# L11a294

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a294 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^5+t(1)^3 t(2)^4-3 t(1)^2 t(2)^4+2 t(1) t(2)^4-2 t(1)^3 t(2)^3+5 t(1)^2 t(2)^3-4 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-4 t(1)^2 t(2)^2+5 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-3 t(1) t(2)+t(2)+t(1)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $q^{3/2}-3 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{11}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z^5 a^7+4 z^3 a^7+5 z a^7+2 a^7 z^{-1} -z^7 a^5-5 z^5 a^5-10 z^3 a^5-10 z a^5-3 a^5 z^{-1} -z^7 a^3-4 z^5 a^3-4 z^3 a^3+a^3 z^{-1} +z^5 a+3 z^3 a+z a$ (db) Kauffman polynomial $-z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-2 z^6 a^{10}+4 z^4 a^{10}-z^2 a^{10}-3 z^7 a^9+6 z^5 a^9-4 z^3 a^9+2 z a^9-3 z^8 a^8+5 z^6 a^8-4 z^4 a^8+2 z^2 a^8-3 z^9 a^7+9 z^7 a^7-20 z^5 a^7+21 z^3 a^7-10 z a^7+2 a^7 z^{-1} -z^{10} a^6-3 z^8 a^6+16 z^6 a^6-30 z^4 a^6+18 z^2 a^6-3 a^6-6 z^9 a^5+22 z^7 a^5-39 z^5 a^5+34 z^3 a^5-16 z a^5+3 a^5 z^{-1} -z^{10} a^4-4 z^8 a^4+23 z^6 a^4-36 z^4 a^4+20 z^2 a^4-3 a^4-3 z^9 a^3+7 z^7 a^3-2 z^5 a^3-z^3 a^3-z a^3+a^3 z^{-1} -4 z^8 a^2+13 z^6 a^2-11 z^4 a^2+4 z^2 a^2-a^2-3 z^7 a+10 z^5 a-7 z^3 a+z a-z^6+3 z^4-z^2$ (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
4           1-1
2          2 2
0         31 -2
-2        52  3
-4       74   -3
-6      64    2
-8     67     1
-10    56      -1
-12   36       3
-14  25        -3
-16  3         3
-1812          -1
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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