L11a3

L11a2

L11a4

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Link Presentations

[edit Notes on L11a3's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X14,10,15,9 X8493 X10,5,11,6 X20,12,21,11 X22,20,5,19 X18,13,19,14 X12,22,13,21 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -10, 9, -4, 11, -2, 3, -9, 8, -7, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-5t(2)^{3}+7t(2)^{2}-5t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+5q^{11/2}-10q^{9/2}+16q^{7/2}-22q^{5/2}+24q^{3/2}-25{\sqrt {q}}+{\frac {20}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+a^{-5}z^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-a^{3}z-za^{-3}-3a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+4az^{3}-4z^{3}a^{-1}+2az+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-6az^{9}-13z^{9}a^{-1}-7z^{9}a^{-3}-7a^{2}z^{8}-19z^{8}a^{-2}-11z^{8}a^{-4}-15z^{8}-4a^{3}z^{7}+4az^{7}+13z^{7}a^{-1}-5z^{7}a^{-3}-10z^{7}a^{-5}-a^{4}z^{6}+15a^{2}z^{6}+43z^{6}a^{-2}+12z^{6}a^{-4}-5z^{6}a^{-6}+42z^{6}+9a^{3}z^{5}+14az^{5}+16z^{5}a^{-1}+26z^{5}a^{-3}+14z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-9a^{2}z^{4}-27z^{4}a^{-2}-z^{4}a^{-4}+5z^{4}a^{-6}-32z^{4}-6a^{3}z^{3}-16az^{3}-20z^{3}a^{-1}-15z^{3}a^{-3}-5z^{3}a^{-5}-a^{4}z^{2}+2a^{2}z^{2}+7z^{2}a^{-2}+3z^{2}a^{-4}+7z^{2}+2a^{3}z+4az+za^{-1}-za^{-3}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     4   -4 10                   6 1   5 8                 10 4     -6 6               12 6       6 4             12 10         -2 2           13 12           1 0         9 14             5 -2       6 11               -5 -4     3 9                 6 -6   1 6                   -5 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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