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Flat knot 6.295

Min(phi) over symmetries of the knot is: [-4,0,1,1,1,1,2,1,1,3,4,0,1,1,1,0,0,0,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.295']
Arrow polynomial of the knot is: -2K12 - 4K1K2 - 2K1K3 + 2K1 + 2K2 + 2K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.180', '6.263', '6.295', '6.317', '6.350', '6.473', '6.504']
Outer characteristic polynomial of the knot is: t^7+56t^5+48t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.295']
2-strand cable arrow polynomial of the knot is: -1296K14 + 352K1*3K2K3 + 32K1*3K3K4 - 672K1*3K3 + 96K12K2*2K4 - 1104K12K2*2 - 704K1*2K2K4 + 3072K1*2K2 - 368K12K3*2 - 128K1*2K4*2 - 1892K1*2 - 192K1K22K3 - 256K1K2*2K5 - 64K1K2K3K4 + 2720K1K2K3 - 32K1K2K4K5 + 984K1K3K4 + 288K1K4K5 + 48K1K5K6 - 72K24 - 32K2*3K6 - 16K22K4*2 + 608K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 1792K2*2 + 384K2K3K5 + 104K2K4K6 + 32K2K5K7 + 8K2K6K8 + 32K3*2K6 - 1040K32 - 552K4*2 - 264K5*2 - 72K6*2 - 12K7*2 - 2K8**2 + 1824
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.295']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13383', 'vk6.13464', 'vk6.13653', 'vk6.13763', 'vk6.13919', 'vk6.14016', 'vk6.14179', 'vk6.14207', 'vk6.14418', 'vk6.14458', 'vk6.14990', 'vk6.15113', 'vk6.15647', 'vk6.16101', 'vk6.16131', 'vk6.16754', 'vk6.16775', 'vk6.23163', 'vk6.23178', 'vk6.25388', 'vk6.25666', 'vk6.33138', 'vk6.33189', 'vk6.33738', 'vk6.33815', 'vk6.35149', 'vk6.35165', 'vk6.35196', 'vk6.37516', 'vk6.37763', 'vk6.42664', 'vk6.42681', 'vk6.42723', 'vk6.42789', 'vk6.44705', 'vk6.44717', 'vk6.53567', 'vk6.54957', 'vk6.56612', 'vk6.64588']
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: [-4,0,1,1,1,1,2,1,1,3,4,0,1,1,1,0,0,0,-1,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.295']

Arrow polynomial of the knot is: -2*K1**2 - 4*K1*K2 - 2*K1*K3 + 2*K1 + 2*K2 + 2*K3 + K4 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.180', '6.263', '6.295', '6.317', '6.350', '6.473', '6.504']

Outer characteristic polynomial of the knot is: t^7+56t^5+48t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.295']

2-strand cable arrow polynomial of the knot is: -1296*K1**4 + 352*K1**3*K2*K3 + 32*K1**3*K3*K4 - 672*K1**3*K3 + 96*K1**2*K2**2*K4 - 1104*K1**2*K2**2 - 704*K1**2*K2*K4 + 3072*K1**2*K2 - 368*K1**2*K3**2 - 128*K1**2*K4**2 - 1892*K1**2 - 192*K1*K2**2*K3 - 256*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 2720*K1*K2*K3 - 32*K1*K2*K4*K5 + 984*K1*K3*K4 + 288*K1*K4*K5 + 48*K1*K5*K6 - 72*K2**4 - 32*K2**3*K6 - 16*K2**2*K4**2 + 608*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 1792*K2**2 + 384*K2*K3*K5 + 104*K2*K4*K6 + 32*K2*K5*K7 + 8*K2*K6*K8 + 32*K3**2*K6 - 1040*K3**2 - 552*K4**2 - 264*K5**2 - 72*K6**2 - 12*K7**2 - 2*K8**2 + 1824

