# L11a374

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a374 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u v-u+1) (u v-v+1) \left(u^2 v^2+u v+1\right)}{u^2 v^2}$ (db) Jones polynomial $\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^7 z^7+6 a^7 z^5+12 a^7 z^3+9 a^7 z+a^7 z^{-1} -a^5 z^9-8 a^5 z^7-24 a^5 z^5-33 a^5 z^3-18 a^5 z-a^5 z^{-1} +a^3 z^7+6 a^3 z^5+11 a^3 z^3+6 a^3 z$ (db) Kauffman polynomial $-z^3 a^{13}+z a^{13}-2 z^4 a^{12}+z^2 a^{12}-3 z^5 a^{11}+2 z^3 a^{11}-z a^{11}-4 z^6 a^{10}+6 z^4 a^{10}-4 z^2 a^{10}-4 z^7 a^9+7 z^5 a^9-2 z^3 a^9-4 z^8 a^8+11 z^6 a^8-9 z^4 a^8+5 z^2 a^8-3 z^9 a^7+10 z^7 a^7-12 z^5 a^7+15 z^3 a^7-9 z a^7+a^7 z^{-1} -z^{10} a^6-z^8 a^6+18 z^6 a^6-29 z^4 a^6+16 z^2 a^6-a^6-5 z^9 a^5+26 z^7 a^5-46 z^5 a^5+40 z^3 a^5-18 z a^5+a^5 z^{-1} -z^{10} a^4+2 z^8 a^4+9 z^6 a^4-23 z^4 a^4+12 z^2 a^4-2 z^9 a^3+12 z^7 a^3-24 z^5 a^3+20 z^3 a^3-7 z a^3-z^8 a^2+6 z^6 a^2-11 z^4 a^2+6 z^2 a^2$ (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          1 1
-2         21 -1
-4        41  3
-6       33   0
-8      53    2
-10     33     0
-12    45      -1
-14   23       1
-16  24        -2
-18 13         2
-20 1          -1
-221           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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