# L11a37

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a37 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{3 t(2)^3+4 t(1) t(2)^2-6 t(2)^2-6 t(1) t(2)+4 t(2)+3 t(1)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{5}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{5}{q^{19/2}}+\frac{4}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{13} z^{-1} +3 z a^{11}+2 a^{11} z^{-1} -2 z^3 a^9-2 z a^9-a^9 z^{-1} -2 z^3 a^7+a^7 z^{-1} -2 z^3 a^5-2 z a^5-a^5 z^{-1} -z^3 a^3-z a^3$ (db) Kauffman polynomial $-z^8 a^{14}+6 z^6 a^{14}-12 z^4 a^{14}+10 z^2 a^{14}-3 a^{14}-2 z^9 a^{13}+11 z^7 a^{13}-18 z^5 a^{13}+10 z^3 a^{13}-3 z a^{13}+a^{13} z^{-1} -z^{10} a^{12}+z^8 a^{12}+16 z^6 a^{12}-39 z^4 a^{12}+28 z^2 a^{12}-7 a^{12}-5 z^9 a^{11}+25 z^7 a^{11}-38 z^5 a^{11}+23 z^3 a^{11}-8 z a^{11}+2 a^{11} z^{-1} -z^{10} a^{10}-z^8 a^{10}+19 z^6 a^{10}-31 z^4 a^{10}+18 z^2 a^{10}-4 a^{10}-3 z^9 a^9+11 z^7 a^9-14 z^5 a^9+13 z^3 a^9-5 z a^9+a^9 z^{-1} -3 z^8 a^8+6 z^6 a^8-3 z^7 a^7+3 z^5 a^7+4 z^3 a^7-4 z a^7+a^7 z^{-1} -3 z^6 a^6+2 z^4 a^6+z^2 a^6-a^6-3 z^5 a^5+3 z^3 a^5-3 z a^5+a^5 z^{-1} -2 z^4 a^4+z^2 a^4-z^3 a^3+z a^3$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-2           11
-4          21-1
-6         2  2
-8        32  -1
-10       42   2
-12      44    0
-14     43     1
-16    24      2
-18   34       -1
-20  12        1
-22 13         -2
-24 1          1
-261           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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