L10a36

L10a35

L10a37

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a36 at Knotilus!

Link Presentations

[edit Notes on L10a36's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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