L10a49

L10a48

L10a50

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Link Presentations

[edit Notes on L10a49's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X10,5,11,6 X14,3,15,4 X18,11,19,12 X20,13,5,14 X12,19,13,20 X2,9,3,10 X8,15,9,16
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -10, 9, -4, 6, -8, 7, -5, 10, -2, 3, -6, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}+4t(1)t(2)^{4}-4t(2)^{4}-6t(1)t(2)^{3}+6t(2)^{3}+6t(1)t(2)^{2}-6t(2)^{2}-4t(1)t(2)+4t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {4}{q^{23/2}}}-{\frac {7}{q^{21/2}}}+{\frac {11}{q^{19/2}}}-{\frac {13}{q^{17/2}}}+{\frac {14}{q^{15/2}}}-{\frac {14}{q^{13/2}}}+{\frac {9}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{11}-2a^{11}z^{-1}-z^{5}a^{9}+z^{3}a^{9}+8za^{9}+5a^{9}z^{-1}-3z^{5}a^{7}-10z^{3}a^{7}-10za^{7}-3a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}$ (db)
Kauffman polynomial	$-z^{5}a^{15}+z^{3}a^{15}-4z^{6}a^{14}+7z^{4}a^{14}-2z^{2}a^{14}-a^{14}-6z^{7}a^{13}+10z^{5}a^{13}-3z^{3}a^{13}+za^{13}-5z^{8}a^{12}+5z^{6}a^{12}+z^{4}a^{12}+z^{2}a^{12}-2z^{9}a^{11}-6z^{7}a^{11}+16z^{5}a^{11}-11z^{3}a^{11}+5za^{11}-2a^{11}z^{-1}-10z^{8}a^{10}+20z^{6}a^{10}-13z^{4}a^{10}-4z^{2}a^{10}+5a^{10}-2z^{9}a^{9}-6z^{7}a^{9}+22z^{5}a^{9}-27z^{3}a^{9}+14za^{9}-5a^{9}z^{-1}-5z^{8}a^{8}+8z^{6}a^{8}-2z^{4}a^{8}-7z^{2}a^{8}+5a^{8}-6z^{7}a^{7}+16z^{5}a^{7}-18z^{3}a^{7}+10za^{7}-3a^{7}z^{-1}-3z^{6}a^{6}+5z^{4}a^{6}-z^{5}a^{5}+2z^{3}a^{5}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   3 1 -2 -8                 4     4 -10               5 3     -2 -12             9 4       5 -14           6 6         0 -16         7 8           -1 -18       4 6             2 -20     3 7               -4 -22   1 4                 3 -24   3                   -3 -26 1                     1

Integral Khovanov Homology (db, data source)

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

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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