# L11a75

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a75 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 t(2)^5+3 t(1) t(2)^4-4 t(2)^4-4 t(1) t(2)^3+4 t(2)^3+4 t(1) t(2)^2-4 t(2)^2-4 t(1) t(2)+3 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{4}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{2}{q^{25/2}}+\frac{5}{q^{23/2}}-\frac{7}{q^{21/2}}+\frac{9}{q^{19/2}}-\frac{11}{q^{17/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{13/2}}+\frac{6}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z a^{13}-2 a^{13} z^{-1} +3 z^3 a^{11}+8 z a^{11}+4 a^{11} z^{-1} -2 z^5 a^9-6 z^3 a^9-4 z a^9-a^9 z^{-1} -2 z^5 a^7-6 z^3 a^7-4 z a^7-a^7 z^{-1} -z^5 a^5-3 z^3 a^5-z a^5$ (db) Kauffman polynomial $-z^6 a^{16}+4 z^4 a^{16}-5 z^2 a^{16}+2 a^{16}-2 z^7 a^{15}+6 z^5 a^{15}-3 z^3 a^{15}-z a^{15}-2 z^8 a^{14}+3 z^6 a^{14}+5 z^4 a^{14}-5 z^2 a^{14}+a^{14}-2 z^9 a^{13}+5 z^7 a^{13}-7 z^5 a^{13}+13 z^3 a^{13}-8 z a^{13}+2 a^{13} z^{-1} -z^{10} a^{12}+7 z^6 a^{12}-17 z^4 a^{12}+19 z^2 a^{12}-6 a^{12}-5 z^9 a^{11}+21 z^7 a^{11}-40 z^5 a^{11}+34 z^3 a^{11}-15 z a^{11}+4 a^{11} z^{-1} -z^{10} a^{10}-z^8 a^{10}+13 z^6 a^{10}-29 z^4 a^{10}+18 z^2 a^{10}-5 a^{10}-3 z^9 a^9+11 z^7 a^9-17 z^5 a^9+6 z^3 a^9-2 z a^9+a^9 z^{-1} -3 z^8 a^8+8 z^6 a^8-6 z^4 a^8-2 z^2 a^8+a^8-3 z^7 a^7+9 z^5 a^7-9 z^3 a^7+5 z a^7-a^7 z^{-1} -2 z^6 a^6+5 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-z a^5$ (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          21-1
-8         2  2
-10        42  -2
-12       62   4
-14      55    0
-16     65     1
-18    35      2
-20   46       -2
-22  13        2
-24 14         -3
-26 1          1
-281           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-7$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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