# L11n454

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n454 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u v w x^2-2 u v w x-u v x^2+u v x-u w x^2+2 u w x+u x^2-2 u x-2 v w^2 x+v w^2+2 v w x-v w+w^2 x-w^2-2 w x+2 w}{\sqrt{u} \sqrt{v} w x}$ (db) Jones polynomial $\frac{6}{q^{9/2}}-\frac{7}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{7}{q^{13/2}}-\frac{9}{q^{11/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 \left(-z^3\right)+a^9 z^{-3} -a^9 z+a^9 z^{-1} +a^7 z^5+2 a^7 z^3-3 a^7 z^{-3} -a^7 z-6 a^7 z^{-1} +a^5 z^5+3 a^5 z^3+3 a^5 z^{-3} +7 a^5 z+9 a^5 z^{-1} -2 a^3 z^3-a^3 z^{-3} -5 a^3 z-4 a^3 z^{-1}$ (db) Kauffman polynomial $-z^6 a^{12}+3 z^4 a^{12}-z^2 a^{12}-3 z^7 a^{11}+11 z^5 a^{11}-10 z^3 a^{11}+3 z a^{11}-3 z^8 a^{10}+9 z^6 a^{10}-4 z^4 a^{10}-z^2 a^{10}-z^9 a^9-3 z^7 a^9+21 z^5 a^9-28 z^3 a^9+16 z a^9-5 a^9 z^{-1} +a^9 z^{-3} -5 z^8 a^8+15 z^6 a^8-10 z^4 a^8-7 z^2 a^8-3 a^8 z^{-2} +10 a^8-z^9 a^7-2 z^7 a^7+16 z^5 a^7-31 z^3 a^7+25 z a^7-12 a^7 z^{-1} +3 a^7 z^{-3} -2 z^8 a^6+4 z^6 a^6-z^4 a^6-15 z^2 a^6-6 a^6 z^{-2} +19 a^6-2 z^7 a^5+6 z^5 a^5-16 z^3 a^5+19 z a^5-12 a^5 z^{-1} +3 a^5 z^{-3} -z^6 a^4+2 z^4 a^4-8 z^2 a^4-3 a^4 z^{-2} +10 a^4-3 z^3 a^3+7 z a^3-5 a^3 z^{-1} +a^3 z^{-3}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         22
-4        220
-6       5  5
-8      34  1
-10     63   3
-12    35    2
-14   44     0
-16  25      3
-18 12       -1
-20 2        2
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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