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Flat knot 6.1253

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,0,2,2,2,1,2,2,2,0,1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1253']
Arrow polynomial of the knot is: 4K13 - 6K1*2 - 4K1K2 - 4K1K3 - K1 + 5K2 + K3 + 2*K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1253']
Outer characteristic polynomial of the knot is: t^7+39t^5+53t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1253']
2-strand cable arrow polynomial of the knot is: 160K14K2 - 912K14 + 512K1*3K2K3 - 448K1*3K3 + 192K12K2*3 - 192K1*2K2*2K3*2 - 4112K1*2K2*2 + 192K1*2K2K32 - 128K1*2K2K4 + 5680K1*2K2 - 976K12K3*2 - 32K1*2K5*2 - 5504K1*2 + 672K1K23K3 - 704K1K2*2K3 - 320K1K2*2K5 + 128K1K2K33 - 384K1K2K3K4 - 128K1K2K3K6 + 8128K1K2K3 - 64K1K2K4K5 + 1336K1K3K4 + 264K1K4K5 + 112K1K5K6 - 32K2*6 + 64K2*4K4 - 1016K24 + 32K2*3K3K5 - 32K2*3K6 - 1328K22K3*2 - 32K2*2K3K7 - 16K2*2K4*2 + 1272K2*2K4 - 64K22K5*2 - 48K2*2K6*2 - 4514K2*2 - 32K2K32K4 + 1608K2K3K5 + 200K2K4K6 + 48K2K5K7 + 32K2K6K8 - 32K34 + 152K3*2K6 - 3316K32 + 8K3K4K7 - 766K42 - 508K5*2 - 150K6*2 - 8K7*2 - 4K8**2 + 5008
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1253']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71386', 'vk6.71445', 'vk6.71912', 'vk6.71971', 'vk6.72459', 'vk6.72598', 'vk6.72715', 'vk6.72823', 'vk6.72885', 'vk6.73030', 'vk6.74231', 'vk6.74363', 'vk6.74421', 'vk6.74861', 'vk6.75037', 'vk6.76610', 'vk6.76909', 'vk6.77051', 'vk6.77419', 'vk6.77770', 'vk6.77819', 'vk6.79275', 'vk6.79401', 'vk6.79750', 'vk6.79819', 'vk6.79872', 'vk6.80851', 'vk6.80902', 'vk6.81378', 'vk6.85505', 'vk6.87215', 'vk6.89270']
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: [-2,-2,0,1,1,2,-1,0,2,2,2,1,2,2,2,0,1,0,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.1253']

Arrow polynomial of the knot is: 4*K1**3 - 6*K1**2 - 4*K1*K2 - 4*K1*K3 - K1 + 5*K2 + K3 + 2*K4 + 4

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1253']

Outer characteristic polynomial of the knot is: t^7+39t^5+53t^3+14t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1253']

2-strand cable arrow polynomial of the knot is: 160*K1**4*K2 - 912*K1**4 + 512*K1**3*K2*K3 - 448*K1**3*K3 + 192*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 4112*K1**2*K2**2 + 192*K1**2*K2*K3**2 - 128*K1**2*K2*K4 + 5680*K1**2*K2 - 976*K1**2*K3**2 - 32*K1**2*K5**2 - 5504*K1**2 + 672*K1*K2**3*K3 - 704*K1*K2**2*K3 - 320*K1*K2**2*K5 + 128*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 8128*K1*K2*K3 - 64*K1*K2*K4*K5 + 1336*K1*K3*K4 + 264*K1*K4*K5 + 112*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 1016*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 - 1328*K2**2*K3**2 - 32*K2**2*K3*K7 - 16*K2**2*K4**2 + 1272*K2**2*K4 - 64*K2**2*K5**2 - 48*K2**2*K6**2 - 4514*K2**2 - 32*K2*K3**2*K4 + 1608*K2*K3*K5 + 200*K2*K4*K6 + 48*K2*K5*K7 + 32*K2*K6*K8 - 32*K3**4 + 152*K3**2*K6 - 3316*K3**2 + 8*K3*K4*K7 - 766*K4**2 - 508*K5**2 - 150*K6**2 - 8*K7**2 - 4*K8**2 + 5008

