L10a120

L10a119

L10a121

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a120 at Knotilus!

Link Presentations

[edit Notes on L10a120's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X6,11,7,12 X20,7,11,8 X18,9,19,10 X10,17,1,18 X8,19,9,20 X4,15,5,16
Gauss code	{1, -2, 3, -10, 4, -5, 6, -9, 7, -8}, {5, -1, 2, -3, 10, -4, 8, -7, 9, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u^4 \left(-v^2\right)-u^3 v^3+u^3 v^2-u^3 v-u^2 v^4+u^2 v^3-u^2 v^2+u^2 v-u^2-u v^3+u v^2-u v-v^2}{u^2 v^2}$ (db)
Jones polynomial	$-\frac{2}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{1}{q^{23/2}}-\frac{2}{q^{21/2}}+\frac{3}{q^{19/2}}-\frac{4}{q^{17/2}}+\frac{4}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{11/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11} z^3+3 a^{11} z+a^{11} z^{-1} -a^9 z^5-4 a^9 z^3-4 a^9 z-a^9 z^{-1} -a^7 z^5-3 a^7 z^3-a^7 z-a^5 z^5-4 a^5 z^3-3 a^5 z$ (db)
Kauffman polynomial	$a^{15} z^5-4 a^{15} z^3+3 a^{15} z+a^{14} z^6-3 a^{14} z^4+a^{14} z^2+a^{13} z^7-3 a^{13} z^5+2 a^{13} z^3-a^{13} z+a^{12} z^8-4 a^{12} z^6+6 a^{12} z^4-3 a^{12} z^2+a^{11} z^9-6 a^{11} z^7+15 a^{11} z^5-15 a^{11} z^3+7 a^{11} z-a^{11} z^{-1} +2 a^{10} z^8-10 a^{10} z^6+18 a^{10} z^4-8 a^{10} z^2+a^{10}+a^9 z^9-6 a^9 z^7+15 a^9 z^5-15 a^9 z^3+7 a^9 z-a^9 z^{-1} +a^8 z^8-4 a^8 z^6+6 a^8 z^4-3 a^8 z^2+a^7 z^7-3 a^7 z^5+2 a^7 z^3-a^7 z+a^6 z^6-3 a^6 z^4+a^6 z^2+a^5 z^5-4 a^5 z^3+3 a^5 z$ (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 -1 0 χ -4                     1 1 -6                   1 1 0 -8                 1     1 -10               2 1     -1 -12             2 1       1 -14           2 2         0 -16         2 2           0 -18       1 2             1 -20     1 2               -1 -22     1                 1 -24 1 1                   0 -26 1                     1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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