L11a32

L11a31

L11a33

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a32 at Knotilus!

Link Presentations

[edit Notes on L11a32's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,8,17,7 X22,18,5,17 X18,12,19,11 X12,22,13,21 X20,14,21,13 X14,20,15,19 X8,16,9,15 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -6, 7, -8, 9, -3, 4, -5, 8, -7, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(2)^{3}-6t(2)^{3}-9t(1)t(2)^{2}+12t(2)^{2}+12t(1)t(2)-9t(2)-6t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{15/2}-3q^{13/2}+7q^{11/2}-11q^{9/2}+16q^{7/2}-19q^{5/2}+18q^{3/2}-17{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}-2az^{3}-2z^{3}a^{-1}-z^{3}a^{-3}-2z^{3}a^{-5}+a^{3}z+az-4za^{-1}-za^{-3}+za^{-7}+2az^{-1}-2a^{-1}z^{-1}-a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+2z^{2}a^{-8}+3z^{7}a^{-7}-8z^{5}a^{-7}+5z^{3}a^{-7}-za^{-7}+5z^{8}a^{-6}-14z^{6}a^{-6}+15z^{4}a^{-6}-11z^{2}a^{-6}+4a^{-6}+4z^{9}a^{-5}-5z^{7}a^{-5}-4z^{5}a^{-5}+6z^{3}a^{-5}-za^{-5}-a^{-5}z^{-1}+z^{10}a^{-4}+12z^{8}a^{-4}-42z^{6}a^{-4}+56z^{4}a^{-4}-36z^{2}a^{-4}+9a^{-4}+8z^{9}a^{-3}-11z^{7}a^{-3}+a^{3}z^{5}-z^{5}a^{-3}-2a^{3}z^{3}+7z^{3}a^{-3}+a^{3}z-za^{-3}-a^{-3}z^{-1}+z^{10}a^{-2}+13z^{8}a^{-2}+3a^{2}z^{6}-35z^{6}a^{-2}-4a^{2}z^{4}+38z^{4}a^{-2}+a^{2}z^{2}-19z^{2}a^{-2}+4a^{-2}+4z^{9}a^{-1}+6az^{7}+3z^{7}a^{-1}-11az^{5}-17z^{5}a^{-1}+11az^{3}+19z^{3}a^{-1}-8az-10za^{-1}+2az^{-1}+2a^{-1}z^{-1}+6z^{8}-5z^{6}-4z^{4}+5z^{2}-2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     2   2 12                   5 1   -4 10                 6 2     4 8               10 5       -5 6             9 6         3 4           9 10           1 2         8 9             -1 0       5 10               5 -2     3 7                 -4 -4     5                   5 -6 1 3                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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