# L11a325

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a325 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^3 t(2)^5-t(1)^2 t(2)^5-2 t(1)^3 t(2)^4+4 t(1)^2 t(2)^4-2 t(1) t(2)^4+2 t(1)^3 t(2)^3-4 t(1)^2 t(2)^3+4 t(1) t(2)^3-t(2)^3-t(1)^3 t(2)^2+4 t(1)^2 t(2)^2-4 t(1) t(2)^2+2 t(2)^2-2 t(1)^2 t(2)+4 t(1) t(2)-2 t(2)-t(1)+1}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $\sqrt{q}-\frac{3}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{13}{q^{11/2}}-\frac{11}{q^{13/2}}+\frac{9}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^7 z^7+5 a^7 z^5+8 a^7 z^3+5 a^7 z-a^5 z^9-7 a^5 z^7-18 a^5 z^5-21 a^5 z^3-9 a^5 z+a^5 z^{-1} +a^3 z^7+5 a^3 z^5+7 a^3 z^3+2 a^3 z-a^3 z^{-1}$ (db) Kauffman polynomial $-z^3 a^{13}-3 z^4 a^{12}-6 z^5 a^{11}+4 z^3 a^{11}-z a^{11}-9 z^6 a^{10}+13 z^4 a^{10}-5 z^2 a^{10}-10 z^7 a^9+19 z^5 a^9-7 z^3 a^9+z a^9-9 z^8 a^8+21 z^6 a^8-10 z^4 a^8+4 z^2 a^8-6 z^9 a^7+14 z^7 a^7-2 z^5 a^7-z^3 a^7-2 z a^7-2 z^{10} a^6-3 z^8 a^6+33 z^6 a^6-43 z^4 a^6+16 z^2 a^6-9 z^9 a^5+40 z^7 a^5-55 z^5 a^5+29 z^3 a^5-7 z a^5-a^5 z^{-1} -2 z^{10} a^4+5 z^8 a^4+8 z^6 a^4-25 z^4 a^4+11 z^2 a^4+a^4-3 z^9 a^3+16 z^7 a^3-28 z^5 a^3+18 z^3 a^3-3 z a^3-a^3 z^{-1} -z^8 a^2+5 z^6 a^2-8 z^4 a^2+4 z^2 a^2$ (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 \
-8-7-6-5-4-3-2-10123χ
2           1-1
0          2 2
-2         31 -2
-4        62  4
-6       54   -1
-8      85    3
-10     55     0
-12    68      -2
-14   46       2
-16  25        -3
-18 14         3
-20 2          -2
-221           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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