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Flat knot 6.274

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,0,3,1,3,4,2,1,1,1,0,0,2,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.274']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 6K12 - 2K1K2 - 2K1K3 - 2K1 + 2*K2 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.172', '6.274', '6.286', '6.423', '6.461']
Outer characteristic polynomial of the knot is: t^7+88t^5+146t^3+16t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.274']
2-strand cable arrow polynomial of the knot is: -400K14 - 32K1*3K3 - 512K12K2*4 + 992K1*2K2*3 + 64K1*2K2*2K4 - 4352K12K2*2 - 224K1*2K2K4 + 4760K1*2K2 - 16K12K3*2 - 48K1*2K4*2 - 3848K1*2 + 1472K1K23K3 + 96K1K2*2K3K4 - 768K1K22K3 + 128K1K2*2K4K5 - 608K1K22K5 - 224K1K2K3K4 + 5056K1K2K3 + 368K1K3K4 + 144K1K4K5 - 736K2*6 - 256K2*4K3*2 - 32K2*4K4*2 + 800K2*4K4 - 2616K24 + 320K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 1088K22K3*2 - 344K2*2K4*2 + 2152K2*2K4 - 160K22K5*2 - 8K2*2K6*2 - 1820K2*2 + 712K2K3K5 + 32K2K4K6 - 1460K3*2 - 460K4*2 - 116K5*2 - 4K6**2 + 3034
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.274']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71618', 'vk6.71776', 'vk6.72195', 'vk6.72345', 'vk6.73372', 'vk6.73535', 'vk6.75277', 'vk6.75545', 'vk6.77240', 'vk6.77321', 'vk6.77568', 'vk6.77679', 'vk6.78260', 'vk6.78511', 'vk6.80076', 'vk6.80226', 'vk6.81116', 'vk6.81172', 'vk6.81191', 'vk6.81238', 'vk6.81331', 'vk6.81518', 'vk6.82007', 'vk6.82420', 'vk6.82742', 'vk6.85433', 'vk6.86344', 'vk6.86915', 'vk6.87121', 'vk6.88100', 'vk6.88657', 'vk6.88762']
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,-1,0,1,1,3,0,3,1,3,4,2,1,1,1,0,0,2,-1,0,1]

Flat knots (up to 7 crossings) with same phi are :['6.274']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.172', '6.274', '6.286', '6.423', '6.461']

Outer characteristic polynomial of the knot is: t^7+88t^5+146t^3+16t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.274']

2-strand cable arrow polynomial of the knot is: -400*K1**4 - 32*K1**3*K3 - 512*K1**2*K2**4 + 992*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 4352*K1**2*K2**2 - 224*K1**2*K2*K4 + 4760*K1**2*K2 - 16*K1**2*K3**2 - 48*K1**2*K4**2 - 3848*K1**2 + 1472*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 768*K1*K2**2*K3 + 128*K1*K2**2*K4*K5 - 608*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 5056*K1*K2*K3 + 368*K1*K3*K4 + 144*K1*K4*K5 - 736*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 800*K2**4*K4 - 2616*K2**4 + 320*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 1088*K2**2*K3**2 - 344*K2**2*K4**2 + 2152*K2**2*K4 - 160*K2**2*K5**2 - 8*K2**2*K6**2 - 1820*K2**2 + 712*K2*K3*K5 + 32*K2*K4*K6 - 1460*K3**2 - 460*K4**2 - 116*K5**2 - 4*K6**2 + 3034

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.274']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71618', 'vk6.71776', 'vk6.72195', 'vk6.72345', 'vk6.73372', 'vk6.73535', 'vk6.75277', 'vk6.75545', 'vk6.77240', 'vk6.77321', 'vk6.77568', 'vk6.77679', 'vk6.78260', 'vk6.78511', 'vk6.80076', 'vk6.80226', 'vk6.81116', 'vk6.81172', 'vk6.81191', 'vk6.81238', 'vk6.81331', 'vk6.81518', 'vk6.82007', 'vk6.82420', 'vk6.82742', 'vk6.85433', 'vk6.86344', 'vk6.86915', 'vk6.87121', 'vk6.88100', 'vk6.88657', 'vk6.88762']

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	O1O2O3O4O5U1U4O6U5U2U3U6
R3 orbit	{'O1O2O3O4O5U1U4O6U5U2U3U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U6U3U4U1O6U2U5
Gauss code of K*	O1O2O3O4U5U2U3U6U1O5O6U4
Gauss code of -K*	O1O2O3O4U1O5O6U4U5U2U3U6
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 1 0 1 3],[ 4 0 3 4 1 2 3],[ 1 -3 0 1 -1 1 3],[-1 -4 -1 0 -1 1 2],[ 0 -1 1 1 0 1 1],[-1 -2 -1 -1 -1 0 1],[-3 -3 -3 -2 -1 -1 0]]
Primitive based matrix	[[ 0 3 1 1 0 -1 -4],[-3 0 -1 -2 -1 -3 -3],[-1 1 0 -1 -1 -1 -2],[-1 2 1 0 -1 -1 -4],[ 0 1 1 1 0 1 -1],[ 1 3 1 1 -1 0 -3],[ 4 3 2 4 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,0,1,4,1,2,1,3,3,1,1,1,2,1,1,4,-1,1,3]
Phi over symmetry	[-4,-1,0,1,1,3,0,3,1,3,4,2,1,1,1,0,0,2,-1,0,1]
Phi of -K	[-4,-1,0,1,1,3,0,3,1,3,4,2,1,1,1,0,0,2,-1,0,1]
Phi of K*	[-3,-1,-1,0,1,4,0,1,2,1,4,1,0,1,1,0,1,3,2,3,0]
Phi of -K*	[-4,-1,0,1,1,3,3,1,2,4,3,-1,1,1,3,1,1,1,-1,1,2]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	-4w^4z^2+8w^3z^2-8w^3z+25w^2z+19w
Inner characteristic polynomial	t^6+60t^4+44t^2+1
Outer characteristic polynomial	t^7+88t^5+146t^3+16t
Flat arrow polynomial	4K13 + 4K1*2K2 - 6K12 - 2K1K2 - 2K1K3 - 2K1 + 2*K2 + 3
2-strand cable arrow polynomial	-400K14 - 32K1*3K3 - 512K12K2*4 + 992K1*2K2*3 + 64K1*2K2*2K4 - 4352K12K2*2 - 224K1*2K2K4 + 4760K1*2K2 - 16K12K3*2 - 48K1*2K4*2 - 3848K1*2 + 1472K1K23K3 + 96K1K2*2K3K4 - 768K1K22K3 + 128K1K2*2K4K5 - 608K1K22K5 - 224K1K2K3K4 + 5056K1K2K3 + 368K1K3K4 + 144K1K4K5 - 736K2*6 - 256K2*4K3*2 - 32K2*4K4*2 + 800K2*4K4 - 2616K24 + 320K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 1088K22K3*2 - 344K2*2K4*2 + 2152K2*2K4 - 160K22K5*2 - 8K2*2K6*2 - 1820K2*2 + 712K2K3K5 + 32K2K4K6 - 1460K3*2 - 460K4*2 - 116K5*2 - 4K6**2 + 3034
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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {1, 5}, {4}, {3}, {2}], [{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}]]
If K is slice	False

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