# L11n312

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n312 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^2 w^3-2 u v^2 w^2+2 u v^2 w-u v^2+u v w^4-2 u v w^3+3 u v w^2-3 u v w+u v+u w-v^2 w^3-v w^4+3 v w^3-3 v w^2+2 v w-v+w^4-2 w^3+2 w^2-w}{\sqrt{u} v w^2}$ (db) Jones polynomial $1-3 q^{-1} +7 q^{-2} -8 q^{-3} +12 q^{-4} -11 q^{-5} +11 q^{-6} -8 q^{-7} +5 q^{-8} -2 q^{-9}$ (db) Signature -4 (db) HOMFLY-PT polynomial $-a^{10} z^{-2} -a^{10}+z^4 a^8+4 z^2 a^8+4 a^8 z^{-2} +7 a^8-z^6 a^6-4 z^4 a^6-9 z^2 a^6-5 a^6 z^{-2} -12 a^6-z^6 a^4-2 z^4 a^4+2 z^2 a^4+2 a^4 z^{-2} +5 a^4+z^4 a^2+2 z^2 a^2+a^2$ (db) Kauffman polynomial $3 z^3 a^{11}-4 z a^{11}+a^{11} z^{-1} +z^6 a^{10}+4 z^4 a^{10}-6 z^2 a^{10}-a^{10} z^{-2} +3 a^{10}+5 z^7 a^9-15 z^5 a^9+28 z^3 a^9-19 z a^9+5 a^9 z^{-1} +6 z^8 a^8-21 z^6 a^8+39 z^4 a^8-32 z^2 a^8-4 a^8 z^{-2} +15 a^8+2 z^9 a^7+4 z^7 a^7-27 z^5 a^7+46 z^3 a^7-33 z a^7+9 a^7 z^{-1} +10 z^8 a^6-33 z^6 a^6+44 z^4 a^6-37 z^2 a^6-5 a^6 z^{-2} +20 a^6+2 z^9 a^5+2 z^7 a^5-20 z^5 a^5+25 z^3 a^5-18 z a^5+5 a^5 z^{-1} +4 z^8 a^4-10 z^6 a^4+6 z^4 a^4-8 z^2 a^4-2 a^4 z^{-2} +8 a^4+3 z^7 a^3-8 z^5 a^3+4 z^3 a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^2$ (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 \
-7-6-5-4-3-2-1012χ
1         11
-1        2 -2
-3       51 4
-5      54  -1
-7     73   4
-9    56    1
-11   66     0
-13  25      3
-15 36       -3
-17 3        3
-192         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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