L11n338

L11n337

L11n339

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Link Presentations

[edit Notes on L11n338's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,18,12,19 X7,14,8,15 X13,8,14,9 X22,20,13,19 X20,16,21,15 X16,22,17,21 X17,12,18,5 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, 7, -8, -9, 3, 6, -7, 8, -6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(2)-1)\left(t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}+2t(1)t(2)t(3)-2t(2)t(3)+t(2)^{2}+t(2)\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{2}+3q-5+6q^{-1}-6q^{-2}+6q^{-3}-4q^{-4}+3q^{-5}+q^{-6}+q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$2a^{8}z^{-2}+a^{8}-2a^{6}z^{2}-5a^{6}z^{-2}-7a^{6}+3a^{4}z^{2}+4a^{4}z^{-2}+6a^{4}+a^{2}z^{6}+4a^{2}z^{4}+5a^{2}z^{2}-a^{2}z^{-2}+a^{2}-z^{4}-2z^{2}-1$ (db)
Kauffman polynomial	$a^{8}z^{8}-8a^{8}z^{6}+21a^{8}z^{4}-24a^{8}z^{2}-2a^{8}z^{-2}+12a^{8}+a^{7}z^{7}-10a^{7}z^{5}+25a^{7}z^{3}-21a^{7}z+5a^{7}z^{-1}+a^{6}z^{8}-11a^{6}z^{6}+35a^{6}z^{4}-41a^{6}z^{2}-5a^{6}z^{-2}+23a^{6}+2a^{5}z^{7}-15a^{5}z^{5}+42a^{5}z^{3}-35a^{5}z+9a^{5}z^{-1}+a^{4}z^{8}-3a^{4}z^{6}+9a^{4}z^{4}-12a^{4}z^{2}-4a^{4}z^{-2}+12a^{4}+4a^{3}z^{7}-11a^{3}z^{5}+17a^{3}z^{3}-15a^{3}z+5a^{3}z^{-1}+a^{2}z^{8}+3a^{2}z^{6}-12a^{2}z^{4}+8a^{2}z^{2}-a^{2}z^{-2}-a^{2}+3az^{7}-5az^{5}+z^{5}a^{-1}-2az^{3}-2z^{3}a^{-1}+az^{-1}+za^{-1}+3z^{6}-7z^{4}+3z^{2}-1$ (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                     2   2 1                   3 1   -2 -1                 3 2     1 -3               4 4       0 -5           1 3 2         0 -7           2 4           2 -9       1 3 3             -1 -11         4               4 -13     1                   1 -15 1                       1 -17 1                       1

Integral Khovanov Homology (db, data source)

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

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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