot is: -544*K1**4 + 128*K1**2*K2**5 - 896*K1**2*K2**4 + 672*K1**2*K2**3 - 1584*K1**2*K2**2 + 1656*K1**2*K2 - 32*K1**2*K3**2 - 768*K1**2 + 128*K1*K2**5*K3 + 576*K1*K2**3*K3 + 1064*K1*K2*K3 + 72*K1*K3*K4 - 128*K2**8 + 128*K2**6*K4 - 672*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 352*K2**4*K4 - 352*K2**4 + 32*K2**3*K3*K5 - 176*K2**2*K3**2 - 40*K2**2*K4**2 + 296*K2**2*K4 - 176*K2**2 + 64*K2*K3*K5 - 252*K3**2 - 64*K4**2 - 12*K5**2 + 686

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.152']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16903', 'vk6.17145', 'vk6.20203', 'vk6.21483', 'vk6.23291', 'vk6.23590', 'vk6.27375', 'vk6.29009', 'vk6.35301', 'vk6.35739', 'vk6.38802', 'vk6.40973', 'vk6.42808', 'vk6.43090', 'vk6.45553', 'vk6.47340', 'vk6.55048', 'vk6.55291', 'vk6.57042', 'vk6.58136', 'vk6.59436', 'vk6.59723', 'vk6.61530', 'vk6.62715', 'vk6.64891', 'vk6.65104', 'vk6.66654', 'vk6.67474', 'vk6.68200', 'vk6.68344', 'vk6.69300', 'vk6.70065', 'vk6.81966', 'vk6.82701', 'vk6.85843', 'vk6.85844', 'vk6.88479', 'vk6.89100', 'vk6.89645', 'vk6.90064']

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	O1O2O3O4O5U1O6U3U4U5U6U2
R3 orbit	{'O1O2O3O4O5U1O6U3U4U5U6U2', 'O1O2O3O4O5U1U2O6U4U5U3U6'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U4U6U1U2U3O6U5
Gauss code of K*	O1O2O3O4O5U6U5U1U2U3O6U4
Gauss code of -K*	O1O2O3O4O5U2O6U3U4U5U1U6
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 -4 1 -2 0 2 3],[ 4 0 4 1 2 3 3],[-1 -4 0 -3 -1 1 3],[ 2 -1 3 0 1 2 3],[ 0 -2 1 -1 0 1 2],[-2 -3 -1 -2 -1 0 1],[-3 -3 -3 -3 -2 -1 0]]
Primitive based matrix	[[ 0 3 2 1 0 -2 -4],[-3 0 -1 -3 -2 -3 -3],[-2 1 0 -1 -1 -2 -3],[-1 3 1 0 -1 -3 -4],[ 0 2 1 1 0 -1 -2],[ 2 3 2 3 1 0 -1],[ 4 3 3 4 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,0,2,4,1,3,2,3,3,1,1,2,3,1,3,4,1,2,1]
Phi over symmetry	[-4,-2,0,1,2,3,1,2,1,3,4,1,0,2,2,0,1,1,0,-1,0]
Phi of -K	[-4,-2,0,1,2,3,1,2,1,3,4,1,0,2,2,0,1,1,0,-1,0]
Phi of K*	[-3,-2,-1,0,2,4,0,-1,1,2,4,0,1,2,3,0,0,1,1,2,1]
Phi of -K*	[-4,-2,0,1,2,3,1,2,4,3,3,1,3,2,3,1,1,2,1,3,1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	5z+11
Enhanced Jones-Krushkal polynomial	4w^4z-6w^3z+7w^2z+11w
Inner characteristic polynomial	t^6+79t^4+17t^2
Outer characteristic polynomial	t^7+113t^5+71t^3
Flat arrow polynomial	-8K14 + 4K1*3 + 4K1*2K2 - 2K1K2 - 2K1 + 2K2 + 3
2-strand cable arrow polynomial	-544K14 + 128K1*2K2*5 - 896K1*2K2*4 + 672K1*2K2*3 - 1584K1*2K2*2 + 1656K1*2K2 - 32K12K3*2 - 768K1*2 + 128K1K25K3 + 576K1K2*3K3 + 1064K1K2K3 + 72K1K3K4 - 128K28 + 128K2*6K4 - 672K26 - 128K2*4K3*2 - 32K2*4K4*2 + 352K2*4K4 - 352K24 + 32K2*3K3K5 - 176K2*2K3*2 - 40K2*2K4*2 + 296K2*2K4 - 176K22 + 64K2K3K5 - 252K32 - 64K4*2 - 12K5**2 + 686
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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