# L11n175

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n175 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-3 u^2 v^2+3 u^2 v-u^2+2 u v^4-5 u v^3+5 u v^2-5 u v+2 u-v^4+3 v^3-3 v^2}{u v^2}$ (db) Jones polynomial $-\frac{2}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{11}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 \left(-z^3\right)+a^9 z^{-1} +a^7 z^5+a^7 z^3-a^7 z-2 a^7 z^{-1} +a^5 z^5+a^5 z^3+a^5 z+2 a^5 z^{-1} -2 a^3 z^3-3 a^3 z-a^3 z^{-1}$ (db) Kauffman polynomial $a^{12} z^6-2 a^{12} z^4+4 a^{11} z^7-12 a^{11} z^5+7 a^{11} z^3+a^{11} z+5 a^{10} z^8-15 a^{10} z^6+11 a^{10} z^4-3 a^{10} z^2+2 a^9 z^9+2 a^9 z^7-19 a^9 z^5+17 a^9 z^3-5 a^9 z+a^9 z^{-1} +9 a^8 z^8-27 a^8 z^6+27 a^8 z^4-11 a^8 z^2+2 a^7 z^9+a^7 z^7-12 a^7 z^5+17 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +4 a^6 z^8-10 a^6 z^6+16 a^6 z^4-9 a^6 z^2+a^6+3 a^5 z^7-5 a^5 z^5+10 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +a^4 z^6+2 a^4 z^4-a^4 z^2+3 a^3 z^3-4 a^3 z+a^3 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         22
-4        31-2
-6       51 4
-8      53  -2
-10     65   1
-12    55    0
-14   46     -2
-16  36      3
-18 13       -2
-20 3        3
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

### Computer Talk

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