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Flat knot 6.1681

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,0,2,2,2,1,1,3,2,0,1,1,-1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.1681']
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+47t^5+108t^3+15t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1681']
2-strand cable arrow polynomial of the knot is: -512K14K2*2 + 352K1*4K2 - 496K14 + 128K1*3K2*3K3 + 800K13K2K3 - 256K1*3K3 - 576K12K2*4 - 384K1*2K2*3K4 + 1152K12K2*3 - 128K1*2K2*2K3*2 - 5616K1*2K2*2 - 800K1*2K2K4 + 4120K1*2K2 - 912K12K3*2 - 3028K1*2 + 1536K1K23K3 + 736K1K2*2K3K4 - 896K1K22K3 - 96K1K2*2K5 + 32K1K2K33 - 96K1K2K3K4 - 160K1K2K3K6 + 6264K1K2K3 + 1280K1K3K4 + 8K1K4K5 + 24K1K5K6 - 32K2*6 + 256K2*4K4 - 1288K24 - 928K2*2K3*2 - 432K2*2K4*2 + 1152K2*2K4 - 1806K22 + 248K2K3K5 + 112K2K4K6 - 1776K3*2 - 538K4*2 - 20K5*2 - 18K6**2 + 2592
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1681']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16908', 'vk6.17152', 'vk6.20517', 'vk6.21903', 'vk6.23300', 'vk6.23601', 'vk6.27958', 'vk6.29435', 'vk6.35314', 'vk6.35752', 'vk6.39370', 'vk6.41552', 'vk6.42819', 'vk6.43103', 'vk6.45937', 'vk6.47622', 'vk6.55055', 'vk6.55302', 'vk6.57386', 'vk6.58554', 'vk6.59451', 'vk6.59742', 'vk6.62041', 'vk6.63037', 'vk6.64896', 'vk6.65111', 'vk6.66935', 'vk6.67789', 'vk6.68205', 'vk6.68351', 'vk6.69540', 'vk6.70243']
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,-2,0,1,1,2,-1,0,2,2,2,1,1,3,2,0,1,1,-1,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.1681']

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+47t^5+108t^3+15t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1681']

2-strand cable arrow polynomial of the knot is: -512*K1**4*K2**2 + 352*K1**4*K2 - 496*K1**4 + 128*K1**3*K2**3*K3 + 800*K1**3*K2*K3 - 256*K1**3*K3 - 576*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 1152*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 - 5616*K1**2*K2**2 - 800*K1**2*K2*K4 + 4120*K1**2*K2 - 912*K1**2*K3**2 - 3028*K1**2 + 1536*K1*K2**3*K3 + 736*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 96*K1*K2**2*K5 + 32*K1*K2*K3**3 - 96*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 6264*K1*K2*K3 + 1280*K1*K3*K4 + 8*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 256*K2**4*K4 - 1288*K2**4 - 928*K2**2*K3**2 - 432*K2**2*K4**2 + 1152*K2**2*K4 - 1806*K2**2 + 248*K2*K3*K5 + 112*K2*K4*K6 - 1776*K3**2 - 538*K4**2 - 20*K5**2 - 18*K6**2 + 2592

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1681']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16908', 'vk6.17152', 'vk6.20517', 'vk6.21903', 'vk6.23300', 'vk6.23601', 'vk6.27958', 'vk6.29435', 'vk6.35314', 'vk6.35752', 'vk6.39370', 'vk6.41552', 'vk6.42819', 'vk6.43103', 'vk6.45937', 'vk6.47622', 'vk6.55055', 'vk6.55302', 'vk6.57386', 'vk6.58554', 'vk6.59451', 'vk6.59742', 'vk6.62041', 'vk6.63037', 'vk6.64896', 'vk6.65111', 'vk6.66935', 'vk6.67789', 'vk6.68205', 'vk6.68351', 'vk6.69540', 'vk6.70243']

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	O1O2O3U4O5U1U2U6O4O6U5U3
R3 orbit	{'O1O2O3U4O5U1U2U6O4O6U5U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3U1U4O5O6U5U2U3O4U6
Gauss code of K*	O1O2O3U4U3O5O6U1U2U6O4U5
Gauss code of -K*	O1O2O3U4O5U6U2U3O6O4U1U5
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 0 2 -2 1 1],[ 2 0 1 2 1 1 3],[ 0 -1 0 1 0 0 1],[-2 -2 -1 0 -2 -1 -1],[ 2 -1 0 2 0 2 2],[-1 -1 0 1 -2 0 -1],[-1 -3 -1 1 -2 1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -2 -2],[-2 0 -1 -1 -1 -2 -2],[-1 1 0 1 -1 -2 -3],[-1 1 -1 0 0 -2 -1],[ 0 1 1 0 0 0 -1],[ 2 2 2 2 0 0 -1],[ 2 2 3 1 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,2,2,1,1,1,2,2,-1,1,2,3,0,2,1,0,1,1]
Phi over symmetry	[-2,-2,0,1,1,2,-1,0,2,2,2,1,1,3,2,0,1,1,-1,1,1]
Phi of -K	[-2,-2,0,1,1,2,-1,1,0,2,2,2,1,1,2,0,1,1,-1,0,0]
Phi of K*	[-2,-1,-1,0,2,2,0,0,1,2,2,-1,1,1,2,0,1,0,2,1,-1]
Phi of -K*	[-2,-2,0,1,1,2,-1,0,2,2,2,1,1,3,2,0,1,1,-1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^2-2t
Normalized Jones-Krushkal polynomial	4z^2+17z+19
Enhanced Jones-Krushkal polynomial	4w^3z^2-8w^3z+25w^2z+19w
Inner characteristic polynomial	t^6+33t^4+61t^2+4
Outer characteristic polynomial	t^7+47t^5+108t^3+15t
Flat arrow polynomial	4K13 - 2K1*2 - 4K1*K2 - K1 + K2 + K3 + 2
2-strand cable arrow polynomial	-512K14K2*2 + 352K1*4K2 - 496K14 + 128K1*3K2*3K3 + 800K13K2K3 - 256K1*3K3 - 576K12K2*4 - 384K1*2K2*3K4 + 1152K12K2*3 - 128K1*2K2*2K3*2 - 5616K1*2K2*2 - 800K1*2K2K4 + 4120K1*2K2 - 912K12K3*2 - 3028K1*2 + 1536K1K23K3 + 736K1K2*2K3K4 - 896K1K22K3 - 96K1K2*2K5 + 32K1K2K33 - 96K1K2K3K4 - 160K1K2K3K6 + 6264K1K2K3 + 1280K1K3K4 + 8K1K4K5 + 24K1K5K6 - 32K2*6 + 256K2*4K4 - 1288K24 - 928K2*2K3*2 - 432K2*2K4*2 + 1152K2*2K4 - 1806K22 + 248K2K3K5 + 112K2K4K6 - 1776K3*2 - 538K4*2 - 20K5*2 - 18K6**2 + 2592
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}], [{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}, {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}, {3, 4}, {1, 2}]]
If K is slice	False

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