# L8a21

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a21 at Knotilus! L8a21 is a closed four-link chain. It is $8^4_{1}$ in the Rolfsen table of links.  Floor decoration in the Biblioteca Medicea Laurenziana, Florence, intended to contain four rings interlinked in this maner (but there is an interlacing error at upper right).

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2) t(1)-t(2) t(3) t(1)+t(3) t(1)-t(2) t(4) t(1)-t(3) t(4) t(1)+2 t(4) t(1)-t(1)-t(2)+2 t(2) t(3)-t(3)+t(2) t(4)-t(2) t(3) t(4)+t(3) t(4)-t(4)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}}$ (db) Jones polynomial $\frac{4}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{7}{q^{11/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^{11} z^{-3} -3 a^9 z^{-3} -4 a^9 z^{-1} +3 a^7 z^{-3} +6 a^7 z+8 a^7 z^{-1} -3 a^5 z^3-a^5 z^{-3} -6 a^5 z-4 a^5 z^{-1} -a^3 z^3$ (db) Kauffman polynomial $-z^5 a^{11}+4 z^3 a^{11}-6 z a^{11}+4 a^{11} z^{-1} -a^{11} z^{-3} -z^6 a^{10}+6 z^2 a^{10}+3 a^{10} z^{-2} -8 a^{10}-z^7 a^9-2 z^5 a^9+11 z^3 a^9-14 z a^9+9 a^9 z^{-1} -3 a^9 z^{-3} -5 z^6 a^8+5 z^4 a^8+12 z^2 a^8+6 a^8 z^{-2} -15 a^8-z^7 a^7-7 z^5 a^7+17 z^3 a^7-14 z a^7+9 a^7 z^{-1} -3 a^7 z^{-3} -4 z^6 a^6+2 z^4 a^6+6 z^2 a^6+3 a^6 z^{-2} -8 a^6-6 z^5 a^5+9 z^3 a^5-6 z a^5+4 a^5 z^{-1} -a^5 z^{-3} -3 z^4 a^4-z^3 a^3$ (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 \
-8-7-6-5-4-3-2-10χ
-2        11
-4       31-2
-6      3  3
-8     13  2
-10    63   3
-12   47    3
-14  1      1
-16  4      4
-1811       0
-201        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{7}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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