L11n151

L11n150

L11n152

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Link Presentations

[edit Notes on L11n151's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X3,10,4,11 X4,15,5,16 X11,16,12,17 X20,13,21,14 X14,21,15,22 X17,2,18,3
Gauss code	{1, 11, -6, -7, 5, -3}, {3, -1, 2, 6, -8, -5, 9, -10, 7, 8, -11, -2, 4, -9, 10, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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