L10a97

L10a96

L10a98

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a97 at Knotilus!

Link Presentations

[edit Notes on L10a97's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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