# L11a392

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a392 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 t(1) t(3)^3+2 t(2) t(3)^3-2 t(3)^3-7 t(1) t(3)^2+4 t(1) t(2) t(3)^2-7 t(2) t(3)^2+5 t(3)^2+7 t(1) t(3)-5 t(1) t(2) t(3)+7 t(2) t(3)-4 t(3)-2 t(1)+2 t(1) t(2)-2 t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $q^{-2} -4 q^{-3} +10 q^{-4} -13 q^{-5} +18 q^{-6} -18 q^{-7} +19 q^{-8} -14 q^{-9} +10 q^{-10} -6 q^{-11} +2 q^{-12} - q^{-13}$ (db) Signature -4 (db) HOMFLY-PT polynomial $-a^{14} z^{-2} +3 a^{12} z^{-2} +4 a^{12}-6 z^2 a^{10}-2 a^{10} z^{-2} -7 a^{10}+3 z^4 a^8+2 z^2 a^8-a^8 z^{-2} -a^8+4 z^4 a^6+7 z^2 a^6+a^6 z^{-2} +4 a^6+z^4 a^4$ (db) Kauffman polynomial $z^7 a^{15}-5 z^5 a^{15}+9 z^3 a^{15}-7 z a^{15}+2 a^{15} z^{-1} +2 z^8 a^{14}-7 z^6 a^{14}+7 z^4 a^{14}-2 z^2 a^{14}-a^{14} z^{-2} +a^{14}+2 z^9 a^{13}-z^7 a^{13}-17 z^5 a^{13}+35 z^3 a^{13}-27 z a^{13}+8 a^{13} z^{-1} +z^{10} a^{12}+6 z^8 a^{12}-25 z^6 a^{12}+24 z^4 a^{12}-7 z^2 a^{12}-3 a^{12} z^{-2} +5 a^{12}+7 z^9 a^{11}-7 z^7 a^{11}-27 z^5 a^{11}+50 z^3 a^{11}-34 z a^{11}+10 a^{11} z^{-1} +z^{10} a^{10}+15 z^8 a^{10}-40 z^6 a^{10}+27 z^4 a^{10}-8 z^2 a^{10}-2 a^{10} z^{-2} +4 a^{10}+5 z^9 a^9+8 z^7 a^9-35 z^5 a^9+26 z^3 a^9-10 z a^9+2 a^9 z^{-1} +11 z^8 a^8-12 z^6 a^8-2 z^4 a^8+4 z^2 a^8+a^8 z^{-2} -3 a^8+13 z^7 a^7-16 z^5 a^7+2 z^3 a^7+4 z a^7-2 a^7 z^{-1} +10 z^6 a^6-11 z^4 a^6+7 z^2 a^6+a^6 z^{-2} -4 a^6+4 z^5 a^5+z^4 a^4$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-3           11
-5          41-3
-7         6  6
-9        74  -3
-11       116   5
-13      1010    0
-15     98     1
-17    510      5
-19   59       -4
-21  15        4
-23 15         -4
-25 1          1
-271           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-7$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-6$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{11}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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