# L11a352

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a352 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u^3 v^3-4 u^3 v^2+3 u^3 v-u^3-5 u^2 v^3+13 u^2 v^2-11 u^2 v+4 u^2+4 u v^3-11 u v^2+13 u v-5 u-v^3+3 v^2-4 v+2}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $\frac{28}{q^{9/2}}-\frac{28}{q^{7/2}}+\frac{23}{q^{5/2}}+q^{3/2}-\frac{17}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{12}{q^{15/2}}+\frac{19}{q^{13/2}}-\frac{25}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z^5 a^7+z^3 a^7-z^7 a^5-2 z^5 a^5+2 z a^5+a^5 z^{-1} -z^7 a^3-3 z^5 a^3-5 z^3 a^3-5 z a^3-a^3 z^{-1} +z^5 a+2 z^3 a+z a$ (db) Kauffman polynomial $a^{11} z^5+5 a^{10} z^6-3 a^{10} z^4+12 a^9 z^7-15 a^9 z^5+5 a^9 z^3-a^9 z+16 a^8 z^8-24 a^8 z^6+11 a^8 z^4-3 a^8 z^2+11 a^7 z^9-3 a^7 z^7-20 a^7 z^5+13 a^7 z^3-2 a^7 z+3 a^6 z^{10}+24 a^6 z^8-58 a^6 z^6+36 a^6 z^4-6 a^6 z^2+18 a^5 z^9-24 a^5 z^7-8 a^5 z^5+19 a^5 z^3-7 a^5 z+a^5 z^{-1} +3 a^4 z^{10}+15 a^4 z^8-43 a^4 z^6+31 a^4 z^4-5 a^4 z^2-a^4+7 a^3 z^9-5 a^3 z^7-13 a^3 z^5+18 a^3 z^3-8 a^3 z+a^3 z^{-1} +7 a^2 z^8-13 a^2 z^6+7 a^2 z^4-a^2 z^2+4 a z^7-9 a z^5+7 a z^3-2 a z+z^6-2 z^4+z^2$ (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 \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          3 3
0         61 -5
-2        113  8
-4       137   -6
-6      1510    5
-8     1313     0
-10    1215      -3
-12   814       6
-14  411        -7
-16 18         7
-18 4          -4
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-4$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=-3$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=-2$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15}$ ${\mathbb Z}^{15}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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