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

Flat knot 6.102

Min(phi) over symmetries of the knot is: [-4,-2,0,1,2,3,0,2,4,2,4,1,2,1,2,1,1,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.102']
Arrow polynomial of the knot is: -4*K1**2 - 2*K1*K2 + K1 - 2*K2**2 + 2*K2 + K3 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.102']
Outer characteristic polynomial of the knot is: t^7+91t^5+36t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.102']
2-strand cable arrow polynomial of the knot is: -112*K1**4 + 32*K1**2*K2**3 - 304*K1**2*K2**2 + 776*K1**2*K2 - 64*K1**2*K3**2 - 32*K1**2*K4**2 - 820*K1**2 + 448*K1*K2*K3 + 296*K1*K3*K4 + 72*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**4 - 8*K2**2*K4**2 + 64*K2**2*K4 - 490*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 16*K3**4 - 32*K3**2*K4**2 + 16*K3**2*K6 - 284*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 172*K4**2 - 64*K5**2 - 14*K6**2 - 2*K8**2 + 644
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.102']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.16506', 'vk6.16597', 'vk6.18107', 'vk6.18443', 'vk6.22933', 'vk6.23028', 'vk6.23491', 'vk6.23828', 'vk6.24554', 'vk6.24971', 'vk6.35013', 'vk6.35636', 'vk6.36689', 'vk6.37111', 'vk6.39466', 'vk6.41665', 'vk6.42471', 'vk6.42582', 'vk6.43965', 'vk6.44280', 'vk6.46050', 'vk6.47716', 'vk6.54733', 'vk6.54828', 'vk6.56207', 'vk6.57464', 'vk6.59193', 'vk6.59256', 'vk6.59641', 'vk6.59987', 'vk6.60802', 'vk6.62135', 'vk6.64802', 'vk6.65037', 'vk6.65557', 'vk6.65867', 'vk6.68037', 'vk6.68100', 'vk6.68635', 'vk6.68848', 'vk6.73732', 'vk6.73851', 'vk6.78299', 'vk6.78503', 'vk6.78648', 'vk6.78843', 'vk6.85148', 'vk6.89430']
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 O1O2O3O4O5O6U2U6U4U1U5U3
R3 orbit {'O1O2O3O4O5O6U2U6U4U1U5U3', 'O1O2O3O4O5U1U5U6U2U4O6U3', 'O1O2O3O4O5U1U6U4U2O6U5U3'}
R3 orbit length 3
Gauss code of -K O1O2O3O4O5O6U4U2U6U3U1U5
Gauss code of K* O1O2O3O4O5O6U4U1U6U3U5U2
Gauss code of -K* O1O2O3O4O5O6U5U2U4U1U6U3
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 -2 -4 2 0 3 1],[ 2 0 -2 3 1 3 1],[ 4 2 0 4 2 3 1],[-2 -3 -4 0 -1 1 0],[ 0 -1 -2 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 -1 -2],[ 2 3 3 1 1 0 -2],[ 4 3 4 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,2,4,1,0,1,3,3,0,1,3,4,0,1,1,1,2,2]
Phi over symmetry [-4,-2,0,1,2,3,0,2,4,2,4,1,2,1,2,1,1,2,1,2,0]
Phi of -K [-4,-2,0,1,2,3,0,2,4,2,4,1,2,1,2,1,1,2,1,2,0]
Phi of K* [-3,-2,-1,0,2,4,0,2,2,2,4,1,1,1,2,1,2,4,1,2,0]
Phi of -K* [-4,-2,0,1,2,3,2,2,1,4,3,1,1,3,3,0,1,1,0,0,1]
Symmetry type of based matrix c
u-polynomial t^4-t^3-t
Normalized Jones-Krushkal polynomial 7z+15
Enhanced Jones-Krushkal polynomial 7w^2z+15w
Inner characteristic polynomial t^6+57t^4+12t^2
Outer characteristic polynomial t^7+91t^5+36t^3
Flat arrow polynomial -4*K1**2 - 2*K1*K2 + K1 - 2*K2**2 + 2*K2 + K3 + K4 + 4
2-strand cable arrow polynomial -112*K1**4 + 32*K1**2*K2**3 - 304*K1**2*K2**2 + 776*K1**2*K2 - 64*K1**2*K3**2 - 32*K1**2*K4**2 - 820*K1**2 + 448*K1*K2*K3 + 296*K1*K3*K4 + 72*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**4 - 8*K2**2*K4**2 + 64*K2**2*K4 - 490*K2**2 + 56*K2*K3*K5 + 8*K2*K4*K6 - 16*K3**4 - 32*K3**2*K4**2 + 16*K3**2*K6 - 284*K3**2 + 16*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 172*K4**2 - 64*K5**2 - 14*K6**2 - 2*K8**2 + 644
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
Contact