# L11a367

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a367 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^4 v^4-u^4 v^3-u^3 v^4+3 u^3 v^3-u^3 v^2-u^2 v^3+3 u^2 v^2-u^2 v-u v^2+3 u v-u-v+1}{u^2 v^2}$ (db) Jones polynomial $\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{4}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^7 z^7+6 a^7 z^5+11 a^7 z^3+7 a^7 z+a^7 z^{-1} -a^5 z^9-8 a^5 z^7-23 a^5 z^5-29 a^5 z^3-14 a^5 z-a^5 z^{-1} +a^3 z^7+6 a^3 z^5+10 a^3 z^3+4 a^3 z$ (db) Kauffman polynomial $a^{13} z^3-a^{13} z+2 a^{12} z^4-2 a^{12} z^2+2 a^{11} z^5-a^{11} z^3+2 a^{10} z^6-2 a^{10} z^4+a^{10} z^2+2 a^9 z^7-4 a^9 z^5+3 a^9 z^3-a^9 z+2 a^8 z^8-6 a^8 z^6+5 a^8 z^4-2 a^8 z^2+2 a^7 z^9-9 a^7 z^7+15 a^7 z^5-16 a^7 z^3+8 a^7 z-a^7 z^{-1} +a^6 z^{10}-3 a^6 z^8-2 a^6 z^6+9 a^6 z^4-7 a^6 z^2+a^6+4 a^5 z^9-24 a^5 z^7+48 a^5 z^5-42 a^5 z^3+15 a^5 z-a^5 z^{-1} +a^4 z^{10}-4 a^4 z^8+10 a^4 z^4-6 a^4 z^2+2 a^3 z^9-13 a^3 z^7+27 a^3 z^5-21 a^3 z^3+5 a^3 z+a^2 z^8-6 a^2 z^6+10 a^2 z^4-4 a^2 z^2$ (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 \
-8-7-6-5-4-3-2-10123χ
2           1-1
0          1 1
-2         11 0
-4        31  2
-6       22   0
-8      32    1
-10     22     0
-12    33      0
-14   12       1
-16  23        -1
-18 12         1
-20 1          -1
-221           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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