L11n25

L11n24

L11n26

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n25 at Knotilus!

Link Presentations

[edit Notes on L11n25's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X20,9,21,10 X8493 X18,22,19,21 X11,14,12,15 X5,13,6,12 X13,5,14,22 X10,19,11,20 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, -1, 2, -5, 4, -10, -7, 8, -9, 7, 11, -2, 3, -6, 10, -4, 6, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (v-2) (v-1) (2 v-1)}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+6 q^{5/2}-10 q^{3/2}+12 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{2}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^5 z^{-1} +a^3 z^3+z^3 a^{-3} -2 a^3 z-2 a^3 z^{-1} - a^{-3} z^{-1} -a z^5-z^5 a^{-1} -z^3 a^{-1} +2 a z+a z^{-1} + a^{-1} z^{-1}$ (db)
Kauffman polynomial	$3 a^5 z^3-3 a^5 z+a^5 z^{-1} +a^4 z^6+z^6 a^{-4} +4 a^4 z^4-2 z^4 a^{-4} -3 a^4 z^2+a^4+4 a^3 z^7+4 z^7 a^{-3} -7 a^3 z^5-12 z^5 a^{-3} +14 a^3 z^3+8 z^3 a^{-3} -11 a^3 z+z a^{-3} +2 a^3 z^{-1} - a^{-3} z^{-1} +5 a^2 z^8+5 z^8 a^{-2} -12 a^2 z^6-14 z^6 a^{-2} +19 a^2 z^4+10 z^4 a^{-2} -13 a^2 z^2-4 z^2 a^{-2} +3 a^2+ a^{-2} +2 a z^9+2 z^9 a^{-1} +4 a z^7+4 z^7 a^{-1} -19 a z^5-24 z^5 a^{-1} +22 a z^3+19 z^3 a^{-1} -10 a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1} +10 z^8-28 z^6+27 z^4-14 z^2+2$ (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 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 3   3 6               3 1   -2 4             7 3     4 2           5 3       -2 0         7 7         0 -2       6 7           1 -4     3 5             -2 -6   2 6               4 -8   3                 -3 -10 2                   2

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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). Back to the top.

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