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Flat knot 6.145

Min(phi) over symmetries of the knot is: [-4,-3,0,2,2,3,0,2,2,3,4,2,1,3,3,0,1,1,0,-1,0]
Flat knots (up to 7 crossings) with same phi are :['6.145']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 10K12 - 4K1K2 - 2K1K3 - 4K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.145']
Outer characteristic polynomial of the knot is: t^7+125t^5+152t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.145']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 544K1*4K2 - 1168K14 + 128K1*3K2*3K3 - 384K13K2*2K3 + 640K13K2K3 - 352K1*3K3 - 512K12K2*4 - 128K1*2K2*3K4 + 2656K12K2*3 - 256K1*2K2*2K3*2 + 128K1*2K2*2K4 - 9024K12K2*2 + 160K1*2K2K32 + 32K1*2K2K3K5 - 672K12K2K4 + 7848K1*2K2 - 368K12K3*2 - 5196K1*2 - 128K1K24K3 + 2592K1K2*3K3 + 512K1K2*2K3K4 - 2048K1K22K3 + 96K1K2*2K4K5 - 384K1K22K5 + 64K1K2K33 - 288K1K2K3K4 - 32K1K2K3K6 + 8304K1K2K3 - 128K1K2K4K5 + 1000K1K3K4 + 96K1K4K5 - 64K26 - 128K2*4K3*2 - 32K2*4K4*2 + 352K2*4K4 - 3480K24 + 160K2*3K3K5 + 32K2*3K4K6 - 1872K2*2K3*2 - 432K2*2K4*2 + 2416K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 2452K2*2 - 96K2K32K4 + 832K2K3K5 + 160K2K4K6 + 8K32K6 - 2216K32 - 640K4*2 - 132K5*2 - 12K6**2 + 4158
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.145']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73282', 'vk6.73423', 'vk6.74011', 'vk6.74543', 'vk6.75198', 'vk6.75426', 'vk6.76019', 'vk6.76757', 'vk6.78155', 'vk6.78388', 'vk6.78986', 'vk6.79537', 'vk6.79980', 'vk6.80133', 'vk6.80501', 'vk6.80977', 'vk6.81879', 'vk6.82151', 'vk6.82178', 'vk6.82593', 'vk6.83578', 'vk6.83758', 'vk6.84038', 'vk6.84617', 'vk6.84944', 'vk6.85584', 'vk6.85702', 'vk6.85940', 'vk6.86729', 'vk6.87669', 'vk6.88938', 'vk6.89967']
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,-3,0,2,2,3,0,2,2,3,4,2,1,3,3,0,1,1,0,-1,0]

Flat knots (up to 7 crossings) with same phi are :['6.145']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 10*K1**2 - 4*K1*K2 - 2*K1*K3 - 4*K1 + 4*K2 + 5

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.145']

Outer characteristic polynomial of the knot is: t^7+125t^5+152t^3+13t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.145']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 544*K1**4*K2 - 1168*K1**4 + 128*K1**3*K2**3*K3 - 384*K1**3*K2**2*K3 + 640*K1**3*K2*K3 - 352*K1**3*K3 - 512*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 2656*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 9024*K1**2*K2**2 + 160*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 672*K1**2*K2*K4 + 7848*K1**2*K2 - 368*K1**2*K3**2 - 5196*K1**2 - 128*K1*K2**4*K3 + 2592*K1*K2**3*K3 + 512*K1*K2**2*K3*K4 - 2048*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 384*K1*K2**2*K5 + 64*K1*K2*K3**3 - 288*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 8304*K1*K2*K3 - 128*K1*K2*K4*K5 + 1000*K1*K3*K4 + 96*K1*K4*K5 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 352*K2**4*K4 - 3480*K2**4 + 160*K2**3*K3*K5 + 32*K2**3*K4*K6 - 1872*K2**2*K3**2 - 432*K2**2*K4**2 + 2416*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 2452*K2**2 - 96*K2*K3**2*K4 + 832*K2*K3*K5 + 160*K2*K4*K6 + 8*K3**2*K6 - 2216*K3**2 - 640*K4**2 - 132*K5**2 - 12*K6**2 + 4158

