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Flat knot 6.475

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,1,2,2,4,3,1,1,2,2,0,0,1,-1,1,2]
Flat knots (up to 7 crossings) with same phi are :['6.475']
Arrow polynomial of the knot is: -8K14 + 8K1*3 + 4K1*2K2 + 4K12 - 6K1K2 - 3K1 + K3 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.475']
Outer characteristic polynomial of the knot is: t^7+101t^5+100t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.475']
2-strand cable arrow polynomial of the knot is: -32K14 - 64K1*3K3 - 1024K12K2*6 + 2688K1*2K2*5 - 5504K1*2K2*4 - 128K1*2K2*3K4 + 4352K12K2*3 - 5120K1*2K2*2 - 160K1*2K2K4 + 2960K1*2K2 - 32K12K3*2 - 1944K1*2 + 1280K1K25K3 - 1536K1K2*4K3 - 128K1K2*4K5 + 4352K1K2*3K3 + 64K1K2*2K3K4 - 1216K1K22K3 - 160K1K2*2K5 - 64K1K2K3K4 + 3216K1K2K3 + 144K1K3K4 - 128K28 + 128K2*6K4 - 2816K26 - 384K2*4K3*2 - 32K2*4K4*2 + 1888K2*4K4 - 1792K24 + 64K2*3K3K5 - 32K2*3K6 - 1088K22K3*2 - 200K2*2K4*2 + 1288K2*2K4 + 426K22 + 160K2K3K5 + 24K2K4K6 - 696K3*2 - 108K4*2 - 2K6**2 + 1354
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.475']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73971', 'vk6.73973', 'vk6.74482', 'vk6.74486', 'vk6.75932', 'vk6.75940', 'vk6.76700', 'vk6.76704', 'vk6.78938', 'vk6.78942', 'vk6.79478', 'vk6.79485', 'vk6.80464', 'vk6.80468', 'vk6.80946', 'vk6.80948', 'vk6.82984', 'vk6.83035', 'vk6.83748', 'vk6.83750', 'vk6.83917', 'vk6.85531', 'vk6.85533', 'vk6.85698', 'vk6.85777', 'vk6.85808', 'vk6.85812', 'vk6.86721', 'vk6.86723', 'vk6.87817', 'vk6.89583', 'vk6.89949']
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,-2,0,1,2,3,1,2,2,4,3,1,1,2,2,0,0,1,-1,1,2]

Flat knots (up to 7 crossings) with same phi are :['6.475']

Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 4*K1**2*K2 + 4*K1**2 - 6*K1*K2 - 3*K1 + K3 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.475']

Outer characteristic polynomial of the knot is: t^7+101t^5+100t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.475']

2-strand cable arrow polynomial of the knot is: -32*K1**4 - 64*K1**3*K3 - 1024*K1**2*K2**6 + 2688*K1**2*K2**5 - 5504*K1**2*K2**4 - 128*K1**2*K2**3*K4 + 4352*K1**2*K2**3 - 5120*K1**2*K2**2 - 160*K1**2*K2*K4 + 2960*K1**2*K2 - 32*K1**2*K3**2 - 1944*K1**2 + 1280*K1*K2**5*K3 - 1536*K1*K2**4*K3 - 128*K1*K2**4*K5 + 4352*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 1216*K1*K2**2*K3 - 160*K1*K2**2*K5 - 64*K1*K2*K3*K4 + 3216*K1*K2*K3 + 144*K1*K3*K4 - 128*K2**8 + 128*K2**6*K4 - 2816*K2**6 - 384*K2**4*K3**2 - 32*K2**4*K4**2 + 1888*K2**4*K4 - 1792*K2**4 + 64*K2**3*K3*K5 - 32*K2**3*K6 - 1088*K2**2*K3**2 - 200*K2**2*K4**2 + 1288*K2**2*K4 + 426*K2**2 + 160*K2*K3*K5 + 24*K2*K4*K6 - 696*K3**2 - 108*K4**2 - 2*K6**2 + 1354

