# L11a92

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a92 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^5+3 t(1) t(2)^4-6 t(2)^4-10 t(1) t(2)^3+11 t(2)^3+11 t(1) t(2)^2-10 t(2)^2-6 t(1) t(2)+3 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{20}{q^{9/2}}-\frac{20}{q^{11/2}}+\frac{17}{q^{13/2}}-\frac{13}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{11} z^{-1} +4 z a^9+4 a^9 z^{-1} -6 z^3 a^7-11 z a^7-6 a^7 z^{-1} +3 z^5 a^5+8 z^3 a^5+10 z a^5+5 a^5 z^{-1} +z^5 a^3-z^3 a^3-4 z a^3-2 a^3 z^{-1} -z^3 a-z a$ (db) Kauffman polynomial $a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+8 a^{11} z^3-4 a^{11} z+a^{11} z^{-1} +4 a^{10} z^8-5 a^{10} z^6-5 a^{10} z^4+9 a^{10} z^2-3 a^{10}+3 a^9 z^9+5 a^9 z^7-27 a^9 z^5+31 a^9 z^3-18 a^9 z+4 a^9 z^{-1} +a^8 z^{10}+11 a^8 z^8-22 a^8 z^6+4 a^8 z^4+7 a^8 z^2-3 a^8+7 a^7 z^9+4 a^7 z^7-41 a^7 z^5+52 a^7 z^3-29 a^7 z+6 a^7 z^{-1} +a^6 z^{10}+14 a^6 z^8-28 a^6 z^6+15 a^6 z^4-a^6+4 a^5 z^9+8 a^5 z^7-33 a^5 z^5+41 a^5 z^3-23 a^5 z+5 a^5 z^{-1} +7 a^4 z^8-9 a^4 z^6+5 a^4 z^4-a^4+6 a^3 z^7-10 a^3 z^5+10 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +3 a^2 z^6-4 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z$ (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          2 -2
-2         61 5
-4        83  -5
-6       105   5
-8      108    -2
-10     1010     0
-12    811      3
-14   59       -4
-16  28        6
-18 15         -4
-20 2          2
-221           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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