L10a26

L10a25

L10a27

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a26 at Knotilus!

Link Presentations

[edit Notes on L10a26's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(2)^5+2 t(1) t(2)^4-4 t(2)^4-6 t(1) t(2)^3+8 t(2)^3+8 t(1) t(2)^2-6 t(2)^2-4 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{10}{q^{9/2}}+\frac{12}{q^{7/2}}+q^{5/2}-\frac{15}{q^{5/2}}-4 q^{3/2}+\frac{13}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{6}{q^{11/2}}+8 \sqrt{q}-\frac{11}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z a^7-a^7 z^{-1} +3 z^3 a^5+6 z a^5+4 a^5 z^{-1} -2 z^5 a^3-6 z^3 a^3-9 z a^3-4 a^3 z^{-1} -z^5 a+2 z a+a z^{-1} +z^3 a^{-1}$ (db)
Kauffman polynomial	$-a^5 z^9-a^3 z^9-3 a^6 z^8-8 a^4 z^8-5 a^2 z^8-3 a^7 z^7-10 a^5 z^7-16 a^3 z^7-9 a z^7-a^8 z^6+3 a^6 z^6+8 a^4 z^6-4 a^2 z^6-8 z^6+9 a^7 z^5+34 a^5 z^5+41 a^3 z^5+12 a z^5-4 z^5 a^{-1} +3 a^8 z^4+9 a^6 z^4+16 a^4 z^4+20 a^2 z^4-z^4 a^{-2} +9 z^4-9 a^7 z^3-34 a^5 z^3-34 a^3 z^3-7 a z^3+2 z^3 a^{-1} -3 a^8 z^2-13 a^6 z^2-21 a^4 z^2-14 a^2 z^2-3 z^2+4 a^7 z+16 a^5 z+15 a^3 z+3 a z+a^8+4 a^6+7 a^4+4 a^2+1-a^7 z^{-1} -4 a^5 z^{-1} -4 a^3 z^{-1} -a z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                     1 -1 4                   3   3 2                 5 1   -4 0               6 3     3 -2             8 6       -2 -4           7 5         2 -6         5 8           3 -8       5 7             -2 -10     2 6               4 -12   1 4                 -3 -14   2                   2 -16 1                     -1

Integral Khovanov Homology (db, data source)

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

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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