# L10n85

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## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n85 at Knotilus!
 Link L10n85. A graph, L10n85. A part of a link and a part of a graph.

### Link Presentations

 Planar diagram presentation X6172 X10,3,11,4 X20,14,15,13 X7,16,8,17 X15,8,16,9 X18,12,19,11 X12,20,13,19 X14,18,5,17 X2536 X4,9,1,10 Gauss code {1, -9, 2, -10}, {-5, 4, 8, -6, 7, -3}, {9, -1, -4, 5, 10, -2, 6, -7, 3, -8}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(w-1) \left(u v^2 w-u v w+2 u v-u-v^2 w+2 v w-v+1\right)}{\sqrt{u} v w}$ (db) Jones polynomial $q^4+2 q^{-4} -3 q^3-3 q^{-3} +5 q^2+6 q^{-2} -6 q-6 q^{-1} +8$ (db) Signature 0 (db) HOMFLY-PT polynomial $-z^6+a^2 z^4+z^4 a^{-2} -4 z^4+a^2 z^2+2 z^2 a^{-2} -5 z^2+a^4-2 a^2+ a^{-2} +a^4 z^{-2} -2 a^2 z^{-2} + z^{-2}$ (db) Kauffman polynomial $3 a^4 z^4+z^4 a^{-4} -7 a^4 z^2-z^2 a^{-4} -a^4 z^{-2} +4 a^4+a^3 z^7-a^3 z^5+3 z^5 a^{-3} +2 a^3 z^3-4 z^3 a^{-3} -5 a^3 z+2 a^3 z^{-1} +a^2 z^8-a^2 z^6+4 z^6 a^{-2} +4 a^2 z^4-6 z^4 a^{-2} -8 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +6 a^2- a^{-2} +4 a z^7+3 z^7 a^{-1} -7 a z^5-3 z^5 a^{-1} +7 a z^3+z^3 a^{-1} -5 a z+2 a z^{-1} +z^8+3 z^6-6 z^4+3 z^2- z^{-2} +2$ (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 \
-4-3-2-101234χ
9        11
7       2 -2
5      31 2
3     43  -1
1    42   2
-1   35    2
-3  33     0
-5 14      3
-712       -1
-92        2
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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

### Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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