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Flat knot 6.243

Min(phi) over symmetries of the knot is: [-4,-1,-1,1,2,3,0,2,1,4,4,0,0,1,1,0,2,2,0,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.243']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 6K12 - 6K1K2 - 2K2*2 + K2 + 2K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.243']
Outer characteristic polynomial of the knot is: t^7+80t^5+60t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.243']
2-strand cable arrow polynomial of the knot is: -64K16 + 480K1*4K2 - 1424K14 + 192K1*3K2K3 + 32K1*3K3K4 - 352K1*3K3 - 128K12K2*4 + 256K1*2K2*3 + 160K1*2K2*2K4 - 1856K12K2*2 + 128K1*2K2K32 + 64K1*2K2K42 - 448K1*2K2K4 + 3208K1*2K2 - 848K12K3*2 - 96K1*2K3K5 - 176K1*2K4*2 - 1816K1*2 + 224K1K23K3 - 384K1K2*2K3 - 32K1K2*2K5 + 64K1K2K33 + 32K1K2K3K42 - 224K1K2K3K4 + 3008K1K2K3 - 32K1K3*2K5 + 984K1K3K4 + 96K1K4K5 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 376K24 + 32K2*3K3K5 + 64K2*2K3*2K4 - 368K22K3*2 + 32K2*2K4*3 - 184K2*2K4*2 + 536K2*2K4 - 1532K22 - 32K2K3K4K5 + 248K2K3K5 + 48K2K4K6 - 96K3*4 - 48K3*2K4*2 + 64K3*2K6 - 832K32 + 16K3K4K7 - 8K44 - 242K4*2 - 32K5*2 - 12K6**2 + 1672
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.243']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13870', 'vk6.13965', 'vk6.14125', 'vk6.14346', 'vk6.14941', 'vk6.15064', 'vk6.15575', 'vk6.16045', 'vk6.16305', 'vk6.16330', 'vk6.16358', 'vk6.16401', 'vk6.17441', 'vk6.22616', 'vk6.22649', 'vk6.22784', 'vk6.23949', 'vk6.25972', 'vk6.26362', 'vk6.33681', 'vk6.34130', 'vk6.34643', 'vk6.34733', 'vk6.34735', 'vk6.36214', 'vk6.36245', 'vk6.38073', 'vk6.38079', 'vk6.42293', 'vk6.44553', 'vk6.44561', 'vk6.53860', 'vk6.54401', 'vk6.54617', 'vk6.55581', 'vk6.56523', 'vk6.56531', 'vk6.59060', 'vk6.59156', 'vk6.64552']
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,-1,-1,1,2,3,0,2,1,4,4,0,0,1,1,0,2,2,0,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.243']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.243']

Outer characteristic polynomial of the knot is: t^7+80t^5+60t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.243']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 480*K1**4*K2 - 1424*K1**4 + 192*K1**3*K2*K3 + 32*K1**3*K3*K4 - 352*K1**3*K3 - 128*K1**2*K2**4 + 256*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 1856*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 448*K1**2*K2*K4 + 3208*K1**2*K2 - 848*K1**2*K3**2 - 96*K1**2*K3*K5 - 176*K1**2*K4**2 - 1816*K1**2 + 224*K1*K2**3*K3 - 384*K1*K2**2*K3 - 32*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 224*K1*K2*K3*K4 + 3008*K1*K2*K3 - 32*K1*K3**2*K5 + 984*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 376*K2**4 + 32*K2**3*K3*K5 + 64*K2**2*K3**2*K4 - 368*K2**2*K3**2 + 32*K2**2*K4**3 - 184*K2**2*K4**2 + 536*K2**2*K4 - 1532*K2**2 - 32*K2*K3*K4*K5 + 248*K2*K3*K5 + 48*K2*K4*K6 - 96*K3**4 - 48*K3**2*K4**2 + 64*K3**2*K6 - 832*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 242*K4**2 - 32*K5**2 - 12*K6**2 + 1672

