L11a465

L11a464

L11a466

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Link Presentations

[edit Notes on L11a465's Link Presentations]

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

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	$-q^{8}+4q^{7}-9q^{6}+13q^{5}-16q^{4}+19q^{3}+q^{-3}-17q^{2}-2q^{-2}+15q+6q^{-1}-9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}-a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}+z^{6}a^{-2}+2z^{4}a^{-2}+a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+2a^{2}+3a^{-2}-2z^{4}-5z^{2}-2z^{-2}-5$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+7z^{9}a^{-3}+5z^{9}a^{-5}+5z^{8}a^{-2}+12z^{8}a^{-4}+9z^{8}a^{-6}+2z^{8}+2az^{7}+z^{7}a^{-1}-11z^{7}a^{-3}-2z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-9z^{6}a^{-2}-31z^{6}a^{-4}-18z^{6}a^{-6}+4z^{6}a^{-8}+z^{6}-5az^{5}-5z^{5}a^{-1}+8z^{5}a^{-3}-7z^{5}a^{-5}-14z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+31z^{4}a^{-4}+15z^{4}a^{-6}-5z^{4}a^{-8}-15z^{4}+2az^{3}-4z^{3}a^{-1}-5z^{3}a^{-3}+8z^{3}a^{-5}+6z^{3}a^{-7}-z^{3}a^{-9}+6a^{2}z^{2}+7z^{2}a^{-2}-13z^{2}a^{-4}-7z^{2}a^{-6}+19z^{2}+3az+7za^{-1}+3za^{-3}-3za^{-5}-2za^{-7}-4a^{2}-6a^{-2}+2a^{-4}+2a^{-6}-9-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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                     3   3 13                   6 1   -5 11                 7 3     4 9               9 6       -3 7             10 7         3 5           7 9           2 3         8 10             -2 1       5 11               6 -1     1 4                 -3 -3   1 5                   4 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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