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Flat knot 6.168

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,2,1,4,3,1,1,2,2,1,1,2,-1,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.168']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 8K12 - 6K1K2 - 2K2*2 + 2K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.168', '6.222', '6.288']
Outer characteristic polynomial of the knot is: t^7+80t^5+52t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.168']
2-strand cable arrow polynomial of the knot is: -64K16 + 544K1*4K2 - 1392K14 + 192K1*3K2K3 + 32K1*3K3K4 - 480K1*3K3 - 128K12K2*4 + 320K1*2K2*3 + 160K1*2K2*2K4 - 1904K12K2*2 + 192K1*2K2K32 + 64K1*2K2K42 - 288K1*2K2K4 + 3552K1*2K2 - 784K12K3*2 - 176K1*2K4*2 - 2148K1*2 + 224K1K23K3 - 640K1K2*2K3 - 64K1K2*2K5 + 64K1K2K33 + 32K1K2K3K42 - 384K1K2K3K4 - 32K1K2K3K6 + 3192K1K2K3 + 976K1K3K4 + 96K1K4K5 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 416K24 + 32K2*3K3K5 - 32K2*3K6 + 64K22K3*2K4 - 432K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 152K2*2K4*2 + 648K2*2K4 - 1748K22 - 32K2K32K4 + 312K2K3K5 + 48K2K4K6 - 64K34 - 48K3*2K4*2 + 48K3*2K6 - 932K32 + 16K3K4K7 - 8K44 - 276K4*2 - 24K5*2 - 4K6**2 + 1842
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.168']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11467', 'vk6.11770', 'vk6.12785', 'vk6.13120', 'vk6.16360', 'vk6.16402', 'vk6.19183', 'vk6.19192', 'vk6.19476', 'vk6.19487', 'vk6.22371', 'vk6.22761', 'vk6.22783', 'vk6.25980', 'vk6.25993', 'vk6.26370', 'vk6.28326', 'vk6.31220', 'vk6.31569', 'vk6.34630', 'vk6.34645', 'vk6.34714', 'vk6.34734', 'vk6.35564', 'vk6.36013', 'vk6.38086', 'vk6.39950', 'vk6.40125', 'vk6.42348', 'vk6.42360', 'vk6.42967', 'vk6.43262', 'vk6.44557', 'vk6.44574', 'vk6.46639', 'vk6.52232', 'vk6.56527', 'vk6.59145', 'vk6.59155', 'vk6.66258']
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,-1,0,1,1,3,1,2,1,4,3,1,1,2,2,1,1,2,-1,0,2]

Flat knots (up to 7 crossings) with same phi are :['6.168']

Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 6*K1*K2 - 2*K2**2 + 2*K2 + 2*K3 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.168', '6.222', '6.288']

Outer characteristic polynomial of the knot is: t^7+80t^5+52t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.168']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 544*K1**4*K2 - 1392*K1**4 + 192*K1**3*K2*K3 + 32*K1**3*K3*K4 - 480*K1**3*K3 - 128*K1**2*K2**4 + 320*K1**2*K2**3 + 160*K1**2*K2**2*K4 - 1904*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 288*K1**2*K2*K4 + 3552*K1**2*K2 - 784*K1**2*K3**2 - 176*K1**2*K4**2 - 2148*K1**2 + 224*K1*K2**3*K3 - 640*K1*K2**2*K3 - 64*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 384*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 3192*K1*K2*K3 + 976*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 416*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 432*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 152*K2**2*K4**2 + 648*K2**2*K4 - 1748*K2**2 - 32*K2*K3**2*K4 + 312*K2*K3*K5 + 48*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 48*K3**2*K6 - 932*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 276*K4**2 - 24*K5**2 - 4*K6**2 + 1842

