# L11a534

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a534 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(4)^2 t(3)^2+t(2) t(4)^2 t(3)^2-t(4)^2 t(3)^2+t(1) t(3)^2-t(1) t(2) t(3)^2+2 t(2) t(3)^2-2 t(1) t(4) t(3)^2+t(1) t(2) t(4) t(3)^2-3 t(2) t(4) t(3)^2+2 t(4) t(3)^2-t(3)^2-3 t(1) t(4)^2 t(3)+t(1) t(2) t(4)^2 t(3)-2 t(2) t(4)^2 t(3)+2 t(4)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-3 t(2) t(3)+6 t(1) t(4) t(3)-4 t(1) t(2) t(4) t(3)+6 t(2) t(4) t(3)-4 t(4) t(3)+t(3)+2 t(1) t(4)^2-t(1) t(2) t(4)^2+t(2) t(4)^2-t(4)^2+t(1)-t(1) t(2)+t(2)-3 t(1) t(4)+2 t(1) t(2) t(4)-2 t(2) t(4)+t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $\frac{21}{q^{9/2}}-\frac{21}{q^{7/2}}+\frac{14}{q^{5/2}}-\frac{10}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{14}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{23}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{11} z^{-1} +4 z a^9+4 a^9 z^{-1} +a^9 z^{-3} -6 z^3 a^7-11 z a^7-9 a^7 z^{-1} -3 a^7 z^{-3} +3 z^5 a^5+8 z^3 a^5+13 z a^5+10 a^5 z^{-1} +3 a^5 z^{-3} +z^5 a^3-2 z^3 a^3-6 z a^3-4 a^3 z^{-1} -a^3 z^{-3} -z^3 a$ (db) Kauffman polynomial $a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+9 a^{11} z^3-6 a^{11} z+2 a^{11} z^{-1} +4 a^{10} z^8-4 a^{10} z^6-8 a^{10} z^4+14 a^{10} z^2-6 a^{10}+3 a^9 z^9+7 a^9 z^7-33 a^9 z^5+42 a^9 z^3-a^9 z^{-3} -30 a^9 z+11 a^9 z^{-1} +a^8 z^{10}+13 a^8 z^8-24 a^8 z^6-4 a^8 z^4+28 a^8 z^2+3 a^8 z^{-2} -18 a^8+8 a^7 z^9+7 a^7 z^7-56 a^7 z^5+74 a^7 z^3-3 a^7 z^{-3} -49 a^7 z+18 a^7 z^{-1} +a^6 z^{10}+19 a^6 z^8-37 a^6 z^6+9 a^6 z^4+24 a^6 z^2+6 a^6 z^{-2} -21 a^6+5 a^5 z^9+12 a^5 z^7-48 a^5 z^5+56 a^5 z^3-3 a^5 z^{-3} -35 a^5 z+14 a^5 z^{-1} +10 a^4 z^8-14 a^4 z^6+4 a^4 z^4+7 a^4 z^2+3 a^4 z^{-2} -9 a^4+9 a^3 z^7-16 a^3 z^5+14 a^3 z^3-a^3 z^{-3} -10 a^3 z+5 a^3 z^{-1} +4 a^2 z^6-4 a^2 z^4+a z^5-a z^3$ (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 \
-9-8-7-6-5-4-3-2-1012χ
2           11
0          3 -3
-2         71 6
-4        84  -4
-6       136   7
-8      1111    0
-10     1210     2
-12    814      6
-14   69       -3
-16  29        7
-18 15         -4
-20 2          2
-221           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-4$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{12}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{13}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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