L9a53

L9a52

L9a54

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a53 at Knotilus!
	L9a53 is $9_{12}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a53's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,10,15,9 X8493 X16,5,17,6 X14,17,5,18 X10,16,11,15 X2,12,3,11
Gauss code	{1, -9, 5, -3}, {8, -6, 7, -4}, {6, -1, 2, -5, 4, -8, 9, -2, 3, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+8q^{2}-9q+12-10q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}-a^{4}+2a^{2}z^{4}+z^{4}a^{-2}+4a^{2}z^{2}+z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+3a^{2}+a^{-2}-z^{6}-3z^{4}-4z^{2}-2z^{-2}-3$ (db)
Kauffman polynomial	$2a^{2}z^{8}+2z^{8}+4a^{3}z^{7}+11az^{7}+7z^{7}a^{-1}+3a^{4}z^{6}+7a^{2}z^{6}+8z^{6}a^{-2}+12z^{6}+a^{5}z^{5}-5a^{3}z^{5}-17az^{5}-7z^{5}a^{-1}+4z^{5}a^{-3}-6a^{4}z^{4}-22a^{2}z^{4}-11z^{4}a^{-2}+z^{4}a^{-4}-28z^{4}-2a^{5}z^{3}-a^{3}z^{3}+3az^{3}-2z^{3}a^{-3}+5a^{4}z^{2}+17a^{2}z^{2}+5z^{2}a^{-2}+17z^{2}+a^{5}z+3a^{3}z+3az+za^{-1}-2a^{4}-6a^{2}-2a^{-2}-5-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               5 1   4 3             4 3     -1 1           8 5       3 -1         6 8         2 -3       4 4           0 -5     2 6             4 -7   1 4               -3 -9   2                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

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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