# L11n411

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n411 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(w-1) (w+1) \left(2 u v w^2-2 u v w+u v+w^3-2 w^2+2 w\right)}{\sqrt{u} \sqrt{v} w^{5/2}}$ (db) Jones polynomial $- q^{-13} +2 q^{-12} -2 q^{-11} + q^{-10} + q^{-8} +2 q^{-7} - q^{-6} +2 q^{-5} - q^{-4} + q^{-3}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-a^{14} z^{-2} +4 a^{12} z^{-2} +5 a^{12}-z^4 a^{10}-8 z^2 a^{10}-5 a^{10} z^{-2} -12 a^{10}+z^6 a^8+5 z^4 a^8+6 z^2 a^8+2 a^8 z^{-2} +5 a^8+z^6 a^6+5 z^4 a^6+6 z^2 a^6+2 a^6$ (db) Kauffman polynomial $a^{15} z^7-5 a^{15} z^5+6 a^{15} z^3-4 a^{15} z+a^{15} z^{-1} +2 a^{14} z^8-11 a^{14} z^6+15 a^{14} z^4-8 a^{14} z^2-a^{14} z^{-2} +3 a^{14}+a^{13} z^9-3 a^{13} z^7-10 a^{13} z^5+30 a^{13} z^3-21 a^{13} z+5 a^{13} z^{-1} +4 a^{12} z^8-26 a^{12} z^6+47 a^{12} z^4-35 a^{12} z^2-4 a^{12} z^{-2} +16 a^{12}+a^{11} z^9-2 a^{11} z^7-20 a^{11} z^5+54 a^{11} z^3-39 a^{11} z+9 a^{11} z^{-1} +3 a^{10} z^8-21 a^{10} z^6+44 a^{10} z^4-43 a^{10} z^2-5 a^{10} z^{-2} +21 a^{10}+3 a^9 z^7-19 a^9 z^5+32 a^9 z^3-22 a^9 z+5 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+7 a^8 z^4-10 a^8 z^2-2 a^8 z^{-2} +7 a^8+a^7 z^7-4 a^7 z^5+2 a^7 z^3+a^6 z^6-5 a^6 z^4+6 a^6 z^2-2 a^6$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-5           11
-7          110
-9        12  1
-11       111  1
-13      252   1
-15     223    3
-17    252     1
-19   221      1
-21  133       -1
-23 11         0
-25 1          1
-271           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-7$ $i=-5$ $i=-3$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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