L9a37

L9a36

L9a38

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a37 at Knotilus!
	L9a37 is $9_{7}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a37's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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