# L11a273

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a273 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-2 u^3 v^4+2 u^3 v^3-u^3 v^2-u^2 v^5+3 u^2 v^4-5 u^2 v^3+3 u^2 v^2-u^2 v-u v^4+3 u v^3-5 u v^2+3 u v-u-v^3+2 v^2-2 v}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $-\frac{1}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{11}{q^{17/2}}-\frac{11}{q^{19/2}}+\frac{10}{q^{21/2}}-\frac{8}{q^{23/2}}+\frac{5}{q^{25/2}}-\frac{3}{q^{27/2}}+\frac{1}{q^{29/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $a^{13} \left(-z^3\right)-3 a^{13} z-a^{13} z^{-1} +3 a^{11} z^5+13 a^{11} z^3+14 a^{11} z+3 a^{11} z^{-1} -2 a^9 z^7-11 a^9 z^5-19 a^9 z^3-12 a^9 z-2 a^9 z^{-1} -a^7 z^7-5 a^7 z^5-7 a^7 z^3-3 a^7 z$ (db) Kauffman polynomial $-z^4 a^{18}+z^2 a^{18}-3 z^5 a^{17}+4 z^3 a^{17}-z a^{17}-4 z^6 a^{16}+4 z^4 a^{16}-4 z^7 a^{15}+3 z^5 a^{15}+2 z^3 a^{15}-2 z a^{15}-4 z^8 a^{14}+7 z^6 a^{14}-7 z^4 a^{14}+4 z^2 a^{14}-a^{14}-3 z^9 a^{13}+6 z^7 a^{13}-5 z^5 a^{13}+3 z^3 a^{13}-3 z a^{13}+a^{13} z^{-1} -z^{10} a^{12}-4 z^8 a^{12}+25 z^6 a^{12}-40 z^4 a^{12}+24 z^2 a^{12}-3 a^{12}-6 z^9 a^{11}+25 z^7 a^{11}-39 z^5 a^{11}+35 z^3 a^{11}-18 z a^{11}+3 a^{11} z^{-1} -z^{10} a^{10}-2 z^8 a^{10}+22 z^6 a^{10}-35 z^4 a^{10}+20 z^2 a^{10}-3 a^{10}-3 z^9 a^9+14 z^7 a^9-23 z^5 a^9+23 z^3 a^9-13 z a^9+2 a^9 z^{-1} -2 z^8 a^8+8 z^6 a^8-7 z^4 a^8+z^2 a^8-z^7 a^7+5 z^5 a^7-7 z^3 a^7+3 z a^7$ (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         3  3
-12        32  -1
-14       63   3
-16      53    -2
-18     66     0
-20    45      1
-22   46       -2
-24  25        3
-26 13         -2
-28 2          2
-301           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-7$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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