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.168']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11467', 'vk6.11770', 'vk6.12785', 'vk6.13120', 'vk6.16360', 'vk6.16402', 'vk6.19183', 'vk6.19192', 'vk6.19476', 'vk6.19487', 'vk6.22371', 'vk6.22761', 'vk6.22783', 'vk6.25980', 'vk6.25993', 'vk6.26370', 'vk6.28326', 'vk6.31220', 'vk6.31569', 'vk6.34630', 'vk6.34645', 'vk6.34714', 'vk6.34734', 'vk6.35564', 'vk6.36013', 'vk6.38086', 'vk6.39950', 'vk6.40125', 'vk6.42348', 'vk6.42360', 'vk6.42967', 'vk6.43262', 'vk6.44557', 'vk6.44574', 'vk6.46639', 'vk6.52232', 'vk6.56527', 'vk6.59145', 'vk6.59155', 'vk6.66258']

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	O1O2O3O4O5U1O6U4U6U3U5U2
R3 orbit	{'O1O2O3O4O5U1O6U4U6U3U5U2', 'O1O2O3O4O5U1U3O6U4U6U5U2'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U4U1U3U6U2O6U5
Gauss code of K*	O1O2O3O4O5U6U5U3U1U4O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U2U5U3U1U6
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 0 -1 3 1],[ 4 0 4 2 1 3 1],[-1 -4 0 -1 -2 2 1],[ 0 -2 1 0 -1 2 1],[ 1 -1 2 1 0 2 1],[-3 -3 -2 -2 -2 0 0],[-1 -1 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 3 1 1 0 -1 -4],[-3 0 0 -2 -2 -2 -3],[-1 0 0 -1 -1 -1 -1],[-1 2 1 0 -1 -2 -4],[ 0 2 1 1 0 -1 -2],[ 1 2 1 2 1 0 -1],[ 4 3 1 4 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,0,1,4,0,2,2,2,3,1,1,1,1,1,2,4,1,2,1]
Phi over symmetry	[-4,-1,0,1,1,3,1,2,1,4,3,1,1,2,2,1,1,2,-1,0,2]
Phi of -K	[-4,-1,0,1,1,3,2,2,1,4,4,0,0,1,2,0,0,1,-1,0,2]
Phi of K*	[-3,-1,-1,0,1,4,0,2,1,2,4,1,0,0,1,0,1,4,0,2,2]
Phi of -K*	[-4,-1,0,1,1,3,1,2,1,4,3,1,1,2,2,1,1,2,-1,0,2]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	2z^2+15z+23
Enhanced Jones-Krushkal polynomial	2w^3z^2+15w^2z+23w
Inner characteristic polynomial	t^6+52t^4+8t^2
Outer characteristic polynomial	t^7+80t^5+52t^3+3t
Flat arrow polynomial	4K13 + 4K1*2K2 - 8K12 - 6K1K2 - 2K2*2 + 2K2 + 2*K3 + 5
2-strand cable arrow polynomial	-64K16 + 544K1*4K2 - 1392K14 + 192K1*3K2K3 + 32K1*3K3K4 - 480K1*3K3 - 128K12K2*4 + 320K1*2K2*3 + 160K1*2K2*2K4 - 1904K12K2*2 + 192K1*2K2K32 + 64K1*2K2K42 - 288K1*2K2K4 + 3552K1*2K2 - 784K12K3*2 - 176K1*2K4*2 - 2148K1*2 + 224K1K23K3 - 640K1K2*2K3 - 64K1K2*2K5 + 64K1K2K33 + 32K1K2K3K42 - 384K1K2K3K4 - 32K1K2K3K6 + 3192K1K2K3 + 976K1K3K4 + 96K1K4K5 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 416K24 + 32K2*3K3K5 - 32K2*3K6 + 64K22K3*2K4 - 432K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 152K2*2K4*2 + 648K2*2K4 - 1748K22 - 32K2K32K4 + 312K2K3K5 + 48K2K4K6 - 64K34 - 48K3*2K4*2 + 48K3*2K6 - 932K32 + 16K3K4K7 - 8K44 - 276K4*2 - 24K5*2 - 4K6**2 + 1842
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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