# L11a351

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a351 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-3 t(2) t(1)^2+2 t(1)^2-2 t(2)^2 t(1)+7 t(2) t(1)-2 t(1)+2 t(2)^2-3 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-q^{7/2}+5 q^{5/2}-13 q^{3/2}+20 \sqrt{q}-\frac{27}{\sqrt{q}}+\frac{30}{q^{3/2}}-\frac{30}{q^{5/2}}+\frac{25}{q^{7/2}}-\frac{18}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)+3 a^5 z^3+2 a^5 z-3 a^3 z^5-5 a^3 z^3-3 a^3 z+a z^7+3 a z^5-z^5 a^{-1} +6 a z^3-z^3 a^{-1} +4 a z+a z^{-1} -2 z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-8 a^7 z^5+5 a^7 z^3-a^7 z+8 a^6 z^8-16 a^6 z^6+12 a^6 z^4-4 a^6 z^2+9 a^5 z^9-13 a^5 z^7+3 a^5 z^5+2 a^5 z^3+4 a^4 z^{10}+14 a^4 z^8-44 a^4 z^6+33 a^4 z^4-7 a^4 z^2+22 a^3 z^9-36 a^3 z^7+9 a^3 z^5+z^5 a^{-3} +3 a^3 z^3+4 a^2 z^{10}+24 a^2 z^8-59 a^2 z^6+5 z^6 a^{-2} +33 a^2 z^4-2 z^4 a^{-2} -4 a^2 z^2+13 a z^9-6 a z^7+13 z^7 a^{-1} -19 a z^5-16 z^5 a^{-1} +13 a z^3+7 z^3 a^{-1} -4 a z-3 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +18 z^8-27 z^6+12 z^4-2 z^2-1$ (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
8           11
6          4 -4
4         91 8
2        114  -7
0       169   7
-2      1613    -3
-4     1414     0
-6    1116      5
-8   714       -7
-10  311        8
-12 17         -6
-14 3          3
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-2$ ${\mathbb Z}^{16}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=-1$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16}$ ${\mathbb Z}^{16}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{16}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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