# L10a153

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a153 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(w-1) \left(2 u v^2 w^2-u v w^2+2 u v w-u w+u+v^2 w^3-v^2 w^2+2 v w^2-v w+2 w\right)}{\sqrt{u} v w^2}$ (db) Jones polynomial $q^{-3} -2 q^{-4} +5 q^{-5} -6 q^{-6} +9 q^{-7} -9 q^{-8} +9 q^{-9} -7 q^{-10} +5 q^{-11} -2 q^{-12} + q^{-13}$ (db) Signature -6 (db) HOMFLY-PT polynomial $a^{12} z^2+a^{12} z^{-2} +3 a^{12}-3 a^{10} z^4-11 a^{10} z^2-2 a^{10} z^{-2} -10 a^{10}+2 a^8 z^6+9 a^8 z^4+12 a^8 z^2+a^8 z^{-2} +6 a^8+a^6 z^6+4 a^6 z^4+4 a^6 z^2+a^6$ (db) Kauffman polynomial $a^{16} z^4-2 a^{16} z^2+a^{16}+2 a^{15} z^5-2 a^{15} z^3+3 a^{14} z^6-3 a^{14} z^4+a^{14} z^2+3 a^{13} z^7-2 a^{13} z^5+3 a^{12} z^8-6 a^{12} z^6+11 a^{12} z^4-11 a^{12} z^2-a^{12} z^{-2} +6 a^{12}+a^{11} z^9+4 a^{11} z^7-17 a^{11} z^5+24 a^{11} z^3-13 a^{11} z+2 a^{11} z^{-1} +6 a^{10} z^8-21 a^{10} z^6+34 a^{10} z^4-32 a^{10} z^2-2 a^{10} z^{-2} +14 a^{10}+a^9 z^9+3 a^9 z^7-19 a^9 z^5+25 a^9 z^3-13 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+15 a^8 z^4-14 a^8 z^2-a^8 z^{-2} +7 a^8+2 a^7 z^7-6 a^7 z^5+3 a^7 z^3+a^6 z^6-4 a^6 z^4+4 a^6 z^2-a^6$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-5          11
-7         21-1
-9        3  3
-11       32  -1
-13      63   3
-15     44    0
-17    55     0
-19   35      2
-21  24       -2
-23  3        3
-2512         -1
-271          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-7$ $i=-5$ $r=-10$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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