L9a36

L9a35

L9a37

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a36 at Knotilus!
	L9a36 is $9_{1}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a36's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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