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Flat knot 6.1081

Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,-1,0,1,2,2,0,1,1,1,1,1,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1081']
Arrow polynomial of the knot is: 4K13 - 2K1*2 - 4K1*K2 - K1 + K2 + K3 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878']
Outer characteristic polynomial of the knot is: t^7+26t^5+45t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1081']
2-strand cable arrow polynomial of the knot is: 1152K14K2 - 3024K14 - 384K1*3K2*2K3 + 1120K13K2K3 + 32K1*3K3K4 - 992K1*3K3 + 416K12K2*3 + 224K1*2K2*2K4 - 4640K12K2*2 + 256K1*2K2K32 - 256K1*2K2K4 + 8312K1*2K2 - 912K12K3*2 - 32K1*2K3K5 - 64K1*2K4*2 - 5220K1*2 + 576K1K23K3 - 1792K1K2*2K3 - 160K1K2*2K5 - 192K1K2K3K4 + 6832K1K2K3 + 1088K1K3K4 + 88K1K4K5 - 32K2*6 + 64K2*4K4 - 728K24 - 32K2*3K6 - 368K22K3*2 - 16K2*2K4*2 + 1304K2*2K4 - 4478K22 + 168K2K3K5 + 16K2K4K6 - 2192K3*2 - 502K4*2 - 36K5*2 - 2K6**2 + 4404
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1081']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11497', 'vk6.11809', 'vk6.12825', 'vk6.13152', 'vk6.17076', 'vk6.17317', 'vk6.20895', 'vk6.21059', 'vk6.22307', 'vk6.22487', 'vk6.23796', 'vk6.28367', 'vk6.31258', 'vk6.31617', 'vk6.32829', 'vk6.35592', 'vk6.36049', 'vk6.40013', 'vk6.40298', 'vk6.42071', 'vk6.43284', 'vk6.46547', 'vk6.46761', 'vk6.48018', 'vk6.52260', 'vk6.53415', 'vk6.57707', 'vk6.57717', 'vk6.58893', 'vk6.59953', 'vk6.64427', 'vk6.69747']
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,-1,0,1,1,1,-1,0,1,2,2,0,1,1,1,1,1,1,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.1081']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.568', '6.806', '6.1000', '6.1049', '6.1081', '6.1101', '6.1112', '6.1122', '6.1193', '6.1195', '6.1208', '6.1235', '6.1263', '6.1517', '6.1528', '6.1537', '6.1542', '6.1545', '6.1558', '6.1569', '6.1575', '6.1644', '6.1650', '6.1681', '6.1692', '6.1702', '6.1706', '6.1728', '6.1734', '6.1739', '6.1799', '6.1813', '6.1820', '6.1834', '6.1840', '6.1851', '6.1861', '6.1878']

Outer characteristic polynomial of the knot is: t^7+26t^5+45t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1081']

2-strand cable arrow polynomial of the knot is: 1152*K1**4*K2 - 3024*K1**4 - 384*K1**3*K2**2*K3 + 1120*K1**3*K2*K3 + 32*K1**3*K3*K4 - 992*K1**3*K3 + 416*K1**2*K2**3 + 224*K1**2*K2**2*K4 - 4640*K1**2*K2**2 + 256*K1**2*K2*K3**2 - 256*K1**2*K2*K4 + 8312*K1**2*K2 - 912*K1**2*K3**2 - 32*K1**2*K3*K5 - 64*K1**2*K4**2 - 5220*K1**2 + 576*K1*K2**3*K3 - 1792*K1*K2**2*K3 - 160*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 6832*K1*K2*K3 + 1088*K1*K3*K4 + 88*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 728*K2**4 - 32*K2**3*K6 - 368*K2**2*K3**2 - 16*K2**2*K4**2 + 1304*K2**2*K4 - 4478*K2**2 + 168*K2*K3*K5 + 16*K2*K4*K6 - 2192*K3**2 - 502*K4**2 - 36*K5**2 - 2*K6**2 + 4404

