# L11a123

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a123 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)^5-2 t(2)^5-4 t(1) t(2)^4+4 t(2)^4+4 t(1) t(2)^3-4 t(2)^3-4 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-4 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{1}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{12}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{8}{q^{17/2}}+\frac{5}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^{11} (-z)-2 a^{11} z^{-1} +3 a^9 z^3+9 a^9 z+5 a^9 z^{-1} -3 a^7 z^5-11 a^7 z^3-10 a^7 z-3 a^7 z^{-1} +a^5 z^7+4 a^5 z^5+4 a^5 z^3+a^5 z-a^3 z^5-3 a^3 z^3-a^3 z$ (db) Kauffman polynomial $-z^4 a^{14}+2 z^2 a^{14}-a^{14}-2 z^5 a^{13}+2 z^3 a^{13}-3 z^6 a^{12}+2 z^4 a^{12}-4 z^7 a^{11}+6 z^5 a^{11}-8 z^3 a^{11}+5 z a^{11}-2 a^{11} z^{-1} -3 z^8 a^{10}+9 z^4 a^{10}-13 z^2 a^{10}+5 a^{10}-2 z^9 a^9-2 z^7 a^9+14 z^5 a^9-20 z^3 a^9+15 z a^9-5 a^9 z^{-1} -z^{10} a^8-2 z^8 a^8+6 z^6 a^8+4 z^4 a^8-9 z^2 a^8+5 a^8-5 z^9 a^7+14 z^7 a^7-8 z^5 a^7-4 z^3 a^7+9 z a^7-3 a^7 z^{-1} -z^{10} a^6-2 z^8 a^6+16 z^6 a^6-17 z^4 a^6+5 z^2 a^6-3 z^9 a^5+11 z^7 a^5-10 z^5 a^5+2 z^3 a^5-3 z^8 a^4+13 z^6 a^4-15 z^4 a^4+3 z^2 a^4-z^7 a^3+4 z^5 a^3-4 z^3 a^3+z a^3$ (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
0           11
-2          2 -2
-4         31 2
-6        53  -2
-8       62   4
-10      55    0
-12     76     1
-14    46      2
-16   46       -2
-18  14        3
-20 14         -3
-22 1          1
-241           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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