L10a25

L10a24

L10a26

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a25 at Knotilus!

Link Presentations

[edit Notes on L10a25's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u v^4-6 u v^3+6 u v^2-4 u v+u+v^5-4 v^4+6 v^3-6 v^2+2 v}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$\frac{13}{q^{9/2}}-\frac{12}{q^{7/2}}+\frac{9}{q^{5/2}}-\frac{7}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{9}{q^{13/2}}-\frac{12}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a^9+a^9 z^{-1} -3 z^3 a^7-6 z a^7-3 a^7 z^{-1} +2 z^5 a^5+6 z^3 a^5+8 z a^5+4 a^5 z^{-1} +z^5 a^3-4 z a^3-2 a^3 z^{-1} -z^3 a-z a$ (db)
Kauffman polynomial	$-z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+6 z^4 a^{10}-3 z^2 a^{10}+a^{10}-4 z^7 a^9+6 z^5 a^9-2 z^3 a^9+2 z a^9-a^9 z^{-1} -3 z^8 a^8-z^6 a^8+11 z^4 a^8-10 z^2 a^8+3 a^8-z^9 a^7-9 z^7 a^7+25 z^5 a^7-27 z^3 a^7+13 z a^7-3 a^7 z^{-1} -7 z^8 a^6+9 z^6 a^6+2 z^4 a^6-10 z^2 a^6+3 a^6-z^9 a^5-10 z^7 a^5+29 z^5 a^5-33 z^3 a^5+18 z a^5-4 a^5 z^{-1} -4 z^8 a^4+4 z^6 a^4+2 z^4 a^4-4 z^2 a^4+2 a^4-5 z^7 a^3+10 z^5 a^3-8 z^3 a^3+7 z a^3-2 a^3 z^{-1} -3 z^6 a^2+5 z^4 a^2-z^2 a^2-z^5 a+2 z^3 a-z a$ (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 2                     1 1 0                   2   -2 -2                 5 1   4 -4               5 3     -2 -6             7 4       3 -8           6 5         -1 -10         6 7           -1 -12       4 7             3 -14     2 5               -3 -16   1 4                 3 -18   2                   -2 -20 1                     1

Integral Khovanov Homology (db, data source)

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

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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