L11n308

L11n307

L11n309

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Link Presentations

[edit Notes on L11n308's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,17,8,16 X9,21,10,20 X15,9,16,8 X19,5,20,10 X13,19,14,18 X17,11,18,22 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 5, -4, 6}, {11, -2, -7, 9, -5, 3, -8, 7, -6, 4, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)t(3)^{4}-2t(2)^{2}t(3)^{3}+2t(2)t(3)^{3}+t(3)^{3}+t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)+2t(1)t(3)-2t(1)t(2)t(3)+t(1)t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{7}+q^{6}-q^{4}+3q^{3}+q^{-3}-3q^{2}-2q^{-2}+5q+4q^{-1}-3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-6}-a^{-6}z^{-2}-2a^{-6}+z^{4}a^{-4}+6z^{2}a^{-4}+4a^{-4}z^{-2}+9a^{-4}-2z^{4}a^{-2}+a^{2}z^{2}-8z^{2}a^{-2}-5a^{-2}z^{-2}+a^{2}-11a^{-2}-z^{4}-z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$z^{7}a^{-7}-6z^{5}a^{-7}+9z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+z^{8}a^{-6}-7z^{6}a^{-6}+13z^{4}a^{-6}-9z^{2}a^{-6}-a^{-6}z^{-2}+3a^{-6}+z^{7}a^{-5}-10z^{5}a^{-5}+23z^{3}a^{-5}-19za^{-5}+5a^{-5}z^{-1}+2z^{8}a^{-4}-15z^{6}a^{-4}+33z^{4}a^{-4}-29z^{2}a^{-4}-4a^{-4}z^{-2}+15a^{-4}+z^{9}a^{-3}-3z^{7}a^{-3}-10z^{5}a^{-3}+37z^{3}a^{-3}-33za^{-3}+9a^{-3}z^{-1}+4z^{8}a^{-2}+a^{2}z^{6}-23z^{6}a^{-2}-4a^{2}z^{4}+43z^{4}a^{-2}+3a^{2}z^{2}-38z^{2}a^{-2}-5a^{-2}z^{-2}-a^{2}+20a^{-2}+z^{9}a^{-1}+2az^{7}-z^{7}a^{-1}-7az^{5}-13z^{5}a^{-1}+3az^{3}+26z^{3}a^{-1}-18za^{-1}+5a^{-1}z^{-1}+3z^{8}-14z^{6}+19z^{4}-15z^{2}-2z^{-2}+8$ (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 15                       1 -1 13                         0 11                 1 1 1   1 9               2 1       -1 7             2 1 1       2 5           4 5 1         0 3         2 1 1           2 1       2 5 1             2 -1     2 2 1               1 -3     2                   2 -5 1 2                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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