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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n1 at Knotilus!

Link L11n1.
A graph, L11n1.
A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11n1's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X11,20,12,21 X17,5,18,22 X21,19,22,18 X19,12,20,13 X9,16,10,17 X13,2,14,3
Gauss code {1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -6, 9, -11, -2, 3, 10, -7, 8, -9, 6, -8, 7}
A Braid Representative
A Morse Link Presentation L11n1 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^5-4 t(1) t(2)^4+2 t(1) t(2)^3+2 t(2)^2-4 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +a^9 z^3+5 a^9 z+3 a^9 z^{-1} -2 a^7 z^5-9 a^7 z^3-9 a^7 z-3 a^7 z^{-1} +a^5 z^7+5 a^5 z^5+7 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-3 a^3 z-a^3 z^{-1} (db)
Kauffman polynomial -z^2 a^{12}+a^{12}-2 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -z^6 a^{10}+3 z^4 a^{10}-5 z^2 a^{10}+2 a^{10}-2 z^7 a^9+9 z^5 a^9-16 z^3 a^9+12 z a^9-3 a^9 z^{-1} -2 z^8 a^8+8 z^6 a^8-9 z^4 a^8+4 z^2 a^8-z^9 a^7+z^7 a^7+12 z^5 a^7-25 z^3 a^7+17 z a^7-3 a^7 z^{-1} -4 z^8 a^6+19 z^6 a^6-25 z^4 a^6+12 z^2 a^6-2 a^6-z^9 a^5+2 z^7 a^5+8 z^5 a^5-18 z^3 a^5+11 z a^5-2 a^5 z^{-1} -2 z^8 a^4+10 z^6 a^4-13 z^4 a^4+4 z^2 a^4-z^7 a^3+5 z^5 a^3-7 z^3 a^3+4 z a^3-a^3 z^{-1} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
j \
0         11
-2        1 -1
-4       21 1
-6     132  0
-8     31   2
-10   133    1
-12  143     0
-14  12      1
-16 23       -1
-18 1        1
-201         -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4 i=-2
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4} {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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