L11a450

L11a449

L11a451

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Link Presentations

[edit Notes on L11a450's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(v+w-1)(vw-v-w)(uvw-uw+u-vw+v-1)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+11q^{5}-15q^{4}+18q^{3}+q^{-3}-16q^{2}-3q^{-2}+16q+7q^{-1}-11$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-2z^{2}a^{-2}+5z^{2}a^{-4}-2z^{2}a^{-6}-3z^{2}+a^{2}-3a^{-2}+3a^{-4}-a^{-6}-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-6z^{4}a^{-8}+2z^{2}a^{-8}+5z^{7}a^{-7}-10z^{5}a^{-7}+7z^{3}a^{-7}-2za^{-7}+6z^{8}a^{-6}-13z^{6}a^{-6}+15z^{4}a^{-6}-8z^{2}a^{-6}+3a^{-6}+4z^{9}a^{-5}-2z^{7}a^{-5}-9z^{5}a^{-5}+19z^{3}a^{-5}-7za^{-5}+z^{10}a^{-4}+11z^{8}a^{-4}-36z^{6}a^{-4}+52z^{4}a^{-4}-36z^{2}a^{-4}-a^{-4}z^{-2}+11a^{-4}+7z^{9}a^{-3}-9z^{7}a^{-3}-6z^{5}a^{-3}+19z^{3}a^{-3}-12za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+9z^{8}a^{-2}+a^{2}z^{6}-28z^{6}a^{-2}-3a^{2}z^{4}+34z^{4}a^{-2}+3a^{2}z^{2}-29z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}+11a^{-2}+3z^{9}a^{-1}+3az^{7}+z^{7}a^{-1}-8az^{5}-16z^{5}a^{-1}+6az^{3}+15z^{3}a^{-1}-az-8za^{-1}+2a^{-1}z^{-1}+4z^{8}-7z^{6}-z^{-2}+3$ (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 17                       1 -1 15                     2   2 13                   4 1   -3 11                 7 2     5 9               9 5       -4 7             9 6         3 5           7 9           2 3         9 9             0 1       5 10               5 -1     2 6                 -4 -3   1 5                   4 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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