# L11a423

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a423 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1) t(3)^4+t(1) t(2) t(3)^4-2 t(2) t(3)^4+t(3)^4+t(1) t(2)^2 t(3)^3-3 t(2)^2 t(3)^3+3 t(1) t(3)^3-4 t(1) t(2) t(3)^3+6 t(2) t(3)^3-2 t(3)^3-2 t(1) t(2)^2 t(3)^2+4 t(2)^2 t(3)^2-4 t(1) t(3)^2+7 t(1) t(2) t(3)^2-7 t(2) t(3)^2+2 t(3)^2+2 t(1) t(2)^2 t(3)-3 t(2)^2 t(3)+3 t(1) t(3)-6 t(1) t(2) t(3)+4 t(2) t(3)-t(3)-t(1) t(2)^2+t(2)^2+2 t(1) t(2)-t(2)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $-q^8+4 q^7-10 q^6+16 q^5-21 q^4+25 q^3+ q^{-3} -23 q^2-3 q^{-2} +21 q+9 q^{-1} -14$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-6} -z^2 a^{-6} - a^{-6} z^{-2} -2 a^{-6} +z^6 a^{-4} +2 z^4 a^{-4} +6 z^2 a^{-4} +4 a^{-4} z^{-2} +9 a^{-4} +z^6 a^{-2} -z^4 a^{-2} +a^2 z^2-8 z^2 a^{-2} -5 a^{-2} z^{-2} +a^2-11 a^{-2} -2 z^4-z^2+2 z^{-2} +3$ (db) Kauffman polynomial $z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -4 z^4 a^{-8} +9 z^7 a^{-7} -14 z^5 a^{-7} +9 z^3 a^{-7} -4 z a^{-7} + a^{-7} z^{-1} +12 z^8 a^{-6} -22 z^6 a^{-6} +19 z^4 a^{-6} -9 z^2 a^{-6} - a^{-6} z^{-2} +3 a^{-6} +8 z^9 a^{-5} -2 z^7 a^{-5} -20 z^5 a^{-5} +29 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +2 z^{10} a^{-4} +20 z^8 a^{-4} -54 z^6 a^{-4} +52 z^4 a^{-4} -29 z^2 a^{-4} -4 a^{-4} z^{-2} +15 a^{-4} +13 z^9 a^{-3} -14 z^7 a^{-3} -21 z^5 a^{-3} +45 z^3 a^{-3} -33 z a^{-3} +9 a^{-3} z^{-1} +2 z^{10} a^{-2} +14 z^8 a^{-2} +a^2 z^6-43 z^6 a^{-2} -3 a^2 z^4+48 z^4 a^{-2} +3 a^2 z^2-38 z^2 a^{-2} -5 a^{-2} z^{-2} -a^2+20 a^{-2} +5 z^9 a^{-1} +3 a z^7-6 a z^5-22 z^5 a^{-1} +3 a z^3+29 z^3 a^{-1} -18 z a^{-1} +5 a^{-1} z^{-1} +6 z^8-14 z^6+16 z^4-15 z^2-2 z^{-2} +8$ (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 \
-4-3-2-101234567χ
17           1-1
15          3 3
13         71 -6
11        93  6
9       127   -5
7      139    4
5     1214     2
3    911      -2
1   613       7
-1  38        -5
-3  6         6
-513          -2
-71           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ $r=3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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