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.145']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73282', 'vk6.73423', 'vk6.74011', 'vk6.74543', 'vk6.75198', 'vk6.75426', 'vk6.76019', 'vk6.76757', 'vk6.78155', 'vk6.78388', 'vk6.78986', 'vk6.79537', 'vk6.79980', 'vk6.80133', 'vk6.80501', 'vk6.80977', 'vk6.81879', 'vk6.82151', 'vk6.82178', 'vk6.82593', 'vk6.83578', 'vk6.83758', 'vk6.84038', 'vk6.84617', 'vk6.84944', 'vk6.85584', 'vk6.85702', 'vk6.85940', 'vk6.86729', 'vk6.87669', 'vk6.88938', 'vk6.89967']

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	O1O2O3O4O5U1O6U2U4U5U6U3
R3 orbit	{'O1O2O3O4O5U1O6U2U4U5U6U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U6U1U2U4O6U5
Gauss code of K*	O1O2O3O4O5U6U1U5U2U3O6U4
Gauss code of -K*	O1O2O3O4O5U2O6U3U4U1U5U6
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 -3 2 0 2 3],[ 4 0 1 4 2 3 3],[ 3 -1 0 4 1 2 3],[-2 -4 -4 0 -2 0 2],[ 0 -2 -1 2 0 1 2],[-2 -3 -2 0 -1 0 1],[-3 -3 -3 -2 -2 -1 0]]
Primitive based matrix	[[ 0 3 2 2 0 -3 -4],[-3 0 -1 -2 -2 -3 -3],[-2 1 0 0 -1 -2 -3],[-2 2 0 0 -2 -4 -4],[ 0 2 1 2 0 -1 -2],[ 3 3 2 4 1 0 -1],[ 4 3 3 4 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-2,0,3,4,1,2,2,3,3,0,1,2,3,2,4,4,1,2,1]
Phi over symmetry	[-4,-3,0,2,2,3,0,2,2,3,4,2,1,3,3,0,1,1,0,-1,0]
Phi of -K	[-4,-3,0,2,2,3,0,2,2,3,4,2,1,3,3,0,1,1,0,-1,0]
Phi of K*	[-3,-2,-2,0,3,4,-1,0,1,3,4,0,0,1,2,1,3,3,2,2,0]
Phi of -K*	[-4,-3,0,2,2,3,1,2,3,4,3,1,2,4,3,1,2,2,0,1,2]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^2
Normalized Jones-Krushkal polynomial	6z^2+27z+31
Enhanced Jones-Krushkal polynomial	6w^3z^2+27w^2z+31w
Inner characteristic polynomial	t^6+83t^4+42t^2
Outer characteristic polynomial	t^7+125t^5+152t^3+13t
Flat arrow polynomial	8K13 + 4K1*2K2 - 10K12 - 4K1K2 - 2K1K3 - 4K1 + 4*K2 + 5
2-strand cable arrow polynomial	-192K14K2*2 + 544K1*4K2 - 1168K14 + 128K1*3K2*3K3 - 384K13K2*2K3 + 640K13K2K3 - 352K1*3K3 - 512K12K2*4 - 128K1*2K2*3K4 + 2656K12K2*3 - 256K1*2K2*2K3*2 + 128K1*2K2*2K4 - 9024K12K2*2 + 160K1*2K2K32 + 32K1*2K2K3K5 - 672K12K2K4 + 7848K1*2K2 - 368K12K3*2 - 5196K1*2 - 128K1K24K3 + 2592K1K2*3K3 + 512K1K2*2K3K4 - 2048K1K22K3 + 96K1K2*2K4K5 - 384K1K22K5 + 64K1K2K33 - 288K1K2K3K4 - 32K1K2K3K6 + 8304K1K2K3 - 128K1K2K4K5 + 1000K1K3K4 + 96K1K4K5 - 64K26 - 128K2*4K3*2 - 32K2*4K4*2 + 352K2*4K4 - 3480K24 + 160K2*3K3K5 + 32K2*3K4K6 - 1872K2*2K3*2 - 432K2*2K4*2 + 2416K2*2K4 - 96K22K5*2 - 8K2*2K6*2 - 2452K2*2 - 96K2K32K4 + 832K2K3K5 + 160K2K4K6 + 8K32K6 - 2216K32 - 640K4*2 - 132K5*2 - 12K6**2 + 4158
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}, {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}, {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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