# L10n112

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n112 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1) t(2)+t(1) t(3) t(2)-t(3) t(2)+t(1) t(4) t(2)-t(1) t(3) t(4) t(2)+t(3) t(4) t(2)+2 t(1) t(5) t(2)-t(1) t(3) t(5) t(2)+t(3) t(5) t(2)-t(1) t(4) t(5) t(2)-t(5) t(2)+t(3)-t(1) t(4)+t(1) t(3) t(4)-2 t(3) t(4)+t(4)-t(1) t(5)-t(3) t(5)+t(1) t(4) t(5)+t(3) t(4) t(5)-t(4) t(5)+t(5)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)} \sqrt{t(5)}}$ (db) Jones polynomial $q^9-q^8+6 q^7-5 q^6+11 q^5-5 q^4+10 q^3-5 q^2+4 q$ (db) Signature 2 (db) HOMFLY-PT polynomial $a^{-10} z^{-4} + a^{-10} z^{-2} -4 a^{-8} z^{-4} -7 a^{-8} z^{-2} -4 a^{-8} +6 a^{-6} z^{-4} +6 z^2 a^{-6} +15 a^{-6} z^{-2} +14 a^{-6} -3 z^4 a^{-4} -4 a^{-4} z^{-4} -10 z^2 a^{-4} -13 a^{-4} z^{-2} -16 a^{-4} + a^{-2} z^{-4} +4 z^2 a^{-2} +4 a^{-2} z^{-2} +6 a^{-2}$ (db) Kauffman polynomial $z^8 a^{-6} +z^8 a^{-8} +5 z^7 a^{-5} +6 z^7 a^{-7} +z^7 a^{-9} +10 z^6 a^{-4} +12 z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +6 z^5 a^{-3} -6 z^5 a^{-7} -25 z^4 a^{-4} -39 z^4 a^{-6} -19 z^4 a^{-8} -5 z^4 a^{-10} -10 z^3 a^{-3} -30 z^3 a^{-5} -30 z^3 a^{-7} -10 z^3 a^{-9} +10 z^2 a^{-2} +30 z^2 a^{-4} +40 z^2 a^{-6} +30 z^2 a^{-8} +10 z^2 a^{-10} +20 z a^{-3} +55 z a^{-5} +55 z a^{-7} +20 z a^{-9} -10 a^{-2} -25 a^{-4} -31 a^{-6} -25 a^{-8} -10 a^{-10} -15 a^{-3} z^{-1} -41 a^{-5} z^{-1} -41 a^{-7} z^{-1} -15 a^{-9} z^{-1} +5 a^{-2} z^{-2} +14 a^{-4} z^{-2} +18 a^{-6} z^{-2} +14 a^{-8} z^{-2} +5 a^{-10} z^{-2} +4 a^{-3} z^{-3} +12 a^{-5} z^{-3} +12 a^{-7} z^{-3} +4 a^{-9} z^{-3} - a^{-2} z^{-4} -4 a^{-4} z^{-4} -6 a^{-6} z^{-4} -4 a^{-8} z^{-4} - a^{-10} z^{-4}$ (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 \
012345678χ
19        11
17       110
15      5  5
13      1  1
11    115   6
9   410    6
7  61     5
51 4      5
356       -1
14        4
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $i=5$ $r=0$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{5}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{11}$ $r=5$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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