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Flat knot 6.1712

Min(phi) over symmetries of the knot is: [-2,-1,1,1,1,-1,1,1,2,0,1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['5.86', '6.1712', '7.37892']
Arrow polynomial of the knot is: 4K13 + 2K1*2 - 4K1*K2 - K1 - K2 + K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.140', '6.569', '6.943', '6.970', '6.1234', '6.1298', '6.1311', '6.1326', '6.1500', '6.1506', '6.1708', '6.1712', '6.1720', '6.1859']
Outer characteristic polynomial of the knot is: t^6+30t^4+26t^2+1
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1712']
2-strand cable arrow polynomial of the knot is: 4608K14K2 - 7552K14 + 1536K1*3K2K3 - 1664K1*3K3 - 384K12K2*4 + 768K1*2K2*3 + 384K1*2K2*2K4 - 7728K12K2*2 - 928K1*2K2K4 + 6840K1*2K2 - 1152K12K3*2 - 96K1*2K4*2 + 904K1*2 + 416K1K23K3 - 480K1K2*2K3 - 160K1K2*2K5 - 224K1K2K3K4 + 4608K1K2K3 + 816K1K3K4 + 96K1K4K5 - 32K2*6 + 64K2*4K4 - 552K24 - 144K2*2K3*2 - 48K2*2K4*2 + 464K2*2K4 - 1102K22 + 120K2K3K5 + 16K2K4K6 - 556K3*2 - 158K4*2 - 28K5*2 - 2K6**2 + 1348
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1712']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3255', 'vk6.3283', 'vk6.3295', 'vk6.3387', 'vk6.3416', 'vk6.3428', 'vk6.3472', 'vk6.3516', 'vk6.4626', 'vk6.5911', 'vk6.6032', 'vk6.7960', 'vk6.8081', 'vk6.9390', 'vk6.17840', 'vk6.17857', 'vk6.19055', 'vk6.19867', 'vk6.24357', 'vk6.25667', 'vk6.25684', 'vk6.26306', 'vk6.26751', 'vk6.37775', 'vk6.43782', 'vk6.43799', 'vk6.45051', 'vk6.48115', 'vk6.48126', 'vk6.48149', 'vk6.48206', 'vk6.50670']
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: [-2,-1,1,1,1,-1,1,1,2,0,1,0,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['5.86', '6.1712', '7.37892']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.140', '6.569', '6.943', '6.970', '6.1234', '6.1298', '6.1311', '6.1326', '6.1500', '6.1506', '6.1708', '6.1712', '6.1720', '6.1859']

Outer characteristic polynomial of the knot is: t^6+30t^4+26t^2+1

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1712']

2-strand cable arrow polynomial of the knot is: 4608*K1**4*K2 - 7552*K1**4 + 1536*K1**3*K2*K3 - 1664*K1**3*K3 - 384*K1**2*K2**4 + 768*K1**2*K2**3 + 384*K1**2*K2**2*K4 - 7728*K1**2*K2**2 - 928*K1**2*K2*K4 + 6840*K1**2*K2 - 1152*K1**2*K3**2 - 96*K1**2*K4**2 + 904*K1**2 + 416*K1*K2**3*K3 - 480*K1*K2**2*K3 - 160*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 4608*K1*K2*K3 + 816*K1*K3*K4 + 96*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 552*K2**4 - 144*K2**2*K3**2 - 48*K2**2*K4**2 + 464*K2**2*K4 - 1102*K2**2 + 120*K2*K3*K5 + 16*K2*K4*K6 - 556*K3**2 - 158*K4**2 - 28*K5**2 - 2*K6**2 + 1348

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1712']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3255', 'vk6.3283', 'vk6.3295', 'vk6.3387', 'vk6.3416', 'vk6.3428', 'vk6.3472', 'vk6.3516', 'vk6.4626', 'vk6.5911', 'vk6.6032', 'vk6.7960', 'vk6.8081', 'vk6.9390', 'vk6.17840', 'vk6.17857', 'vk6.19055', 'vk6.19867', 'vk6.24357', 'vk6.25667', 'vk6.25684', 'vk6.26306', 'vk6.26751', 'vk6.37775', 'vk6.43782', 'vk6.43799', 'vk6.45051', 'vk6.48115', 'vk6.48126', 'vk6.48149', 'vk6.48206', 'vk6.50670']

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	O1O2O3U1U4O5O4U6U5O6U3U2
R3 orbit	{'O1O2O3U1U4O5O4U6U5O6U3U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3U2U1O4U5U4O6O5U6U3
Gauss code of K*	O1O2U1O3O4U5U4U3O5O6U2U6
Gauss code of -K*	O1O2U3O4O3U5U4O5O6U2U1U6
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 -2 1 1 1 0 -1],[ 2 0 2 1 2 0 2],[-1 -2 0 0 0 0 -2],[-1 -1 0 0 0 0 -2],[-1 -2 0 0 0 0 -1],[ 0 0 0 0 0 0 0],[ 1 -2 2 2 1 0 0]]
Primitive based matrix	[[ 0 1 1 1 -1 -2],[-1 0 0 0 -1 -2],[-1 0 0 0 -2 -1],[-1 0 0 0 -2 -2],[ 1 1 2 2 0 -2],[ 2 2 1 2 2 0]]
If based matrix primitive	False
Phi of primitive based matrix	[-1,-1,-1,1,2,0,0,1,2,0,2,1,2,2,2]
Phi over symmetry	[-2,-1,1,1,1,-1,1,1,2,0,1,0,0,0,0]
Phi of -K	[-2,-1,1,1,1,-1,1,1,2,0,1,0,0,0,0]
Phi of K*	[-1,-1,-1,1,2,0,0,0,1,0,0,2,1,1,-1]
Phi of -K*	[-2,-1,1,1,1,2,1,2,2,2,1,2,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	9z^2+30z+25
Enhanced Jones-Krushkal polynomial	9w^3z^2+30w^2z+25w
Inner characteristic polynomial	t^5+22t^3+17t
Outer characteristic polynomial	t^6+30t^4+26t^2+1
Flat arrow polynomial	4K13 + 2K1*2 - 4K1*K2 - K1 - K2 + K3
2-strand cable arrow polynomial	4608K14K2 - 7552K14 + 1536K1*3K2K3 - 1664K1*3K3 - 384K12K2*4 + 768K1*2K2*3 + 384K1*2K2*2K4 - 7728K12K2*2 - 928K1*2K2K4 + 6840K1*2K2 - 1152K12K3*2 - 96K1*2K4*2 + 904K1*2 + 416K1K23K3 - 480K1K2*2K3 - 160K1K2*2K5 - 224K1K2K3K4 + 4608K1K2K3 + 816K1K3K4 + 96K1K4K5 - 32K2*6 + 64K2*4K4 - 552K24 - 144K2*2K3*2 - 48K2*2K4*2 + 464K2*2K4 - 1102K22 + 120K2K3K5 + 16K2K4K6 - 556K3*2 - 158K4*2 - 28K5*2 - 2K6**2 + 1348
Genus of based matrix	1
Fillings of based matrix	[[{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {4, 5}, {1, 2}], [{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}, {2, 5}, {1, 3}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {3, 4}, {1, 2}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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