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

Visit L11n24 at Knotilus!

Link Presentations

[edit Notes on L11n24's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X9,21,10,20 X8493 X21,18,22,19 X11,14,12,15 X5,13,6,12 X13,5,14,22 X19,11,20,10 X2,16,3,15
Gauss code {1, -11, 5, -3}, {-8, -1, 2, -5, -4, 10, -7, 8, -9, 7, 11, -2, 3, 6, -10, 4, -6, 9}
A Braid Representative
A Morse Link Presentation L11n24 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial q^{9/2}+\frac{1}{q^{9/2}}-\frac{3}{q^{7/2}}-q^{5/2}+\frac{4}{q^{5/2}}+3 q^{3/2}-\frac{5}{q^{3/2}}-q^{11/2}-5 \sqrt{q}+\frac{4}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a z^5-a^3 z^3+3 a z^3-2 z^3 a^{-1} +z^3 a^{-3} -a^3 z+3 a z-5 z a^{-1} +4 z a^{-3} -z a^{-5} +2 a z^{-1} -4 a^{-1} z^{-1} +3 a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial z^7 a^{-5} -6 z^5 a^{-5} +9 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +z^8 a^{-4} +a^4 z^6-7 z^6 a^{-4} -3 a^4 z^4+13 z^4 a^{-4} +a^4 z^2-9 z^2 a^{-4} + a^{-4} +3 a^3 z^7+z^7 a^{-3} -11 a^3 z^5-10 z^5 a^{-3} +8 a^3 z^3+21 z^3 a^{-3} -3 a^3 z-14 z a^{-3} +3 a^{-3} z^{-1} +3 a^2 z^8+2 z^8 a^{-2} -11 a^2 z^6-15 z^6 a^{-2} +7 a^2 z^4+31 z^4 a^{-2} -a^2 z^2-20 z^2 a^{-2} +a^2+3 a^{-2} +a z^9+z^9 a^{-1} -3 z^7 a^{-1} -15 a z^5-8 z^5 a^{-1} +23 a z^3+27 z^3 a^{-1} -11 a z-18 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +4 z^8-20 z^6+28 z^4-13 z^2+2 (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 \
12           11
10            0
8        111 -1
6       21   1
4      211   -2
2     431    2
0    34      1
-2   232      1
-4  23        1
-6 12         -1
-8 2          2
-101           -1
Integral Khovanov Homology

(db, data source)

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