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.295']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13383', 'vk6.13464', 'vk6.13653', 'vk6.13763', 'vk6.13919', 'vk6.14016', 'vk6.14179', 'vk6.14207', 'vk6.14418', 'vk6.14458', 'vk6.14990', 'vk6.15113', 'vk6.15647', 'vk6.16101', 'vk6.16131', 'vk6.16754', 'vk6.16775', 'vk6.23163', 'vk6.23178', 'vk6.25388', 'vk6.25666', 'vk6.33138', 'vk6.33189', 'vk6.33738', 'vk6.33815', 'vk6.35149', 'vk6.35165', 'vk6.35196', 'vk6.37516', 'vk6.37763', 'vk6.42664', 'vk6.42681', 'vk6.42723', 'vk6.42789', 'vk6.44705', 'vk6.44717', 'vk6.53567', 'vk6.54957', 'vk6.56612', 'vk6.64588']

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	O1O2O3O4O5U1U5O6U4U6U3U2
R3 orbit	{'O1O2O3O4O5U1U5U3O6U4U6U2', 'O1O2O3O4O5U1U5O6U4U6U3U2'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U4U3U6U2O6U1U5
Gauss code of K*	O1O2O3O4U5U4U3U1U6O5O6U2
Gauss code of -K*	O1O2O3O4U3O5O6U5U4U2U1U6
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 1 0 1 1],[ 4 0 4 3 2 1 1],[-1 -4 0 0 -1 0 1],[-1 -3 0 0 -1 0 1],[ 0 -2 1 1 0 0 1],[-1 -1 0 0 0 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 1 1 1 1 0 -4],[-1 0 1 0 0 -1 -3],[-1 -1 0 0 -1 -1 -1],[-1 0 0 0 0 0 -1],[-1 0 1 0 0 -1 -4],[ 0 1 1 0 1 0 -2],[ 4 3 1 1 4 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-1,-1,-1,-1,0,4,-1,0,0,1,3,0,1,1,1,0,0,1,1,4,2]
Phi over symmetry	[-4,0,1,1,1,1,2,1,1,3,4,0,1,1,1,0,0,0,-1,-1,0]
Phi of -K	[-4,0,1,1,1,1,2,1,2,4,4,0,0,0,1,0,-1,0,-1,0,0]
Phi of K*	[-1,-1,-1,-1,0,4,-1,-1,0,0,4,0,0,0,1,0,0,2,1,4,2]
Phi of -K*	[-4,0,1,1,1,1,2,1,1,3,4,0,1,1,1,0,0,0,-1,-1,0]
Symmetry type of based matrix	c
u-polynomial	t^4-4t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	4w^3z^2+17w^2z+19w
Inner characteristic polynomial	t^6+36t^4+8t^2
Outer characteristic polynomial	t^7+56t^5+48t^3+3t
Flat arrow polynomial	-2K12 - 4K1K2 - 2K1K3 + 2K1 + 2K2 + 2K3 + K4 + 2
2-strand cable arrow polynomial	-1296K14 + 352K1*3K2K3 + 32K1*3K3K4 - 672K1*3K3 + 96K12K2*2K4 - 1104K12K2*2 - 704K1*2K2K4 + 3072K1*2K2 - 368K12K3*2 - 128K1*2K4*2 - 1892K1*2 - 192K1K22K3 - 256K1K2*2K5 - 64K1K2K3K4 + 2720K1K2K3 - 32K1K2K4K5 + 984K1K3K4 + 288K1K4K5 + 48K1K5K6 - 72K24 - 32K2*3K6 - 16K22K4*2 + 608K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 1792K2*2 + 384K2K3K5 + 104K2K4K6 + 32K2K5K7 + 8K2K6K8 + 32K3*2K6 - 1040K32 - 552K4*2 - 264K5*2 - 72K6*2 - 12K7*2 - 2K8**2 + 1824
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}], [{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}], [{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}, {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}, {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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