# L11n451

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n451 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(w-1) (x-1)^2 (u x-v)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}}$ (db) Jones polynomial $-q^{5/2}+q^{3/2}-\sqrt{q}-\frac{4}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+a^5 z^{-3} +7 a^5 z+5 a^5 z^{-1} -2 a^3 z^5-9 a^3 z^3-3 a^3 z^{-3} -15 a^3 z-10 a^3 z^{-1} +a z^5+7 a z^3+3 a z^{-3} -z^3 a^{-1} - a^{-1} z^{-3} +12 a z+9 a z^{-1} -3 z a^{-1} -3 a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^6 a^8+3 z^4 a^8-3 z^2 a^8+a^8-3 z^7 a^7+10 z^5 a^7-9 z^3 a^7+5 z a^7-2 a^7 z^{-1} -2 z^8 a^6+z^6 a^6+15 z^4 a^6-15 z^2 a^6+6 a^6-9 z^7 a^5+35 z^5 a^5-43 z^3 a^5+31 z a^5-11 a^5 z^{-1} +a^5 z^{-3} -2 z^8 a^4+z^6 a^4+19 z^4 a^4-33 z^2 a^4-3 a^4 z^{-2} +18 a^4-7 z^7 a^3+39 z^5 a^3-73 z^3 a^3+54 z a^3-18 a^3 z^{-1} +3 a^3 z^{-3} -z^8 a^2+5 z^6 a^2+2 z^4 a^2-27 z^2 a^2-6 a^2 z^{-2} +21 a^2-2 z^7 a+20 z^5 a-49 z^3 a+38 z a-14 a z^{-1} +3 a z^{-3} -z^8+6 z^6-5 z^4-6 z^2-3 z^{-2} +9-z^7 a^{-1} +6 z^5 a^{-1} -10 z^3 a^{-1} +10 z a^{-1} -5 a^{-1} z^{-1} + a^{-1} z^{-3}$ (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 \
-7-6-5-4-3-2-101234χ
6           11
4            0
2         11 0
0       61   5
-2      461   1
-4     422    4
-6    24      2
-8   541      2
-10  24        2
-12 13         -2
-14 2          2
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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