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1081']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11497', 'vk6.11809', 'vk6.12825', 'vk6.13152', 'vk6.17076', 'vk6.17317', 'vk6.20895', 'vk6.21059', 'vk6.22307', 'vk6.22487', 'vk6.23796', 'vk6.28367', 'vk6.31258', 'vk6.31617', 'vk6.32829', 'vk6.35592', 'vk6.36049', 'vk6.40013', 'vk6.40298', 'vk6.42071', 'vk6.43284', 'vk6.46547', 'vk6.46761', 'vk6.48018', 'vk6.52260', 'vk6.53415', 'vk6.57707', 'vk6.57717', 'vk6.58893', 'vk6.59953', 'vk6.64427', 'vk6.69747']

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	O1O2O3O4U5U4O6U3O5U2U1U6
R3 orbit	{'O1O2O3O4U5U4O6U3O5U2U1U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U4U3O6U2O5U1U6
Gauss code of K*	O1O2O3U2U1U4U5O6O5U3O4U6
Gauss code of -K*	O1O2O3U4O5U1O6O4U6U5U3U2
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 -1 -1 0 1 -1 2],[ 1 0 0 1 1 -1 2],[ 1 0 0 1 1 -1 1],[ 0 -1 -1 0 0 -1 0],[-1 -1 -1 0 0 -1 -1],[ 1 1 1 1 1 0 2],[-2 -2 -1 0 1 -2 0]]
Primitive based matrix	[[ 0 2 1 0 -1 -1 -1],[-2 0 1 0 -1 -2 -2],[-1 -1 0 0 -1 -1 -1],[ 0 0 0 0 -1 -1 -1],[ 1 1 1 1 0 0 -1],[ 1 2 1 1 0 0 -1],[ 1 2 1 1 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,0,1,1,1,-1,0,1,2,2,0,1,1,1,1,1,1,0,1,1]
Phi over symmetry	[-2,-1,0,1,1,1,-1,0,1,2,2,0,1,1,1,1,1,1,0,1,1]
Phi of -K	[-1,-1,-1,0,1,2,-1,-1,0,1,1,0,0,1,1,0,1,2,1,2,2]
Phi of K*	[-2,-1,0,1,1,1,2,2,1,1,2,1,1,1,1,0,0,0,-1,0,1]
Phi of -K*	[-1,-1,-1,0,1,2,-1,0,1,1,1,1,1,1,2,1,1,2,0,0,-1]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	7z^2+28z+29
Enhanced Jones-Krushkal polynomial	7w^3z^2+28w^2z+29w
Inner characteristic polynomial	t^6+18t^4+8t^2
Outer characteristic polynomial	t^7+26t^5+45t^3+3t
Flat arrow polynomial	4K13 - 2K1*2 - 4K1*K2 - K1 + K2 + K3 + 2
2-strand cable arrow polynomial	1152K14K2 - 3024K14 - 384K1*3K2*2K3 + 1120K13K2K3 + 32K1*3K3K4 - 992K1*3K3 + 416K12K2*3 + 224K1*2K2*2K4 - 4640K12K2*2 + 256K1*2K2K32 - 256K1*2K2K4 + 8312K1*2K2 - 912K12K3*2 - 32K1*2K3K5 - 64K1*2K4*2 - 5220K1*2 + 576K1K23K3 - 1792K1K2*2K3 - 160K1K2*2K5 - 192K1K2K3K4 + 6832K1K2K3 + 1088K1K3K4 + 88K1K4K5 - 32K2*6 + 64K2*4K4 - 728K24 - 32K2*3K6 - 368K22K3*2 - 16K2*2K4*2 + 1304K2*2K4 - 4478K22 + 168K2K3K5 + 16K2K4K6 - 2192K3*2 - 502K4*2 - 36K5*2 - 2K6**2 + 4404
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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}, {3, 5}, {4}, {2}, {1}], [{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}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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