# L11a337

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a337 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1) t(2)^5-4 t(1)^2 t(2)^4+5 t(1) t(2)^4-t(2)^4-2 t(1)^3 t(2)^3+9 t(1)^2 t(2)^3-10 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-10 t(1)^2 t(2)^2+9 t(1) t(2)^2-2 t(2)^2-t(1)^3 t(2)+5 t(1)^2 t(2)-4 t(1) t(2)-t(1)^2}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{8}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{4}{q^{25/2}}+\frac{9}{q^{23/2}}-\frac{15}{q^{21/2}}+\frac{19}{q^{19/2}}-\frac{22}{q^{17/2}}+\frac{22}{q^{15/2}}-\frac{19}{q^{13/2}}+\frac{13}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z a^{13}-a^{13} z^{-1} +4 z^3 a^{11}+7 z a^{11}+3 a^{11} z^{-1} -3 z^5 a^9-7 z^3 a^9-5 z a^9-2 a^9 z^{-1} -3 z^5 a^7-7 z^3 a^7-4 z a^7-z^5 a^5-2 z^3 a^5-z a^5$ (db) Kauffman polynomial $-z^6 a^{16}+2 z^4 a^{16}-z^2 a^{16}-4 z^7 a^{15}+9 z^5 a^{15}-6 z^3 a^{15}+z a^{15}-7 z^8 a^{14}+15 z^6 a^{14}-8 z^4 a^{14}+2 z^2 a^{14}-a^{14}-6 z^9 a^{13}+6 z^7 a^{13}+8 z^5 a^{13}-6 z^3 a^{13}+a^{13} z^{-1} -2 z^{10} a^{12}-12 z^8 a^{12}+36 z^6 a^{12}-31 z^4 a^{12}+15 z^2 a^{12}-3 a^{12}-12 z^9 a^{11}+21 z^7 a^{11}-16 z^5 a^{11}+19 z^3 a^{11}-14 z a^{11}+3 a^{11} z^{-1} -2 z^{10} a^{10}-12 z^8 a^{10}+31 z^6 a^{10}-28 z^4 a^{10}+12 z^2 a^{10}-3 a^{10}-6 z^9 a^9+5 z^7 a^9-4 z^5 a^9+8 z^3 a^9-8 z a^9+2 a^9 z^{-1} -7 z^8 a^8+8 z^6 a^8-3 z^4 a^8-z^2 a^8-6 z^7 a^7+10 z^5 a^7-9 z^3 a^7+4 z a^7-3 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+2 z^3 a^5-z a^5$ (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
-4           11
-6          31-2
-8         5  5
-10        83  -5
-12       115   6
-14      118    -3
-16     1111     0
-18    811      3
-20   711       -4
-22  39        6
-24 16         -5
-26 3          3
-281           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-8$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=-7$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-6$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-5$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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