L11a478

L11a477

L11a479

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Link Presentations

[edit Notes on L11a478's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)\left(v^{2}w^{2}-2v^{2}w+vw^{3}-3vw^{2}+3vw-v+2w^{2}-w\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-9}-2q^{-8}+6q^{-7}-10q^{-6}+15q^{-5}-17q^{-4}+19q^{-3}-q^{2}-16q^{-2}+4q+13q^{-1}-8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}z^{-2}+2a^{8}-2z^{4}a^{6}-5z^{2}a^{6}-2a^{6}z^{-2}-6a^{6}+z^{6}a^{4}+2z^{4}a^{4}+3z^{2}a^{4}+a^{4}z^{-2}+3a^{4}+z^{6}a^{2}+2z^{4}a^{2}+2z^{2}a^{2}+a^{2}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$z^{6}a^{10}-4z^{4}a^{10}+5z^{2}a^{10}-2a^{10}+2z^{7}a^{9}-5z^{5}a^{9}+2z^{3}a^{9}+za^{9}+3z^{8}a^{8}-6z^{6}a^{8}+3z^{4}a^{8}-3z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+3z^{9}a^{7}-4z^{7}a^{7}+4z^{3}a^{7}-7za^{7}+2a^{7}z^{-1}+z^{10}a^{6}+8z^{8}a^{6}-27z^{6}a^{6}+37z^{4}a^{6}-27z^{2}a^{6}-2a^{6}z^{-2}+11a^{6}+7z^{9}a^{5}-9z^{7}a^{5}-2z^{5}a^{5}+12z^{3}a^{5}-9za^{5}+2a^{5}z^{-1}+z^{10}a^{4}+12z^{8}a^{4}-32z^{6}a^{4}+36z^{4}a^{4}-21z^{2}a^{4}-a^{4}z^{-2}+7a^{4}+4z^{9}a^{3}+4z^{7}a^{3}-19z^{5}a^{3}+15z^{3}a^{3}-2za^{3}+7z^{8}a^{2}-8z^{6}a^{2}+7z^{7}a-11z^{5}a+4z^{3}a-za+4z^{6}-6z^{4}+2z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     3   3 1                   5 1   -4 -1                 8 3     5 -3               10 7       -3 -5             9 6         3 -7           8 10           2 -9         7 9             -2 -11       3 8               5 -13     3 7                 -4 -15   1 5                   4 -17   1                     -1 -19 1                       1

Integral Khovanov Homology (db, data source)

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

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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