# L11a372

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a372 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^3 t(1)^4-t(2)^2 t(1)^4+t(2)^4 t(1)^3-t(2)^3 t(1)^3+t(2)^2 t(1)^3-t(2) t(1)^3-t(2)^4 t(1)^2+t(2)^3 t(1)^2-t(2)^2 t(1)^2+t(2) t(1)^2-t(1)^2-t(2)^3 t(1)+t(2)^2 t(1)-t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^2 t(2)^2}$ (db) Jones polynomial $-\frac{5}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{4}{q^{5/2}}-q^{3/2}+\frac{3}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{2}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{4}{q^{11/2}}+\sqrt{q}-\frac{2}{\sqrt{q}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z^5 a^7-4 z^3 a^7-3 z a^7+z^7 a^5+5 z^5 a^5+6 z^3 a^5+z a^5+z^7 a^3+6 z^5 a^3+11 z^3 a^3+7 z a^3+a^3 z^{-1} -z^5 a-5 z^3 a-6 z a-a z^{-1}$ (db) Kauffman polynomial $-z^2 a^{12}-2 z^3 a^{11}-3 z^4 a^{10}+2 z^2 a^{10}-4 z^5 a^9+7 z^3 a^9-2 z a^9-4 z^6 a^8+9 z^4 a^8-2 z^2 a^8-4 z^7 a^7+13 z^5 a^7-9 z^3 a^7+2 z a^7-3 z^8 a^6+11 z^6 a^6-8 z^4 a^6-2 z^9 a^5+8 z^7 a^5-5 z^5 a^5-5 z^3 a^5+2 z a^5-z^{10} a^4+4 z^8 a^4-z^6 a^4-7 z^4 a^4+3 z^2 a^4-3 z^9 a^3+20 z^7 a^3-44 z^5 a^3+37 z^3 a^3-11 z a^3+a^3 z^{-1} -z^{10} a^2+7 z^8 a^2-16 z^6 a^2+13 z^4 a^2-2 z^2 a^2-a^2-z^9 a+8 z^7 a-22 z^5 a+24 z^3 a-9 z a+a z^{-1}$ (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 \
-7-6-5-4-3-2-101234χ
4           11
2            0
0         21 1
-2        1   -1
-4       32   1
-6      22    0
-8     32     1
-10    23      1
-12   22       0
-14  12        1
-16 12         -1
-18 1          1
-201           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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