# L11a547

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a547 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(4)^2 t(3)^2+2 t(2) t(4)^2 t(3)^2-2 t(4)^2 t(3)^2+t(1) t(3)^2-t(1) t(2) t(3)^2+t(2) t(3)^2-2 t(1) t(4) t(3)^2+2 t(1) t(2) t(4) t(3)^2-3 t(2) t(4) t(3)^2+3 t(4) t(3)^2-t(3)^2-2 t(1) t(4)^2 t(3)+2 t(1) t(2) t(4)^2 t(3)-3 t(2) t(4)^2 t(3)+3 t(4)^2 t(3)-3 t(1) t(3)+3 t(1) t(2) t(3)-2 t(2) t(3)+5 t(1) t(4) t(3)-6 t(1) t(2) t(4) t(3)+5 t(2) t(4) t(3)-6 t(4) t(3)+2 t(3)+t(1) t(4)^2-t(1) t(2) t(4)^2+t(2) t(4)^2-t(4)^2+2 t(1)-2 t(1) t(2)+t(2)-3 t(1) t(4)+3 t(1) t(2) t(4)-2 t(2) t(4)+2 t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $-\frac{11}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{4}{q^{25/2}}+\frac{9}{q^{23/2}}-\frac{17}{q^{21/2}}+\frac{21}{q^{19/2}}-\frac{27}{q^{17/2}}+\frac{25}{q^{15/2}}-\frac{24}{q^{13/2}}+\frac{16}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z a^{13}+a^{13} z^{-3} +3 z^3 a^{11}+z a^{11}-5 a^{11} z^{-1} -3 a^{11} z^{-3} -2 z^5 a^9+z^3 a^9+11 z a^9+10 a^9 z^{-1} +3 a^9 z^{-3} -4 z^5 a^7-11 z^3 a^7-11 z a^7-5 a^7 z^{-1} -a^7 z^{-3} -z^5 a^5-z^3 a^5$ (db) Kauffman polynomial $a^{16} z^6-2 a^{16} z^4+a^{16} z^2+4 a^{15} z^7-9 a^{15} z^5+8 a^{15} z^3-3 a^{15} z+7 a^{14} z^8-13 a^{14} z^6+6 a^{14} z^4+a^{14} z^2+6 a^{13} z^9-23 a^{13} z^5+25 a^{13} z^3-a^{13} z^{-3} -12 a^{13} z+5 a^{13} z^{-1} +2 a^{12} z^{10}+18 a^{12} z^8-45 a^{12} z^6+26 a^{12} z^4+3 a^{12} z^2+3 a^{12} z^{-2} -10 a^{12}+14 a^{11} z^9-10 a^{11} z^7-26 a^{11} z^5+31 a^{11} z^3-3 a^{11} z^{-3} -21 a^{11} z+12 a^{11} z^{-1} +2 a^{10} z^{10}+23 a^{10} z^8-50 a^{10} z^6+22 a^{10} z^4+15 a^{10} z^2+6 a^{10} z^{-2} -19 a^{10}+8 a^9 z^9+4 a^9 z^7-30 a^9 z^5+32 a^9 z^3-3 a^9 z^{-3} -23 a^9 z+12 a^9 z^{-1} +12 a^8 z^8-15 a^8 z^6+a^8 z^4+12 a^8 z^2+3 a^8 z^{-2} -10 a^8+10 a^7 z^7-17 a^7 z^5+17 a^7 z^3-a^7 z^{-3} -11 a^7 z+5 a^7 z^{-1} +4 a^6 z^6-3 a^6 z^4+a^5 z^5-a^5 z^3$ (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          41-3
-8         7  7
-10        94  -5
-12       157   8
-14      1211    -1
-16     1513     2
-18    1016      6
-20   711       -4
-22  311        8
-24 16         -5
-26 3          3
-281           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-8$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=-7$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-6$ ${\mathbb Z}^{16}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{15}$ $r=-5$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{15}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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