L11a499

L11a498

L11a500

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Link Presentations

[edit Notes on L11a499's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(2t(3)^{2}-3t(3)+2\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+12q^{5}-16q^{4}+19q^{3}-17q^{2}+16q-11+7q^{-1}-2q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-z^{2}a^{-2}+6z^{2}a^{-4}-2z^{2}a^{-6}-4z^{2}+2a^{2}+4a^{-4}-2a^{-6}-4+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-5z^{4}a^{-8}+6z^{7}a^{-7}-13z^{5}a^{-7}+11z^{3}a^{-7}-4za^{-7}+8z^{8}a^{-6}-21z^{6}a^{-6}+27z^{4}a^{-6}-14z^{2}a^{-6}+4a^{-6}+5z^{9}a^{-5}-4z^{7}a^{-5}-9z^{5}a^{-5}+21z^{3}a^{-5}-8za^{-5}+z^{10}a^{-4}+14z^{8}a^{-4}-46z^{6}a^{-4}+62z^{4}a^{-4}-36z^{2}a^{-4}+8a^{-4}+8z^{9}a^{-3}-14z^{7}a^{-3}+9z^{5}a^{-3}-3z^{3}a^{-3}+z^{10}a^{-2}+9z^{8}a^{-2}+a^{2}z^{6}-26z^{6}a^{-2}-4a^{2}z^{4}+27z^{4}a^{-2}+6a^{2}z^{2}-16z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-4a^{2}+3z^{9}a^{-1}+2az^{7}-2z^{7}a^{-1}-4az^{5}-11z^{3}a^{-1}+4az+8za^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-3z^{6}-7z^{4}+12z^{2}+2z^{-2}-7$ (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                 7 2     5 9               9 5       -4 7             10 7         3 5           7 9           2 3         9 10             -1 1       6 11               5 -1     1 5                 -4 -3   1 6                   5 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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