L11a442

L11a441

L11a443

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Link Presentations

[edit Notes on L11a442's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}w^{3}-uv^{3}w^{2}-uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w-uvw^{2}+2uvw-uv-uw+2u-2v^{3}w^{3}+v^{3}w^{2}+v^{2}w^{3}-2v^{2}w^{2}+v^{2}w+vw^{2}-2vw+v+w-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-4q^{7}+6q^{6}-7q^{5}+8q^{4}-7q^{3}+6q^{2}-4q+4-q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-6}-4z^{4}a^{-6}-3z^{2}a^{-6}+z^{8}a^{-4}+6z^{6}a^{-4}+12z^{4}a^{-4}+11z^{2}a^{-4}+a^{-4}z^{-2}+5a^{-4}-2z^{6}a^{-2}-11z^{4}a^{-2}-18z^{2}a^{-2}-2a^{-2}z^{-2}-11a^{-2}+z^{4}+5z^{2}+z^{-2}+6$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+z^{9}a^{-1}+4z^{9}a^{-3}+3z^{9}a^{-5}-3z^{8}a^{-2}+4z^{8}a^{-6}+z^{8}-4z^{7}a^{-1}-18z^{7}a^{-3}-10z^{7}a^{-5}+4z^{7}a^{-7}-5z^{6}a^{-2}-13z^{6}a^{-4}-11z^{6}a^{-6}+4z^{6}a^{-8}-7z^{6}+18z^{5}a^{-3}+8z^{5}a^{-5}-6z^{5}a^{-7}+4z^{5}a^{-9}+22z^{4}a^{-2}+19z^{4}a^{-4}+8z^{4}a^{-6}-3z^{4}a^{-8}+3z^{4}a^{-10}+17z^{4}+12z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}-3z^{3}a^{-9}+z^{3}a^{-11}-25z^{2}a^{-2}-10z^{2}a^{-4}-3z^{2}a^{-6}-2z^{2}a^{-8}-2z^{2}a^{-10}-18z^{2}-11za^{-1}-11za^{-3}+13a^{-2}+5a^{-4}+a^{-8}+8+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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 19                       1 -1 17                     2   2 15                   2 1   -1 13                 4 2     2 11               5 4       -1 9             3 2         1 7           4 5           1 5         2 3             -1 3       3 5               2 1     1 1                 0 -1     3                   3 -3 1 1                     0 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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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