L10a32

L10a31

L10a33

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a32 at Knotilus!

Link Presentations

[edit Notes on L10a32's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-2)(t(2)-1)(2t(2)-1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-6q^{9/2}+9q^{7/2}-12q^{5/2}+12q^{3/2}-11{\sqrt {q}}+{\frac {9}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}-2az^{3}+2z^{3}a^{-3}-z^{3}a^{-5}+a^{3}z-2az-2za^{-1}+4za^{-3}-za^{-5}+a^{3}z^{-1}-az^{-1}-a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-7z^{8}a^{-2}-4z^{8}a^{-4}-3z^{8}-3az^{7}-7z^{7}a^{-1}-9z^{7}a^{-3}-5z^{7}a^{-5}-2a^{2}z^{6}+14z^{6}a^{-2}+6z^{6}a^{-4}-3z^{6}a^{-6}+3z^{6}-a^{3}z^{5}+3az^{5}+19z^{5}a^{-1}+28z^{5}a^{-3}+12z^{5}a^{-5}-z^{5}a^{-7}+3a^{2}z^{4}-16z^{4}a^{-2}-3z^{4}a^{-4}+6z^{4}a^{-6}-4z^{4}+3a^{3}z^{3}+2az^{3}-21z^{3}a^{-1}-33z^{3}a^{-3}-11z^{3}a^{-5}+2z^{3}a^{-7}+11z^{2}a^{-2}+z^{2}a^{-4}-2z^{2}a^{-6}+8z^{2}-3a^{3}z-3az+10za^{-1}+15za^{-3}+5za^{-5}-a^{2}-3a^{-2}-a^{-4}-2+a^{3}z^{-1}+az^{-1}-a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   2   -2 10                 4 1   3 8               5 2     -3 6             7 4       3 4           5 5         0 2         6 7           -1 0       5 7             2 -2     1 4               -3 -4   1 5                 4 -6   1                   -1 -8 1                     1

Integral Khovanov Homology (db, data source)

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

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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