# L10a172

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a172 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1) t(4)^2 t(3)^2+t(1) t(2) t(4)^2 t(3)^2-t(2) t(4)^2 t(3)^2+t(4)^2 t(3)^2-t(2) t(3)^2-t(1) t(2) t(4) t(3)^2+2 t(2) t(4) t(3)^2-t(4) t(3)^2+2 t(1) t(4)^2 t(3)-t(1) t(2) t(4)^2 t(3)-t(4)^2 t(3)-t(1) t(2) t(3)+2 t(2) t(3)-t(1) t(4) t(3)+2 t(1) t(2) t(4) t(3)-t(2) t(4) t(3)+2 t(4) t(3)-t(3)-t(1) t(4)^2-t(1)+t(1) t(2)-t(2)+2 t(1) t(4)-t(1) t(2) t(4)-t(4)+1}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $8 q^{9/2}-11 q^{7/2}+9 q^{5/2}-\frac{1}{q^{5/2}}-11 q^{3/2}+\frac{2}{q^{3/2}}-q^{15/2}+3 q^{13/2}-6 q^{11/2}+6 \sqrt{q}-\frac{6}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-z^5 a^{-5} -3 z^3 a^{-5} - a^{-5} z^{-3} -3 z a^{-5} -2 a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +10 z^3 a^{-3} +3 a^{-3} z^{-3} +11 z a^{-3} +7 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a z-11 z a^{-1} +3 a z^{-1} -8 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^3 a^{-9} +3 z^4 a^{-8} +6 z^5 a^{-7} -5 z^3 a^{-7} +2 z a^{-7} +8 z^6 a^{-6} -10 z^4 a^{-6} +2 z^2 a^{-6} +9 z^7 a^{-5} -20 z^5 a^{-5} +17 z^3 a^{-5} - a^{-5} z^{-3} -12 z a^{-5} +5 a^{-5} z^{-1} +5 z^8 a^{-4} -4 z^6 a^{-4} -17 z^4 a^{-4} +20 z^2 a^{-4} +3 a^{-4} z^{-2} -10 a^{-4} +z^9 a^{-3} +12 z^7 a^{-3} -51 z^5 a^{-3} +61 z^3 a^{-3} -3 a^{-3} z^{-3} -35 z a^{-3} +12 a^{-3} z^{-1} +7 z^8 a^{-2} -19 z^6 a^{-2} +26 z^2 a^{-2} +6 a^{-2} z^{-2} -19 a^{-2} +z^9 a^{-1} +a z^7+4 z^7 a^{-1} -5 a z^5-30 z^5 a^{-1} +10 a z^3+48 z^3 a^{-1} -a z^{-3} -3 a^{-1} z^{-3} -10 a z-31 z a^{-1} +5 a z^{-1} +12 a^{-1} z^{-1} +2 z^8-7 z^6+4 z^4+8 z^2+3 z^{-2} -10$ (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 \
-4-3-2-10123456χ
16          11
14         31-2
12        3  3
10       53  -2
8      63   3
6     57    2
4    64     2
2   49      5
0  22       0
-2 15        4
-4 1         -1
-61          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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