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

Flat knot 6.258

Min(phi) over symmetries of the knot is: [-4,-2,1,1,1,3,1,1,2,3,4,1,1,2,3,-1,-1,-1,1,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.258']
Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.258']
Outer characteristic polynomial of the knot is: t^7+84t^5+108t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.258']
2-strand cable arrow polynomial of the knot is: -1296*K1**4 - 192*K1**3*K3 - 768*K1**2*K2**6 + 1408*K1**2*K2**5 - 3392*K1**2*K2**4 + 4416*K1**2*K2**3 - 8640*K1**2*K2**2 - 416*K1**2*K2*K4 + 7144*K1**2*K2 - 144*K1**2*K3**2 - 3676*K1**2 + 640*K1*K2**5*K3 - 640*K1*K2**4*K3 + 3360*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 384*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5776*K1*K2*K3 + 424*K1*K3*K4 + 40*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 1440*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 992*K2**4*K4 - 3240*K2**4 + 32*K2**3*K3*K5 - 96*K2**3*K6 - 944*K2**2*K3**2 - 168*K2**2*K4**2 + 2256*K2**2*K4 - 816*K2**2 + 280*K2*K3*K5 + 56*K2*K4*K6 - 1120*K3**2 - 318*K4**2 - 20*K5**2 + 2804
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.258']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11411', 'vk6.11692', 'vk6.12707', 'vk6.13066', 'vk6.20270', 'vk6.21591', 'vk6.27536', 'vk6.29108', 'vk6.31144', 'vk6.31467', 'vk6.32288', 'vk6.32735', 'vk6.38937', 'vk6.41164', 'vk6.45701', 'vk6.47413', 'vk6.52162', 'vk6.52389', 'vk6.52973', 'vk6.53307', 'vk6.57099', 'vk6.58269', 'vk6.61676', 'vk6.62837', 'vk6.63740', 'vk6.63838', 'vk6.64156', 'vk6.64354', 'vk6.66734', 'vk6.67606', 'vk6.69385', 'vk6.70118']
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 O1O2O3O4O5U1U2O6U5U6U3U4
R3 orbit {'O1O2O3O4O5U1U2O6U5U6U3U4'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U2U3U6U1O6U4U5
Gauss code of K* O1O2O3O4U5U6U3U4U1O5O6U2
Gauss code of -K* O1O2O3O4U3O5O6U4U1U2U5U6
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 1 3 1 1],[ 4 0 1 3 4 2 1],[ 2 -1 0 2 3 1 1],[-1 -3 -2 0 1 -1 1],[-3 -4 -3 -1 0 -1 1],[-1 -2 -1 1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix [[ 0 3 1 1 1 -2 -4],[-3 0 1 -1 -1 -3 -4],[-1 -1 0 -1 -1 -1 -1],[-1 1 1 0 1 -1 -2],[-1 1 1 -1 0 -2 -3],[ 2 3 1 1 2 0 -1],[ 4 4 1 2 3 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-1,-1,-1,2,4,-1,1,1,3,4,1,1,1,1,-1,1,2,2,3,1]
Phi over symmetry [-4,-2,1,1,1,3,1,1,2,3,4,1,1,2,3,-1,-1,-1,1,1,1]
Phi of -K [-4,-2,1,1,1,3,1,2,3,4,3,1,2,2,2,1,-1,1,-1,1,3]
Phi of K* [-3,-1,-1,-1,2,4,1,1,3,2,3,-1,1,1,2,1,2,3,2,4,1]
Phi of -K* [-4,-2,1,1,1,3,1,1,2,3,4,1,1,2,3,-1,-1,-1,1,1,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3+t^2-3t
Normalized Jones-Krushkal polynomial 4z^2+17z+19
Enhanced Jones-Krushkal polynomial -4w^4z^2+8w^3z^2-4w^3z+21w^2z+19w
Inner characteristic polynomial t^6+52t^4+17t^2
Outer characteristic polynomial t^7+84t^5+108t^3+7t
Flat arrow polynomial -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 2*K1**2 - 2*K1*K2 - 2*K1 + 3*K2 + 4
2-strand cable arrow polynomial -1296*K1**4 - 192*K1**3*K3 - 768*K1**2*K2**6 + 1408*K1**2*K2**5 - 3392*K1**2*K2**4 + 4416*K1**2*K2**3 - 8640*K1**2*K2**2 - 416*K1**2*K2*K4 + 7144*K1**2*K2 - 144*K1**2*K3**2 - 3676*K1**2 + 640*K1*K2**5*K3 - 640*K1*K2**4*K3 + 3360*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 1536*K1*K2**2*K3 - 384*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 5776*K1*K2*K3 + 424*K1*K3*K4 + 40*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 1440*K2**6 - 192*K2**4*K3**2 - 32*K2**4*K4**2 + 992*K2**4*K4 - 3240*K2**4 + 32*K2**3*K3*K5 - 96*K2**3*K6 - 944*K2**2*K3**2 - 168*K2**2*K4**2 + 2256*K2**2*K4 - 816*K2**2 + 280*K2*K3*K5 + 56*K2*K4*K6 - 1120*K3**2 - 318*K4**2 - 20*K5**2 + 2804
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {4, 5}, {2, 3}]]
If K is slice False
Contact