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

Flat knot 6.271

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,3,4,2,4,2,2,1,2,1,1,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.271']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K1*K3 + 4*K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.271', '6.378']
Outer characteristic polynomial of the knot is: t^7+87t^5+43t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.271']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 576*K1**4*K2 - 1952*K1**4 + 192*K1**3*K2*K3 - 384*K1**3*K3 - 256*K1**2*K2**4 + 672*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 3712*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 6384*K1**2*K2 - 480*K1**2*K3**2 - 64*K1**2*K3*K5 - 4512*K1**2 + 480*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 992*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 320*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5584*K1*K2*K3 + 1320*K1*K3*K4 + 224*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 936*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**3*K6 - 496*K2**2*K3**2 - 32*K2**2*K3*K7 - 216*K2**2*K4**2 + 1648*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 3744*K2**2 - 32*K2*K3**2*K4 + 616*K2*K3*K5 + 176*K2*K4*K6 + 16*K2*K5*K7 + 8*K3**2*K6 - 1964*K3**2 - 832*K4**2 - 172*K5**2 - 32*K6**2 + 3910
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.271']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16519', 'vk6.16612', 'vk6.18079', 'vk6.18415', 'vk6.22950', 'vk6.23047', 'vk6.24526', 'vk6.24943', 'vk6.34917', 'vk6.35028', 'vk6.36669', 'vk6.37091', 'vk6.42488', 'vk6.42601', 'vk6.43945', 'vk6.44260', 'vk6.54746', 'vk6.54843', 'vk6.55907', 'vk6.56195', 'vk6.59210', 'vk6.59275', 'vk6.60434', 'vk6.60790', 'vk6.64756', 'vk6.64817', 'vk6.65553', 'vk6.65863', 'vk6.68054', 'vk6.68119', 'vk6.68631', 'vk6.68844']
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 O1O2O3O4O5U1U4O6U2U6U5U3
R3 orbit {'O1O2O3O4O5U1U4O6U2U6U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U3U1U6U4O6U2U5
Gauss code of K* O1O2O3O4U5U1U4U6U3O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U2U5U1U4U6
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 -4 -2 2 0 3 1],[ 4 0 2 4 1 3 1],[ 2 -2 0 3 0 3 1],[-2 -4 -3 0 -1 1 0],[ 0 -1 0 1 0 1 0],[-3 -3 -3 -1 -1 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix [[ 0 3 2 1 0 -2 -4],[-3 0 -1 0 -1 -3 -3],[-2 1 0 0 -1 -3 -4],[-1 0 0 0 0 -1 -1],[ 0 1 1 0 0 0 -1],[ 2 3 3 1 0 0 -2],[ 4 3 4 1 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,2,4,1,0,1,3,3,0,1,3,4,0,1,1,0,1,2]
Phi over symmetry [-4,-2,0,1,2,3,0,3,4,2,4,2,2,1,2,1,1,2,1,2,0]
Phi of -K [-4,-2,0,1,2,3,0,3,4,2,4,2,2,1,2,1,1,2,1,2,0]
Phi of K* [-3,-2,-1,0,2,4,0,2,2,2,4,1,1,1,2,1,2,4,2,3,0]
Phi of -K* [-4,-2,0,1,2,3,2,1,1,4,3,0,1,3,3,0,1,1,0,0,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 2z^2+19z+31
Enhanced Jones-Krushkal polynomial 2w^3z^2+19w^2z+31w
Inner characteristic polynomial t^6+53t^4+11t^2
Outer characteristic polynomial t^7+87t^5+43t^3+4t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 10*K1**2 - 6*K1*K2 - 2*K1*K3 + 4*K2 + 2*K3 + 5
2-strand cable arrow polynomial -192*K1**4*K2**2 + 576*K1**4*K2 - 1952*K1**4 + 192*K1**3*K2*K3 - 384*K1**3*K3 - 256*K1**2*K2**4 + 672*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 3712*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 352*K1**2*K2*K4 + 6384*K1**2*K2 - 480*K1**2*K3**2 - 64*K1**2*K3*K5 - 4512*K1**2 + 480*K1*K2**3*K3 + 128*K1*K2**2*K3*K4 - 992*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 320*K1*K2**2*K5 - 352*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5584*K1*K2*K3 + 1320*K1*K3*K4 + 224*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 192*K2**4*K4 - 936*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 128*K2**3*K6 - 496*K2**2*K3**2 - 32*K2**2*K3*K7 - 216*K2**2*K4**2 + 1648*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 3744*K2**2 - 32*K2*K3**2*K4 + 616*K2*K3*K5 + 176*K2*K4*K6 + 16*K2*K5*K7 + 8*K3**2*K6 - 1964*K3**2 - 832*K4**2 - 172*K5**2 - 32*K6**2 + 3910
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {3, 5}, {2, 4}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {2, 5}, {1, 4}]]
If K is slice False
Contact