# L11n374

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n374 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(v-1) (w-1) \left(2 u w^2-u w+w-2\right)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $- q^{-9} +3 q^{-8} -5 q^{-7} +8 q^{-6} -8 q^{-5} +8 q^{-4} -6 q^{-3} +6 q^{-2} -2 q^{-1} +1$ (db) Signature -4 (db) HOMFLY-PT polynomial $-a^{10}+z^4 a^8+4 z^2 a^8+3 a^8-z^6 a^6-4 z^4 a^6-5 z^2 a^6+a^6 z^{-2} -2 a^6-z^6 a^4-3 z^4 a^4-2 z^2 a^4-2 a^4 z^{-2} -3 a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2$ (db) Kauffman polynomial $z^3 a^{11}-2 z a^{11}+3 z^4 a^{10}-4 z^2 a^{10}+a^{10}+2 z^7 a^9-7 z^5 a^9+14 z^3 a^9-6 z a^9+3 z^8 a^8-12 z^6 a^8+22 z^4 a^8-12 z^2 a^8+3 a^8+z^9 a^7+2 z^7 a^7-14 z^5 a^7+20 z^3 a^7-6 z a^7+5 z^8 a^6-15 z^6 a^6+14 z^4 a^6-5 z^2 a^6+a^6 z^{-2} -a^6+z^9 a^5+2 z^7 a^5-12 z^5 a^5+7 z^3 a^5+2 z a^5-2 a^5 z^{-1} +2 z^8 a^4-2 z^6 a^4-9 z^4 a^4+9 z^2 a^4+2 a^4 z^{-2} -6 a^4+2 z^7 a^3-5 z^5 a^3+4 z a^3-2 a^3 z^{-1} +z^6 a^2-4 z^4 a^2+6 z^2 a^2+a^2 z^{-2} -4 a^2$ (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 \
-7-6-5-4-3-2-1012χ
1         11
-1        1 -1
-3       51 4
-5      33  0
-7     53   2
-9   143    0
-11   55     0
-13  25      3
-15 13       -2
-17 2        2
-191         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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