# L11a405

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a405 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(2)^2 t(3)^4+t(1) t(3)^4-t(1) t(2) t(3)^4+2 t(2) t(3)^4-t(3)^4-t(1) t(2)^2 t(3)^3+2 t(2)^2 t(3)^3-2 t(1) t(3)^3+4 t(1) t(2) t(3)^3-4 t(2) t(3)^3+t(3)^3+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+2 t(1) t(3)^2-4 t(1) t(2) t(3)^2+4 t(2) t(3)^2-t(3)^2-t(1) t(2)^2 t(3)+2 t(2)^2 t(3)-2 t(1) t(3)+4 t(1) t(2) t(3)-4 t(2) t(3)+t(3)+t(1) t(2)^2-t(2)^2+t(1)-2 t(1) t(2)+t(2)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $1-3 q^{-1} +7 q^{-2} -10 q^{-3} +16 q^{-4} -16 q^{-5} +18 q^{-6} -15 q^{-7} +11 q^{-8} -7 q^{-9} +3 q^{-10} - q^{-11}$ (db) Signature -4 (db) HOMFLY-PT polynomial $-z^2 a^{10}-a^{10} z^{-2} -2 a^{10}+3 z^4 a^8+9 z^2 a^8+4 a^8 z^{-2} +9 a^8-2 z^6 a^6-8 z^4 a^6-14 z^2 a^6-5 a^6 z^{-2} -14 a^6-z^6 a^4-z^4 a^4+5 z^2 a^4+2 a^4 z^{-2} +7 a^4+z^4 a^2+2 z^2 a^2$ (db) Kauffman polynomial $a^{13} z^5-2 a^{13} z^3+a^{13} z+3 a^{12} z^6-5 a^{12} z^4+a^{12} z^2+5 a^{11} z^7-8 a^{11} z^5+4 a^{11} z^3-4 a^{11} z+a^{11} z^{-1} +6 a^{10} z^8-11 a^{10} z^6+11 a^{10} z^4-10 a^{10} z^2-a^{10} z^{-2} +5 a^{10}+4 a^9 z^9-18 a^9 z^5+33 a^9 z^3-21 a^9 z+5 a^9 z^{-1} +a^8 z^{10}+13 a^8 z^8-48 a^8 z^6+72 a^8 z^4-48 a^8 z^2-4 a^8 z^{-2} +18 a^8+8 a^7 z^9-14 a^7 z^7-6 a^7 z^5+36 a^7 z^3-29 a^7 z+9 a^7 z^{-1} +a^6 z^{10}+12 a^6 z^8-50 a^6 z^6+77 a^6 z^4-57 a^6 z^2-5 a^6 z^{-2} +21 a^6+4 a^5 z^9-6 a^5 z^7-5 a^5 z^5+13 a^5 z^3-13 a^5 z+5 a^5 z^{-1} +5 a^4 z^8-15 a^4 z^6+18 a^4 z^4-18 a^4 z^2-2 a^4 z^{-2} +9 a^4+3 a^3 z^7-8 a^3 z^5+4 a^3 z^3+a^2 z^6-3 a^2 z^4+2 a^2 z^2$ (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 \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1          2 -2
-3         51 4
-5        63  -3
-7       104   6
-9      88    0
-11     108     2
-13    69      3
-15   59       -4
-17  26        4
-19 15         -4
-21 2          2
-231           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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