Table of flat knot invariants
Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo
Glossary Reference List

Flat knot 6.727

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,-1,1,1,3,4,0,0,0,1,0,1,2,0,-1,-1]
Flat knots (up to 7 crossings) with same phi are :['6.727']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']
Outer characteristic polynomial of the knot is: t^7+76t^5+79t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.727']
2-strand cable arrow polynomial of the knot is: -192*K1**6 + 256*K1**4*K2**3 - 896*K1**4*K2**2 + 1568*K1**4*K2 - 2720*K1**4 + 256*K1**3*K2*K3 - 448*K1**2*K2**4 + 1088*K1**2*K2**3 - 3504*K1**2*K2**2 + 4632*K1**2*K2 - 128*K1**2*K3**2 - 32*K1**2*K4**2 - 1836*K1**2 + 256*K1*K2**3*K3 + 2040*K1*K2*K3 + 192*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 608*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 192*K2**2*K4 - 1384*K2**2 + 16*K2*K3*K5 - 492*K3**2 - 128*K4**2 - 8*K5**2 + 1934
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.727']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4220', 'vk6.4301', 'vk6.5485', 'vk6.5598', 'vk6.7581', 'vk6.7675', 'vk6.9085', 'vk6.9166', 'vk6.11192', 'vk6.12274', 'vk6.12383', 'vk6.19367', 'vk6.19660', 'vk6.19768', 'vk6.26147', 'vk6.26203', 'vk6.26563', 'vk6.26646', 'vk6.30778', 'vk6.31979', 'vk6.38151', 'vk6.38191', 'vk6.44808', 'vk6.44932', 'vk6.48530', 'vk6.49227', 'vk6.49340', 'vk6.50314', 'vk6.52756', 'vk6.63598', 'vk6.66319', 'vk6.66339']
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 O1O2O3O4U1O5U3U6U5O6U4U2
R3 orbit {'O1O2O3O4U1O5U3U6U5O6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U1O5U6U5U2O6U4
Gauss code of K* O1O2O3U2O4O5U6U5U1U4O6U3
Gauss code of -K* O1O2O3U1O4U5U3U6U4O6O5U2
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 1 -1 2 2 -1],[ 3 0 3 1 2 1 3],[-1 -3 0 -2 1 2 -2],[ 1 -1 2 0 2 1 0],[-2 -2 -1 -2 0 1 -3],[-2 -1 -2 -1 -1 0 -2],[ 1 -3 2 0 3 2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 1 -1 -2 -3 -2],[-2 -1 0 -2 -1 -2 -1],[-1 1 2 0 -2 -2 -3],[ 1 2 1 2 0 0 -1],[ 1 3 2 2 0 0 -3],[ 3 2 1 3 1 3 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,-1,1,2,3,2,2,1,2,1,2,2,3,0,1,3]
Phi over symmetry [-3,-1,-1,1,2,2,-1,1,1,3,4,0,0,0,1,0,1,2,0,-1,-1]
Phi of -K [-3,-1,-1,1,2,2,-1,1,1,3,4,0,0,0,1,0,1,2,0,-1,-1]
Phi of K* [-2,-2,-1,1,1,3,-1,-1,1,2,4,0,0,1,3,0,0,1,0,-1,1]
Phi of -K* [-3,-1,-1,1,2,2,1,3,3,1,2,0,2,1,2,2,2,3,2,1,-1]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 13z+27
Enhanced Jones-Krushkal polynomial -4w^3z+17w^2z+27w
Inner characteristic polynomial t^6+56t^4+55t^2
Outer characteristic polynomial t^7+76t^5+79t^3
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial -192*K1**6 + 256*K1**4*K2**3 - 896*K1**4*K2**2 + 1568*K1**4*K2 - 2720*K1**4 + 256*K1**3*K2*K3 - 448*K1**2*K2**4 + 1088*K1**2*K2**3 - 3504*K1**2*K2**2 + 4632*K1**2*K2 - 128*K1**2*K3**2 - 32*K1**2*K4**2 - 1836*K1**2 + 256*K1*K2**3*K3 + 2040*K1*K2*K3 + 192*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 608*K2**4 - 80*K2**2*K3**2 - 8*K2**2*K4**2 + 192*K2**2*K4 - 1384*K2**2 + 16*K2*K3*K5 - 492*K3**2 - 128*K4**2 - 8*K5**2 + 1934
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {5}, {4}, {1, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {3, 4}, {1, 2}]]
If K is slice False
Contact