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1253']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71386', 'vk6.71445', 'vk6.71912', 'vk6.71971', 'vk6.72459', 'vk6.72598', 'vk6.72715', 'vk6.72823', 'vk6.72885', 'vk6.73030', 'vk6.74231', 'vk6.74363', 'vk6.74421', 'vk6.74861', 'vk6.75037', 'vk6.76610', 'vk6.76909', 'vk6.77051', 'vk6.77419', 'vk6.77770', 'vk6.77819', 'vk6.79275', 'vk6.79401', 'vk6.79750', 'vk6.79819', 'vk6.79872', 'vk6.80851', 'vk6.80902', 'vk6.81378', 'vk6.85505', 'vk6.87215', 'vk6.89270']

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	O1O2O3O4U5U1U3O5O6U4U2U6
R3 orbit	{'O1O2O3O4U5U1U3O5O6U4U2U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U3U1O5O6U2U4U6
Gauss code of K*	O1O2O3U1U4O5O6O4U2U6U3U5
Gauss code of -K*	O1O2O3U1U4O5O6O4U3U5U2U6
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 -2 0 1 1 -2 2],[ 2 0 2 1 1 1 2],[ 0 -2 0 0 1 -1 2],[-1 -1 0 0 0 -1 1],[-1 -1 -1 0 0 -1 1],[ 2 -1 1 1 1 0 2],[-2 -2 -2 -1 -1 -2 0]]
Primitive based matrix	[[ 0 2 1 1 0 -2 -2],[-2 0 -1 -1 -2 -2 -2],[-1 1 0 0 0 -1 -1],[-1 1 0 0 -1 -1 -1],[ 0 2 0 1 0 -1 -2],[ 2 2 1 1 1 0 -1],[ 2 2 1 1 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,2,2,1,1,2,2,2,0,0,1,1,1,1,1,1,2,1]
Phi over symmetry	[-2,-2,0,1,1,2,-1,0,2,2,2,1,2,2,2,0,1,0,0,0,0]
Phi of -K	[-2,-2,0,1,1,2,-1,0,2,2,2,1,2,2,2,0,1,0,0,0,0]
Phi of K*	[-2,-1,-1,0,2,2,0,0,0,2,2,0,0,2,2,1,2,2,0,1,1]
Phi of -K*	[-2,-2,0,1,1,2,-1,1,1,1,2,2,1,1,2,0,1,2,0,1,1]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	3z^2+24z+37
Enhanced Jones-Krushkal polynomial	3w^3z^2+24w^2z+37w
Inner characteristic polynomial	t^6+25t^4+12t^2+1
Outer characteristic polynomial	t^7+39t^5+53t^3+14t
Flat arrow polynomial	4K13 - 6K1*2 - 4K1K2 - 4K1K3 - K1 + 5K2 + K3 + 2*K4 + 4
2-strand cable arrow polynomial	160K14K2 - 912K14 + 512K1*3K2K3 - 448K1*3K3 + 192K12K2*3 - 192K1*2K2*2K3*2 - 4112K1*2K2*2 + 192K1*2K2K32 - 128K1*2K2K4 + 5680K1*2K2 - 976K12K3*2 - 32K1*2K5*2 - 5504K1*2 + 672K1K23K3 - 704K1K2*2K3 - 320K1K2*2K5 + 128K1K2K33 - 384K1K2K3K4 - 128K1K2K3K6 + 8128K1K2K3 - 64K1K2K4K5 + 1336K1K3K4 + 264K1K4K5 + 112K1K5K6 - 32K2*6 + 64K2*4K4 - 1016K24 + 32K2*3K3K5 - 32K2*3K6 - 1328K22K3*2 - 32K2*2K3K7 - 16K2*2K4*2 + 1272K2*2K4 - 64K22K5*2 - 48K2*2K6*2 - 4514K2*2 - 32K2K32K4 + 1608K2K3K5 + 200K2K4K6 + 48K2K5K7 + 32K2K6K8 - 32K34 + 152K3*2K6 - 3316K32 + 8K3K4K7 - 766K42 - 508K5*2 - 150K6*2 - 8K7*2 - 4K8**2 + 5008
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}], [{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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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