L10a117

L10a116

L10a118

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Link Presentations

[edit Notes on L10a117's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X4,11,5,12 X20,9,11,10 X16,5,17,6 X18,7,19,8 X6,17,7,18 X8,19,9,20 X10,15,1,16
Gauss code	{1, -2, 3, -4, 6, -8, 7, -9, 5, -10}, {4, -1, 2, -3, 10, -6, 8, -7, 9, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{4}\left(-v^{3}\right)-u^{3}v^{4}+2u^{3}v^{3}-2u^{3}v^{2}-2u^{2}v^{3}+3u^{2}v^{2}-2u^{2}v-2uv^{2}+2uv-u-v}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {6}{q^{17/2}}}-{\frac {6}{q^{19/2}}}+{\frac {5}{q^{21/2}}}-{\frac {4}{q^{23/2}}}+{\frac {2}{q^{25/2}}}-{\frac {1}{q^{27/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$z^{5}a^{11}+4z^{3}a^{11}+4za^{11}+a^{11}z^{-1}-z^{7}a^{9}-5z^{5}a^{9}-7z^{3}a^{9}-3za^{9}-a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-11z^{3}a^{7}-6za^{7}$ (db)
Kauffman polynomial	$-z^{3}a^{17}+za^{17}-2z^{4}a^{16}+z^{2}a^{16}-3z^{5}a^{15}+3z^{3}a^{15}-2za^{15}-3z^{6}a^{14}+3z^{4}a^{14}-z^{2}a^{14}-3z^{7}a^{13}+6z^{5}a^{13}-5z^{3}a^{13}+2za^{13}-2z^{8}a^{12}+4z^{6}a^{12}-2z^{4}a^{12}+2z^{2}a^{12}-z^{9}a^{11}+2z^{7}a^{11}-2z^{5}a^{11}+7z^{3}a^{11}-5za^{11}+a^{11}z^{-1}-3z^{8}a^{10}+11z^{6}a^{10}-10z^{4}a^{10}+3z^{2}a^{10}-a^{10}-z^{9}a^{9}+4z^{7}a^{9}-5z^{5}a^{9}+5z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-z^{8}a^{8}+4z^{6}a^{8}-3z^{4}a^{8}-z^{2}a^{8}-z^{7}a^{7}+6z^{5}a^{7}-11z^{3}a^{7}+6za^{7}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                     1 1 -8                   1 1 0 -10                 2     2 -12               2 1     -1 -14             3 2       1 -16           3 2         -1 -18         3 3           0 -20       2 3             1 -22     2 3               -1 -24     2                 2 -26 1 2                   -1 -28 1                     1

Integral Khovanov Homology (db, data source)

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

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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