# L11a28

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a28 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-10 q^{9/2}+\frac{1}{q^{9/2}}+17 q^{7/2}-\frac{4}{q^{7/2}}-23 q^{5/2}+\frac{9}{q^{5/2}}+26 q^{3/2}-\frac{17}{q^{3/2}}-q^{13/2}+4 q^{11/2}-26 \sqrt{q}+\frac{22}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} -a^3 z^3+4 z^3 a^{-3} -a^3 z+5 z a^{-3} +3 a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +4 a z^3-6 z^3 a^{-1} +4 a z-7 z a^{-1} +2 a z^{-1} -4 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -4 z^4 a^{-6} +z^2 a^{-6} +9 z^7 a^{-5} -13 z^5 a^{-5} +10 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +12 z^8 a^{-4} +a^4 z^6-18 z^6 a^{-4} -2 a^4 z^4+13 z^4 a^{-4} +a^4 z^2-5 z^2 a^{-4} + a^{-4} +8 z^9 a^{-3} +4 a^3 z^7+4 z^7 a^{-3} -9 a^3 z^5-32 z^5 a^{-3} +8 a^3 z^3+34 z^3 a^{-3} -3 a^3 z-16 z a^{-3} +3 a^{-3} z^{-1} +2 z^{10} a^{-2} +7 a^2 z^8+24 z^8 a^{-2} -13 a^2 z^6-58 z^6 a^{-2} +7 a^2 z^4+45 z^4 a^{-2} -a^2 z^2-17 z^2 a^{-2} +a^2+3 a^{-2} +6 a z^9+14 z^9 a^{-1} +a z^7-8 z^7 a^{-1} -27 a z^5-36 z^5 a^{-1} +30 a z^3+45 z^3 a^{-1} -13 a z-22 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +2 z^{10}+19 z^8-50 z^6+37 z^4-13 z^2+2$ (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        103  -7
6       137   6
4      1310    -3
2     1313     0
0    1115      4
-2   611       -5
-4  311        8
-6 16         -5
-8 3          3
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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