# L11a235

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a235 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(2)^6-3 t(1) t(2)^5+3 t(2)^5-3 t(1)^2 t(2)^4+6 t(1) t(2)^4-4 t(2)^4+4 t(1)^2 t(2)^3-7 t(1) t(2)^3+4 t(2)^3-4 t(1)^2 t(2)^2+6 t(1) t(2)^2-3 t(2)^2+3 t(1)^2 t(2)-3 t(1) t(2)-t(1)^2}{t(1) t(2)^3}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{16}{q^{13/2}}+\frac{17}{q^{15/2}}-\frac{18}{q^{17/2}}+\frac{15}{q^{19/2}}-\frac{11}{q^{21/2}}+\frac{7}{q^{23/2}}-\frac{3}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^{13} (-z)-2 a^{13} z^{-1} +4 a^{11} z^3+10 a^{11} z+5 a^{11} z^{-1} -3 a^9 z^5-9 a^9 z^3-8 a^9 z-3 a^9 z^{-1} -3 a^7 z^5-8 a^7 z^3-4 a^7 z-a^5 z^5-2 a^5 z^3$ (db) Kauffman polynomial $a^{16} z^6-3 a^{16} z^4+3 a^{16} z^2-a^{16}+3 a^{15} z^7-8 a^{15} z^5+5 a^{15} z^3+4 a^{14} z^8-7 a^{14} z^6-2 a^{14} z^4+4 a^{14} z^2+4 a^{13} z^9-7 a^{13} z^7+5 a^{13} z^5-10 a^{13} z^3+7 a^{13} z-2 a^{13} z^{-1} +2 a^{12} z^{10}+2 a^{12} z^8-12 a^{12} z^6+17 a^{12} z^4-17 a^{12} z^2+5 a^{12}+10 a^{11} z^9-31 a^{11} z^7+51 a^{11} z^5-47 a^{11} z^3+22 a^{11} z-5 a^{11} z^{-1} +2 a^{10} z^{10}+5 a^{10} z^8-23 a^{10} z^6+36 a^{10} z^4-21 a^{10} z^2+5 a^{10}+6 a^9 z^9-15 a^9 z^7+23 a^9 z^5-18 a^9 z^3+11 a^9 z-3 a^9 z^{-1} +7 a^8 z^8-16 a^8 z^6+15 a^8 z^4-3 a^8 z^2+6 a^7 z^7-14 a^7 z^5+12 a^7 z^3-4 a^7 z+3 a^6 z^6-5 a^6 z^4+a^5 z^5-2 a^5 z^3$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          31-2
-8         4  4
-10        73  -4
-12       94   5
-14      87    -1
-16     109     1
-18    69      3
-20   59       -4
-22  26        4
-24 15         -4
-26 2          2
-281           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-7$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-6$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=-5$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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