# L10a101

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a101 at Knotilus! Contains two L4a1 configurations.  As a Celtic (or pseudo-Celtic) linear decorative knot  Rotated knotwork cross with four L10a101 sub-configurations

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)+t(2)) \left(2 t(2) t(1)^2-t(1)^2+2 t(2)^2 t(1)-4 t(2) t(1)+2 t(1)-t(2)^2+2 t(2)\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{1}{q^{23/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z a^{11}-z^3 a^9+z a^9+a^9 z^{-1} -3 z^3 a^7-3 z a^7-a^7 z^{-1} -3 z^3 a^5-3 z a^5-z^3 a^3$ (db) Kauffman polynomial $a^{13} z^7-5 a^{13} z^5+8 a^{13} z^3-4 a^{13} z+2 a^{12} z^8-9 a^{12} z^6+12 a^{12} z^4-4 a^{12} z^2+a^{11} z^9-11 a^{11} z^5+15 a^{11} z^3-3 a^{11} z+5 a^{10} z^8-15 a^{10} z^6+9 a^{10} z^4+a^{10} z^2+a^9 z^9+5 a^9 z^7-21 a^9 z^5+18 a^9 z^3-7 a^9 z+a^9 z^{-1} +3 a^8 z^8+a^8 z^6-15 a^8 z^4+8 a^8 z^2-a^8+6 a^7 z^7-9 a^7 z^5+4 a^7 z^3-5 a^7 z+a^7 z^{-1} +7 a^6 z^6-9 a^6 z^4+3 a^6 z^2+6 a^5 z^5-6 a^5 z^3+3 a^5 z+3 a^4 z^4+a^3 z^3$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-2          11
-4         31-2
-6        3  3
-8       43  -1
-10      53   2
-12     44    0
-14    45     -1
-16   24      2
-18  24       -2
-20 13        2
-22 1         -1
-241          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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