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.475']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73971', 'vk6.73973', 'vk6.74482', 'vk6.74486', 'vk6.75932', 'vk6.75940', 'vk6.76700', 'vk6.76704', 'vk6.78938', 'vk6.78942', 'vk6.79478', 'vk6.79485', 'vk6.80464', 'vk6.80468', 'vk6.80946', 'vk6.80948', 'vk6.82984', 'vk6.83035', 'vk6.83748', 'vk6.83750', 'vk6.83917', 'vk6.85531', 'vk6.85533', 'vk6.85698', 'vk6.85777', 'vk6.85808', 'vk6.85812', 'vk6.86721', 'vk6.86723', 'vk6.87817', 'vk6.89583', 'vk6.89949']

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	O1O2O3O4O5U1U2U3O6U4U5U6
R3 orbit	{'O1O2O3O4O5U1U2U3O6U4U5U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U1U2O6U3U4U5
Gauss code of K*	O1O2O3U4O5O6O4U1U2U3U5U6
Gauss code of -K*	O1O2O3U1O4O5O6U2U3U4U5U6
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 -2 0 1 3 2],[ 4 0 1 2 3 4 2],[ 2 -1 0 1 2 3 2],[ 0 -2 -1 0 1 2 2],[-1 -3 -2 -1 0 1 2],[-3 -4 -3 -2 -1 0 1],[-2 -2 -2 -2 -2 -1 0]]
Primitive based matrix	[[ 0 3 2 1 0 -2 -4],[-3 0 1 -1 -2 -3 -4],[-2 -1 0 -2 -2 -2 -2],[-1 1 2 0 -1 -2 -3],[ 0 2 2 1 0 -1 -2],[ 2 3 2 2 1 0 -1],[ 4 4 2 3 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,0,2,4,-1,1,2,3,4,2,2,2,2,1,2,3,1,2,1]
Phi over symmetry	[-4,-2,0,1,2,3,1,2,2,4,3,1,1,2,2,0,0,1,-1,1,2]
Phi of -K	[-4,-2,0,1,2,3,1,2,2,4,3,1,1,2,2,0,0,1,-1,1,2]
Phi of K*	[-3,-2,-1,0,2,4,2,1,1,2,3,-1,0,2,4,0,1,2,1,2,1]
Phi of -K*	[-4,-2,0,1,2,3,1,2,3,2,4,1,2,2,3,1,2,2,2,1,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	2z^2+7z+7
Enhanced Jones-Krushkal polynomial	-6w^4z^2+8w^3z^2-10w^3z+17w^2z+7w
Inner characteristic polynomial	t^6+67t^4+20t^2
Outer characteristic polynomial	t^7+101t^5+100t^3+6t
Flat arrow polynomial	-8K14 + 8K1*3 + 4K1*2K2 + 4K12 - 6K1K2 - 3K1 + K3 + 1
2-strand cable arrow polynomial	-32K14 - 64K1*3K3 - 1024K12K2*6 + 2688K1*2K2*5 - 5504K1*2K2*4 - 128K1*2K2*3K4 + 4352K12K2*3 - 5120K1*2K2*2 - 160K1*2K2K4 + 2960K1*2K2 - 32K12K3*2 - 1944K1*2 + 1280K1K25K3 - 1536K1K2*4K3 - 128K1K2*4K5 + 4352K1K2*3K3 + 64K1K2*2K3K4 - 1216K1K22K3 - 160K1K2*2K5 - 64K1K2K3K4 + 3216K1K2K3 + 144K1K3K4 - 128K28 + 128K2*6K4 - 2816K26 - 384K2*4K3*2 - 32K2*4K4*2 + 1888K2*4K4 - 1792K24 + 64K2*3K3K5 - 32K2*3K6 - 1088K22K3*2 - 200K2*2K4*2 + 1288K2*2K4 + 426K22 + 160K2K3K5 + 24K2K4K6 - 696K3*2 - 108K4*2 - 2K6**2 + 1354
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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {1, 3}, {2}]]
If K is slice	False

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