L11n176

L11n175

L11n177

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n176 at Knotilus!

Link Presentations

[edit Notes on L11n176's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X14,5,15,6 X11,18,12,19 X3,10,4,11 X12,7,13,8 X16,13,17,14 X22,17,7,18 X6,15,1,16 X4,21,5,22 X19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {6, -1, 2, 5, -4, -6, 7, -3, 9, -7, 8, 4, -11, -2, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{2 u^2 v^4-u^2 v^3-u v^4+u v^3+u v^2+u v-u-v+2}{u v^2}$ (db)
Jones polynomial	$\frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{23/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13} (-z)+a^{11} z^5+5 a^{11} z^3+4 a^{11} z-a^{11} z^{-1} -a^9 z^7-5 a^9 z^5-5 a^9 z^3+2 a^9 z+3 a^9 z^{-1} -a^7 z^7-6 a^7 z^5-11 a^7 z^3-8 a^7 z-2 a^7 z^{-1}$ (db)
Kauffman polynomial	$-z^7 a^{13}+5 z^5 a^{13}-6 z^3 a^{13}+z a^{13}-2 z^8 a^{12}+11 z^6 a^{12}-17 z^4 a^{12}+8 z^2 a^{12}+a^{12}-z^9 a^{11}+4 z^7 a^{11}-2 z^5 a^{11}-z^3 a^{11}-a^{11} z^{-1} -3 z^8 a^{10}+15 z^6 a^{10}-18 z^4 a^{10}+3 z^2 a^{10}+3 a^{10}-z^9 a^9+4 z^7 a^9-z^5 a^9-6 z^3 a^9+7 z a^9-3 a^9 z^{-1} -z^8 a^8+4 z^6 a^8-z^4 a^8-5 z^2 a^8+3 a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+8 z a^7-2 a^7 z^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                 1 1 -8               1 1 0 -10             2     2 -12         1 1 1     1 -14         3 2       1 -16     1 2 1         0 -18     2 3           -1 -20   1 1             0 -22   1               -1 -24 1                 1

Integral Khovanov Homology (db, data source)

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

Computer Talk

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

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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