L10a12

L10a11

L10a13

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a12 at Knotilus!

Link Presentations

[edit Notes on L10a12's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X18,11,19,12 X20,16,5,15 X16,20,17,19 X12,17,13,18 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 10, -2, 4, -7, 8, -3, 5, -6, 7, -4, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u v^3-6 u v^2+9 u v-3 u-3 v^3+9 v^2-6 v+1}{\sqrt{u} v^{3/2}}$ (db)
Jones polynomial	$-\frac{6}{q^{9/2}}-q^{7/2}+\frac{9}{q^{7/2}}+3 q^{5/2}-\frac{12}{q^{5/2}}-7 q^{3/2}+\frac{13}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+10 \sqrt{q}-\frac{12}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^7 z^{-1} -3 a^5 z-a^5 z^{-1} +3 a^3 z^3+a^3 z-a^3 z^{-1} -z a^{-3} -a z^5+2 z^3 a^{-1} +a z+2 a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-a^3 z^9-a z^9-3 a^4 z^8-7 a^2 z^8-4 z^8-3 a^5 z^7-8 a^3 z^7-10 a z^7-5 z^7 a^{-1} -2 a^6 z^6+2 a^4 z^6+10 a^2 z^6-3 z^6 a^{-2} +3 z^6-a^7 z^5+3 a^5 z^5+20 a^3 z^5+26 a z^5+9 z^5 a^{-1} -z^5 a^{-3} +3 a^6 z^4-a^4 z^4-7 a^2 z^4+5 z^4 a^{-2} +2 z^4+3 a^7 z^3+a^5 z^3-20 a^3 z^3-26 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} +3 a^4 z^2+6 a^2 z^2-2 z^2 a^{-2} +z^2-3 a^7 z-2 a^5 z+9 a^3 z+13 a z+4 z a^{-1} -z a^{-3} -a^6-2 a^4-3 a^2-1+a^7 z^{-1} +a^5 z^{-1} -a^3 z^{-1} -2 a z^{-1} - a^{-1} z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   2   -2 4                 5 1   4 2               5 2     -3 0             7 5       2 -2           7 6         -1 -4         5 6           -1 -6       4 7             3 -8     2 5               -3 -10   1 5                 4 -12   1                   -1 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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). Back to the top.

L10a11 L10a13