# L11n453

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n453 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u v w^2 x^2-u v w^2 x-u v w x^2-u w^2 x^2+u w^2 x+u w x^2-u w x-v w x+v w+v x-v-w-x+2}{\sqrt{u} \sqrt{v} w x}$ (db) Jones polynomial $\frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{25/2}}-\frac{3}{q^{23/2}}+\frac{3}{q^{21/2}}-\frac{6}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{11/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $-z a^{13}+a^{13} z^{-3} +z^5 a^{11}+4 z^3 a^{11}+z a^{11}-5 a^{11} z^{-1} -3 a^{11} z^{-3} -z^7 a^9-4 z^5 a^9+11 z a^9+10 a^9 z^{-1} +3 a^9 z^{-3} -z^7 a^7-6 z^5 a^7-12 z^3 a^7-11 z a^7-5 a^7 z^{-1} -a^7 z^{-3}$ (db) Kauffman polynomial $-z^2 a^{16}-3 z^3 a^{15}+3 z a^{15}-z^6 a^{14}+z^4 a^{14}-z^2 a^{14}-3 z^7 a^{13}+12 z^5 a^{13}-18 z^3 a^{13}+12 z a^{13}-5 a^{13} z^{-1} +a^{13} z^{-3} -3 z^8 a^{12}+12 z^6 a^{12}-12 z^4 a^{12}-3 z^2 a^{12}-3 a^{12} z^{-2} +10 a^{12}-z^9 a^{11}+14 z^5 a^{11}-25 z^3 a^{11}+21 z a^{11}-12 a^{11} z^{-1} +3 a^{11} z^{-3} -4 z^8 a^{10}+16 z^6 a^{10}-10 z^4 a^{10}-15 z^2 a^{10}-6 a^{10} z^{-2} +19 a^{10}-z^9 a^9+2 z^7 a^9+8 z^5 a^9-22 z^3 a^9+23 z a^9-12 a^9 z^{-1} +3 a^9 z^{-3} -z^8 a^8+3 z^6 a^8+3 z^4 a^8-12 z^2 a^8-3 a^8 z^{-2} +10 a^8-z^7 a^7+6 z^5 a^7-12 z^3 a^7+11 z a^7-5 a^7 z^{-1} +a^7 z^{-3}$ (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
-6         11
-8        110
-10       3  3
-12     111  1
-14     63   3
-16   222    2
-18   64     2
-20 124      3
-22 22       0
-24 2        2
-261         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{6}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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