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.243']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13870', 'vk6.13965', 'vk6.14125', 'vk6.14346', 'vk6.14941', 'vk6.15064', 'vk6.15575', 'vk6.16045', 'vk6.16305', 'vk6.16330', 'vk6.16358', 'vk6.16401', 'vk6.17441', 'vk6.22616', 'vk6.22649', 'vk6.22784', 'vk6.23949', 'vk6.25972', 'vk6.26362', 'vk6.33681', 'vk6.34130', 'vk6.34643', 'vk6.34733', 'vk6.34735', 'vk6.36214', 'vk6.36245', 'vk6.38073', 'vk6.38079', 'vk6.42293', 'vk6.44553', 'vk6.44561', 'vk6.53860', 'vk6.54401', 'vk6.54617', 'vk6.55581', 'vk6.56523', 'vk6.56531', 'vk6.59060', 'vk6.59156', 'vk6.64552']

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	O1O2O3O4O5U4O6U1U6U2U5U3
R3 orbit	{'O1O2O3O4U3O5O6U1U6U2U4U5', 'O1O2O3O4O5U4O6U1U6U2U5U3'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U3U1U4U6U5O6U2
Gauss code of K*	O1O2O3O4O5U1U3U5U6U4O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U2U6U1U3U5
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 2 -1 3 1],[ 4 0 2 4 0 4 1],[ 1 -2 0 2 0 2 0],[-2 -4 -2 0 -1 1 0],[ 1 0 0 1 0 1 0],[-3 -4 -2 -1 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix	[[ 0 3 2 1 -1 -1 -4],[-3 0 -1 0 -1 -2 -4],[-2 1 0 0 -1 -2 -4],[-1 0 0 0 0 0 -1],[ 1 1 1 0 0 0 0],[ 1 2 2 0 0 0 -2],[ 4 4 4 1 0 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,1,1,4,1,0,1,2,4,0,1,2,4,0,0,1,0,0,2]
Phi over symmetry	[-4,-1,-1,1,2,3,0,2,1,4,4,0,0,1,1,0,2,2,0,0,1]
Phi of -K	[-4,-1,-1,1,2,3,1,3,4,2,3,0,2,1,2,2,2,3,1,2,0]
Phi of K*	[-3,-2,-1,1,1,4,0,2,2,3,3,1,1,2,2,2,2,4,0,1,3]
Phi of -K*	[-4,-1,-1,1,2,3,0,2,1,4,4,0,0,1,1,0,2,2,0,0,1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t^2+t
Normalized Jones-Krushkal polynomial	2z^2+15z+23
Enhanced Jones-Krushkal polynomial	2w^3z^2+15w^2z+23w
Inner characteristic polynomial	t^6+48t^4+23t^2
Outer characteristic polynomial	t^7+80t^5+60t^3+3t
Flat arrow polynomial	4K13 + 4K1*2K2 - 6K12 - 6K1K2 - 2K2*2 + K2 + 2K3 + 4
2-strand cable arrow polynomial	-64K16 + 480K1*4K2 - 1424K14 + 192K1*3K2K3 + 32K1*3K3K4 - 352K1*3K3 - 128K12K2*4 + 256K1*2K2*3 + 160K1*2K2*2K4 - 1856K12K2*2 + 128K1*2K2K32 + 64K1*2K2K42 - 448K1*2K2K4 + 3208K1*2K2 - 848K12K3*2 - 96K1*2K3K5 - 176K1*2K4*2 - 1816K1*2 + 224K1K23K3 - 384K1K2*2K3 - 32K1K2*2K5 + 64K1K2K33 + 32K1K2K3K42 - 224K1K2K3K4 + 3008K1K2K3 - 32K1K3*2K5 + 984K1K3K4 + 96K1K4K5 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 376K24 + 32K2*3K3K5 + 64K2*2K3*2K4 - 368K22K3*2 + 32K2*2K4*3 - 184K2*2K4*2 + 536K2*2K4 - 1532K22 - 32K2K3K4K5 + 248K2K3K5 + 48K2K4K6 - 96K3*4 - 48K3*2K4*2 + 64K3*2K6 - 832K32 + 16K3K4K7 - 8K44 - 242K4*2 - 32K5*2 - 12K6**2 + 1672
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}], [{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}, {4, 5}, {3}, {2}, {1}], [{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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