# L11a107

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a107 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(2)^7-2 t(1) t(2)^6+2 t(2)^6+2 t(1) t(2)^5-2 t(2)^5-2 t(1) t(2)^4+2 t(2)^4+2 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-2 t(2)-t(1)}{\sqrt{t(1)} t(2)^{7/2}}$ (db) Jones polynomial $\frac{2}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{29/2}}-\frac{2}{q^{27/2}}+\frac{4}{q^{25/2}}-\frac{5}{q^{23/2}}+\frac{7}{q^{21/2}}-\frac{8}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{4}{q^{11/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $-z^3 a^{13}-4 z a^{13}-3 a^{13} z^{-1} +3 z^5 a^{11}+15 z^3 a^{11}+20 z a^{11}+7 a^{11} z^{-1} -2 z^7 a^9-12 z^5 a^9-23 z^3 a^9-17 z a^9-4 a^9 z^{-1} -z^7 a^7-5 z^5 a^7-6 z^3 a^7-z a^7$ (db) Kauffman polynomial $a^{18} z^4-2 a^{18} z^2+a^{18}+2 a^{17} z^5-3 a^{17} z^3+2 a^{16} z^6-a^{16} z^4-2 a^{16} z^2+2 a^{15} z^7-2 a^{15} z^5+a^{15} z^3+2 a^{14} z^8-4 a^{14} z^6+4 a^{14} z^4+2 a^{13} z^9-7 a^{13} z^7+12 a^{13} z^5-12 a^{13} z^3+10 a^{13} z-3 a^{13} z^{-1} +a^{12} z^{10}-a^{12} z^8-9 a^{12} z^6+24 a^{12} z^4-22 a^{12} z^2+7 a^{12}+5 a^{11} z^9-27 a^{11} z^7+55 a^{11} z^5-57 a^{11} z^3+29 a^{11} z-7 a^{11} z^{-1} +a^{10} z^{10}-a^{10} z^8-12 a^{10} z^6+27 a^{10} z^4-23 a^{10} z^2+7 a^{10}+3 a^9 z^9-17 a^9 z^7+34 a^9 z^5-35 a^9 z^3+18 a^9 z-4 a^9 z^{-1} +2 a^8 z^8-9 a^8 z^6+9 a^8 z^4-a^8 z^2+a^7 z^7-5 a^7 z^5+6 a^7 z^3-a^7 z$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-6           11
-8          21-1
-10         2  2
-12        22  0
-14       52   3
-16      33    0
-18     54     1
-20    23      1
-22   35       -2
-24  12        1
-26 13         -2
-28 1          1
-301           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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