L11n109

L11n108

L11n110

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Link Presentations

[edit Notes on L11n109's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,18,8,19 X19,22,20,5 X13,20,14,21 X21,14,22,15 X9,16,10,17 X15,10,16,11 X17,8,18,9 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, -7, 8, 11, -2, -5, 6, -8, 7, -9, 3, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{3}-uv^{2}-uv+u+v^{3}-v^{2}-v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{21/2}}}-{\frac {1}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+2za^{11}+2a^{11}z^{-1}+za^{9}-a^{9}z^{-1}-z^{5}a^{7}-4z^{3}a^{7}-2za^{7}+a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-3za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{14}+7z^{6}a^{14}-15z^{4}a^{14}+11z^{2}a^{14}-3a^{14}-z^{9}a^{13}+7z^{7}a^{13}-15z^{5}a^{13}+12z^{3}a^{13}-4za^{13}+a^{13}z^{-1}-2z^{8}a^{12}+15z^{6}a^{12}-34z^{4}a^{12}+28z^{2}a^{12}-7a^{12}-z^{9}a^{11}+8z^{7}a^{11}-20z^{5}a^{11}+21z^{3}a^{11}-10za^{11}+2a^{11}z^{-1}-z^{8}a^{10}+8z^{6}a^{10}-18z^{4}a^{10}+14z^{2}a^{10}-4a^{10}+2z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-z^{6}a^{8}+4z^{4}a^{8}-3z^{2}a^{8}-z^{7}a^{7}+4z^{5}a^{7}-3z^{3}a^{7}-za^{7}+a^{7}z^{-1}-z^{6}a^{6}+3z^{4}a^{6}-a^{6}-z^{5}a^{5}+4z^{3}a^{5}-3za^{5}+a^{5}z^{-1}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     1 1 0 -8                   1     1 -10               1 1 1     1 -12             1 3 1       1 -14               1         1 -16         1 2 2           -1 -18       1                 1 -20       1 1               0 -22   1 1                   0 -24                         0 -26 1                       -1

Integral Khovanov Homology (db, data source)

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

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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