L6a5 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L6a5 at Knotilus! L6a5 is $6^3_1$ in the Rolfsen table of links. It is a closed three-link chain.

Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2) t(1)+t(3) t(1)-t(1)-t(2)+t(2) t(3)-t(3)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}}$ (db) Jones polynomial $q^{-1} -2 q^{-2} +3 q^{-3} - q^{-4} +3 q^{-5} - q^{-6} + q^{-7}$ (db) Signature -2 (db) HOMFLY-PT polynomial $a^8 z^{-2} -2 a^6 z^{-2} -3 a^6+2 a^4 z^2+a^4 z^{-2} +3 a^4+a^2 z^2$ (db) Kauffman polynomial $z^4 a^8-3 z^2 a^8-a^8 z^{-2} +3 a^8+z^5 a^7-z^3 a^7-3 z a^7+2 a^7 z^{-1} +4 z^4 a^6-9 z^2 a^6-2 a^6 z^{-2} +5 a^6+z^5 a^5+z^3 a^5-3 z a^5+2 a^5 z^{-1} +3 z^4 a^4-5 z^2 a^4-a^4 z^{-2} +3 a^4+2 z^3 a^3+z^2 a^2$ (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 \
-6-5-4-3-2-10χ
-1      11
-3     21-1
-5    1  1
-7    2  2
-9  31   2
-11 13    2
-13       0
-151      1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

Computer Talk

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