L10a92

L10a91

L10a93

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a92 at Knotilus!

Link Presentations

[edit Notes on L10a92's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

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