L11a415

L11a414

L11a416

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Link Presentations

[edit Notes on L11a415's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(3)^{4}+t(1)t(2)t(3)^{4}-2t(2)t(3)^{4}+t(3)^{4}+t(1)t(2)^{2}t(3)^{3}-2t(2)^{2}t(3)^{3}+2t(1)t(3)^{3}-3t(1)t(2)t(3)^{3}+4t(2)t(3)^{3}-t(3)^{3}-t(1)t(2)^{2}t(3)^{2}+2t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}+4t(1)t(2)t(3)^{2}-4t(2)t(3)^{2}+t(3)^{2}+t(1)t(2)^{2}t(3)-2t(2)^{2}t(3)+2t(1)t(3)-4t(1)t(2)t(3)+3t(2)t(3)-t(3)-t(1)t(2)^{2}+t(2)^{2}+2t(1)t(2)-t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+11q^{5}-14q^{4}+17q^{3}+q^{-3}-15q^{2}-2q^{-2}+14q+6q^{-1}-9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}z^{-2}-2a^{-6}+z^{6}a^{-4}+3z^{4}a^{-4}+6z^{2}a^{-4}+4a^{-4}z^{-2}+8a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}-4z^{2}a^{-2}-5a^{-2}z^{-2}+2a^{2}-8a^{-2}-2z^{4}-4z^{2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+5z^{8}a^{-2}+9z^{8}a^{-4}+7z^{8}a^{-6}+3z^{8}+2az^{7}+2z^{7}a^{-1}-8z^{7}a^{-3}-2z^{7}a^{-5}+6z^{7}a^{-7}+a^{2}z^{6}-17z^{6}a^{-2}-30z^{6}a^{-4}-18z^{6}a^{-6}+3z^{6}a^{-8}-7z^{6}-5az^{5}-16z^{5}a^{-1}-9z^{5}a^{-3}-13z^{5}a^{-5}-14z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+27z^{4}a^{-2}+44z^{4}a^{-4}+22z^{4}a^{-6}-5z^{4}a^{-8}+6z^{4}+2az^{3}+21z^{3}a^{-1}+35z^{3}a^{-3}+30z^{3}a^{-5}+12z^{3}a^{-7}-2z^{3}a^{-9}+5a^{2}z^{2}-33z^{2}a^{-2}-33z^{2}a^{-4}-12z^{2}a^{-6}-7z^{2}+az-16za^{-1}-33za^{-3}-21za^{-5}-5za^{-7}-2a^{2}+20a^{-2}+17a^{-4}+4a^{-6}+6+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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 17                       1 -1 15                     2   2 13                   5 1   -4 11                 6 2     4 9               8 5       -3 7             9 6         3 5           8 10           2 3         6 7             -1 1       4 9               5 -1     2 5                 -3 -3     4                   4 -5 1 2                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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