# L11a373

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a373 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^4 v^3-u^4 v^2+u^3 v^4-4 u^3 v^3+6 u^3 v^2-4 u^3 v+u^3-2 u^2 v^4+7 u^2 v^3-11 u^2 v^2+7 u^2 v-2 u^2+u v^4-4 u v^3+6 u v^2-4 u v+u-v^2+v}{u^2 v^2}$ (db) Jones polynomial $-q^{7/2}+4 q^{5/2}-9 q^{3/2}+14 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{21}{q^{5/2}}+\frac{17}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^5 z^5-3 a^5 z^3-3 a^5 z+a^3 z^7+4 a^3 z^5+7 a^3 z^3+5 a^3 z+a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +2 a z^3-2 z^3 a^{-1} -2 a z-z a^{-1} -a z^{-1}$ (db) Kauffman polynomial $-2 a^4 z^{10}-2 a^2 z^{10}-5 a^5 z^9-11 a^3 z^9-6 a z^9-5 a^6 z^8-7 a^4 z^8-11 a^2 z^8-9 z^8-3 a^7 z^7+9 a^5 z^7+21 a^3 z^7+a z^7-8 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+24 a^4 z^6+28 a^2 z^6-4 z^6 a^{-2} +13 z^6+8 a^7 z^5-5 a^5 z^5-16 a^3 z^5+11 a z^5+13 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-9 a^6 z^4-23 a^4 z^4-22 a^2 z^4+5 z^4 a^{-2} -6 z^4-6 a^7 z^3+2 a^5 z^3+9 a^3 z^3-7 a z^3-7 z^3 a^{-1} +z^3 a^{-3} -2 a^8 z^2+3 a^6 z^2+8 a^4 z^2+6 a^2 z^2-z^2 a^{-2} +2 z^2+2 a^7 z-5 a^3 z-a z+2 z a^{-1} -a^2+a^3 z^{-1} +a z^{-1}$ (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 \
-7-6-5-4-3-2-101234χ
8           11
6          3 -3
4         61 5
2        83  -5
0       116   5
-2      119    -2
-4     1010     0
-6    812      4
-8   59       -4
-10  28        6
-12 15         -4
-14 2          2
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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