# L11a453

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a453 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(v w-v+1) (v w-w+1) (u v w-u w+u-v w+v-1)}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $q^7-3 q^6+7 q^5-12 q^4+16 q^3-16 q^2+18 q-14+11 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4}$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-4} +4 z^4 a^{-4} +6 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -z^8 a^{-2} -6 z^6 a^{-2} -a^2 z^4-15 z^4 a^{-2} -3 a^2 z^2-18 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2-9 a^{-2} +2 z^6+9 z^4+14 z^2+ z^{-2} +8$ (db) Kauffman polynomial $z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -2 z^3 a^{-7} +6 z^6 a^{-6} -6 z^4 a^{-6} +4 z^2 a^{-6} - a^{-6} +9 z^7 a^{-5} -14 z^5 a^{-5} +11 z^3 a^{-5} -z a^{-5} +9 z^8 a^{-4} -15 z^6 a^{-4} +11 z^4 a^{-4} -6 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +5 z^9 a^{-3} +a^3 z^7+z^7 a^{-3} -4 a^3 z^5-22 z^5 a^{-3} +5 a^3 z^3+22 z^3 a^{-3} -2 a^3 z-8 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +3 a^2 z^8+15 z^8 a^{-2} -12 a^2 z^6-51 z^6 a^{-2} +16 a^2 z^4+58 z^4 a^{-2} -9 a^2 z^2-37 z^2 a^{-2} -2 a^{-2} z^{-2} +3 a^2+11 a^{-2} +3 a z^9+8 z^9 a^{-1} -6 a z^7-15 z^7 a^{-1} -6 a z^5-7 z^5 a^{-1} +16 a z^3+20 z^3 a^{-1} -7 a z-12 z a^{-1} +2 a^{-1} z^{-1} +z^{10}+9 z^8-42 z^6+56 z^4-35 z^2- z^{-2} +11$ (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 \
-5-4-3-2-10123456χ
15           11
13          2 -2
11         51 4
9        83  -5
7       84   4
5      88    0
3     108     2
1    711      4
-1   47       -3
-3  27        5
-5 14         -3
-7 2          2
-91           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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