L11n85

L11n84

L11n86

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n85 at Knotilus!

Link Presentations

[edit Notes on L11n85's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X9,18,10,19 X22,19,5,20 X20,15,21,16 X16,21,17,22 X17,8,18,9 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 8, -4, -9, 11, -2, 9, -3, 6, -7, -8, 4, 5, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u v^3-5 u v^2+6 u v-3 u-3 v^3+6 v^2-5 v+1}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$\frac{9}{q^{9/2}}-\frac{8}{q^{7/2}}+\frac{5}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{8}{q^{15/2}}+\frac{9}{q^{13/2}}-\frac{11}{q^{11/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11} z^{-1} +3 z a^9+2 a^9 z^{-1} -3 z^3 a^7-3 z a^7+z^5 a^5+z^3 a^5-a^5 z^{-1} -2 z^3 a^3-2 z a^3$ (db)
Kauffman polynomial	$-z^6 a^{12}+4 z^4 a^{12}-5 z^2 a^{12}+2 a^{12}-2 z^7 a^{11}+6 z^5 a^{11}-5 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -2 z^8 a^{10}+2 z^6 a^{10}+8 z^4 a^{10}-12 z^2 a^{10}+5 a^{10}-z^9 a^9-3 z^7 a^9+12 z^5 a^9-8 z^3 a^9+4 z a^9-2 a^9 z^{-1} -5 z^8 a^8+8 z^6 a^8+z^4 a^8-4 z^2 a^8+3 a^8-z^9 a^7-4 z^7 a^7+8 z^5 a^7-z^3 a^7-2 z a^7-3 z^8 a^6+4 z^6 a^6-7 z^4 a^6+6 z^2 a^6-a^6-3 z^7 a^5+2 z^5 a^5-z^3 a^5-2 z a^5+a^5 z^{-1} -z^6 a^4-4 z^4 a^4+3 z^2 a^4-3 z^3 a^3+2 z a^3$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 4 1 -3 -6               4 1   3 -8             5 4     -1 -10           6 4       2 -12         4 6         2 -14       4 5           -1 -16     1 4             3 -18   1 4               -3 -20   1                 1 -22 1                   -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

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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