Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo

Flat knot 6.693

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,1,3,3,0,1,1,2,1,0,1,-1,-1,1]
Flat knots (up to 7 crossings) with same phi are :['6.693']
Arrow polynomial of the knot is: -6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']
Outer characteristic polynomial of the knot is: t^7+46t^5+34t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.693']
2-strand cable arrow polynomial of the knot is: -384K14K2*2 + 832K1*4K2 - 1152K14 + 224K1*3K2K3 - 192K1*3K3 + 416K12K2*3 - 2096K1*2K2*2 - 192K1*2K2K4 + 2816K1*2K2 - 96K12K3*2 - 1516K1*2 + 96K1K23K3 - 128K1K2*2K3 - 64K1K2*2K5 + 1640K1K2K3 + 160K1K3K4 - 216K24 - 64K2*2K3*2 - 8K2*2K4*2 + 216K2*2K4 - 1126K22 + 72K2K3K5 + 8K2K4K6 - 356K3*2 - 82K4*2 - 16K5*2 - 2K6**2 + 1208
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.693']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16524', 'vk6.16615', 'vk6.16966', 'vk6.17207', 'vk6.17532', 'vk6.17587', 'vk6.18856', 'vk6.18935', 'vk6.19206', 'vk6.19501', 'vk6.22246', 'vk6.23050', 'vk6.24133', 'vk6.25484', 'vk6.26017', 'vk6.26403', 'vk6.28304', 'vk6.34927', 'vk6.35039', 'vk6.35406', 'vk6.35826', 'vk6.35850', 'vk6.36313', 'vk6.36384', 'vk6.37577', 'vk6.39914', 'vk6.39929', 'vk6.42501', 'vk6.42611', 'vk6.43157', 'vk6.43491', 'vk6.44598', 'vk6.46464', 'vk6.54767', 'vk6.55120', 'vk6.55377', 'vk6.56566', 'vk6.59822', 'vk6.60193', 'vk6.66090']
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: [-3,-1,0,1,1,2,1,1,1,3,3,0,1,1,2,1,0,1,-1,-1,1]

Flat knots (up to 7 crossings) with same phi are :['6.693']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']

Outer characteristic polynomial of the knot is: t^7+46t^5+34t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.693']

2-strand cable arrow polynomial of the knot is: -384*K1**4*K2**2 + 832*K1**4*K2 - 1152*K1**4 + 224*K1**3*K2*K3 - 192*K1**3*K3 + 416*K1**2*K2**3 - 2096*K1**2*K2**2 - 192*K1**2*K2*K4 + 2816*K1**2*K2 - 96*K1**2*K3**2 - 1516*K1**2 + 96*K1*K2**3*K3 - 128*K1*K2**2*K3 - 64*K1*K2**2*K5 + 1640*K1*K2*K3 + 160*K1*K3*K4 - 216*K2**4 - 64*K2**2*K3**2 - 8*K2**2*K4**2 + 216*K2**2*K4 - 1126*K2**2 + 72*K2*K3*K5 + 8*K2*K4*K6 - 356*K3**2 - 82*K4**2 - 16*K5**2 - 2*K6**2 + 1208

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.693']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16524', 'vk6.16615', 'vk6.16966', 'vk6.17207', 'vk6.17532', 'vk6.17587', 'vk6.18856', 'vk6.18935', 'vk6.19206', 'vk6.19501', 'vk6.22246', 'vk6.23050', 'vk6.24133', 'vk6.25484', 'vk6.26017', 'vk6.26403', 'vk6.28304', 'vk6.34927', 'vk6.35039', 'vk6.35406', 'vk6.35826', 'vk6.35850', 'vk6.36313', 'vk6.36384', 'vk6.37577', 'vk6.39914', 'vk6.39929', 'vk6.42501', 'vk6.42611', 'vk6.43157', 'vk6.43491', 'vk6.44598', 'vk6.46464', 'vk6.54767', 'vk6.55120', 'vk6.55377', 'vk6.56566', 'vk6.59822', 'vk6.60193', 'vk6.66090']

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	O1O2O3O4U3O5U6U1O6U4U2U5
R3 orbit	{'O1O2O3U2O4O5U6U1O6U3U4U5', 'O1O2O3O4U3O5U6U1O6U4U2U5'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4U5U3U1O6U4U6O5U2
Gauss code of K*	O1O2O3U4U2U5U1O5U3O6O4U6
Gauss code of -K*	O1O2O3U4O5O4U1O6U3U6U2U5
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 -1 1 3 -1],[ 2 0 1 0 1 2 2],[ 0 -1 0 -1 1 2 0],[ 1 0 1 0 1 1 1],[-1 -1 -1 -1 0 1 -1],[-3 -2 -2 -1 -1 0 -3],[ 1 -2 0 -1 1 3 0]]
Primitive based matrix	[[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -2 -1 -3 -2],[-1 1 0 -1 -1 -1 -1],[ 0 2 1 0 -1 0 -1],[ 1 1 1 1 0 1 0],[ 1 3 1 0 -1 0 -2],[ 2 2 1 1 0 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,0,1,1,2,1,2,1,3,2,1,1,1,1,1,0,1,-1,0,2]
Phi over symmetry	[-3,-1,0,1,1,2,1,1,1,3,3,0,1,1,2,1,0,1,-1,-1,1]
Phi of -K	[-2,-1,-1,0,1,3,-1,1,1,2,3,1,1,1,1,0,1,3,0,1,1]
Phi of K*	[-3,-1,0,1,1,2,1,1,1,3,3,0,1,1,2,1,0,1,-1,-1,1]
Phi of -K*	[-2,-1,-1,0,1,3,0,2,1,1,2,1,1,1,1,0,1,3,1,2,1]
Symmetry type of based matrix	c
u-polynomial	-t^3+t^2+t
Normalized Jones-Krushkal polynomial	2z^2+15z+23
Enhanced Jones-Krushkal polynomial	2w^3z^2+15w^2z+23w
Inner characteristic polynomial	t^6+30t^4+11t^2
Outer characteristic polynomial	t^7+46t^5+34t^3+3t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-384K14K2*2 + 832K1*4K2 - 1152K14 + 224K1*3K2K3 - 192K1*3K3 + 416K12K2*3 - 2096K1*2K2*2 - 192K1*2K2K4 + 2816K1*2K2 - 96K12K3*2 - 1516K1*2 + 96K1K23K3 - 128K1K2*2K3 - 64K1K2*2K5 + 1640K1K2K3 + 160K1K3K4 - 216K24 - 64K2*2K3*2 - 8K2*2K4*2 + 216K2*2K4 - 1126K22 + 72K2K3K5 + 8K2K4K6 - 356K3*2 - 82K4*2 - 16K5*2 - 2K6**2 + 1208
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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{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}, {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}], [{6}, {5}, {3, 4}, {2}, {1}]]
If K is slice	False

Contact