# L11a48

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a48 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(v^2-4 v+1\right) \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-q^{13/2}+4 q^{11/2}-10 q^{9/2}+15 q^{7/2}-20 q^{5/2}+24 q^{3/2}-23 \sqrt{q}+\frac{19}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+4 a z^3-4 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a^3 z+a z-2 z a^{-1} +3 z a^{-3} -z a^{-5} +a^3 z^{-1} -a z^{-1} - a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1}$ (db) Kauffman polynomial $z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -4 z^4 a^{-6} +9 z^7 a^{-5} -15 z^5 a^{-5} +10 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +11 z^8 a^{-4} +a^4 z^6-19 z^6 a^{-4} -2 a^4 z^4+14 z^4 a^{-4} +a^4 z^2-4 z^2 a^{-4} + a^{-4} +7 z^9 a^{-3} +4 a^3 z^7-10 a^3 z^5-19 z^5 a^{-3} +8 a^3 z^3+23 z^3 a^{-3} -a^3 z-13 z a^{-3} -a^3 z^{-1} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +6 a^2 z^8+16 z^8 a^{-2} -11 a^2 z^6-38 z^6 a^{-2} +3 a^2 z^4+26 z^4 a^{-2} +a^2 z^2-6 z^2 a^{-2} +a^2+3 a^{-2} +5 a z^9+12 z^9 a^{-1} -a z^7-14 z^7 a^{-1} -16 a z^5-9 z^5 a^{-1} +14 a z^3+18 z^3 a^{-1} -3 a z-10 z a^{-1} -a z^{-1} + a^{-1} z^{-1} +2 z^{10}+11 z^8-27 z^6+13 z^4-2 z^2+2$ (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 \
-5-4-3-2-10123456χ
14           11
12          3 -3
10         71 6
8        83  -5
6       127   5
4      128    -4
2     1112     -1
0    1014      4
-2   59       -4
-4  310        7
-6 15         -4
-8 3          3
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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