# L11n354

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n354 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(3)-1) \left(t(2) t(3)^3-t(3)^3+t(1) t(2)^2 t(3)^2-t(1) t(2) t(3)^2-t(2) t(3)^2-t(1) t(2) t(3)-t(2) t(3)+t(3)-t(1) t(2)^2+t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $- q^{-10} +2 q^{-9} -2 q^{-8} +3 q^{-7} - q^{-6} +2 q^{-5} + q^{-2} - q^{-1} +1$ (db) Signature -2 (db) HOMFLY-PT polynomial $-a^{10}+2 a^8 z^2+a^8 z^{-2} +2 a^8+a^6 z^2-2 a^6 z^{-2} -2 a^6-a^4 z^6-5 a^4 z^4-5 a^4 z^2+a^4 z^{-2} -a^4+a^2 z^4+4 a^2 z^2+2 a^2$ (db) Kauffman polynomial $a^{11} z^7-5 a^{11} z^5+6 a^{11} z^3-2 a^{11} z+2 a^{10} z^8-11 a^{10} z^6+17 a^{10} z^4-11 a^{10} z^2+3 a^{10}+a^9 z^9-3 a^9 z^7-6 a^9 z^5+16 a^9 z^3-7 a^9 z+4 a^8 z^8-25 a^8 z^6+46 a^8 z^4-35 a^8 z^2-a^8 z^{-2} +12 a^8+a^7 z^9-4 a^7 z^7-4 a^7 z^5+19 a^7 z^3-14 a^7 z+2 a^7 z^{-1} +2 a^6 z^8-14 a^6 z^6+28 a^6 z^4-26 a^6 z^2-2 a^6 z^{-2} +12 a^6+a^5 z^7-8 a^5 z^5+14 a^5 z^3-10 a^5 z+2 a^5 z^{-1} +a^4 z^6-6 a^4 z^4+4 a^4 z^2-a^4 z^{-2} +2 a^4+a^3 z^7-5 a^3 z^5+5 a^3 z^3-a^3 z+a^2 z^6-5 a^2 z^4+6 a^2 z^2-2 a^2$ (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 \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1            0
-3        121 0
-5       111  1
-7      231   0
-9     222    2
-11    241     1
-13   211      2
-15  131       1
-17 11         0
-19 1          1
-211           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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