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

Visit L11n352 at Knotilus!

Link Presentations

[edit Notes on L11n352's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X11,20,12,21 X7,18,8,19 X9,13,10,22 X21,17,22,16 X17,8,18,9 X15,11,16,10 X19,12,20,5 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, -4, 7, -5, 8, -3, 9}, {11, -2, -8, 6, -7, 4, -9, 3, -6, 5}
A Braid Representative
A Morse Link Presentation L11n352 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^2 w^3-2 u v^2 w^2+2 u v^2 w-u v w^3+2 u v w^2-u v w+u w^2-u w+v^3 w^2-v^3 w+v^2 w^2-2 v^2 w+v^2-2 v w^2+2 v w-v}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial  q^{-8} - q^{-7} + q^{-6} +2 q^{-5} -3 q^{-4} +6 q^{-3} -q^2-5 q^{-2} +3 q+6 q^{-1} -5 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^8 z^{-2} +a^8-2 a^6 z^2-2 a^6 z^{-2} -5 a^6+2 a^4 z^2+a^4 z^{-2} +3 a^4+a^2 z^6+4 a^2 z^4+5 a^2 z^2+2 a^2-z^4-2 z^2-1 (db)
Kauffman polynomial a^7 z^9+a^5 z^9+a^8 z^8+2 a^6 z^8+2 a^4 z^8+a^2 z^8-7 a^7 z^7-7 a^5 z^7+3 a^3 z^7+3 a z^7-7 a^8 z^6-18 a^6 z^6-12 a^4 z^6+2 a^2 z^6+3 z^6+12 a^7 z^5+11 a^5 z^5-7 a^3 z^5-5 a z^5+z^5 a^{-1} +15 a^8 z^4+45 a^6 z^4+28 a^4 z^4-9 a^2 z^4-7 z^4-3 a^7 z^3+5 a^5 z^3+7 a^3 z^3-3 a z^3-2 z^3 a^{-1} -14 a^8 z^2-38 a^6 z^2-23 a^4 z^2+4 a^2 z^2+3 z^2-5 a^7 z-8 a^5 z-3 a^3 z+a z+z a^{-1} +6 a^8+13 a^6+9 a^4-1+2 a^7 z^{-1} +2 a^5 z^{-1} -a^8 z^{-2} -2 a^6 z^{-2} -a^4 z^{-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 \
5           1-1
3          2 2
1         31 -2
-1        32  1
-3      144   1
-5      32    1
-7    124     3
-9   133      -1
-11   14       3
-13 11         0
-15            0
-171           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1 i=1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4} {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\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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