L11a120

L11a119

L11a121

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Link Presentations

[edit Notes on L11a120's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {7uv^{2}-10uv+4u+4v^{3}-10v^{2}+7v}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {9}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{25/2}}}-{\frac {2}{q^{23/2}}}+{\frac {5}{q^{21/2}}}-{\frac {8}{q^{19/2}}}+{\frac {11}{q^{17/2}}}-{\frac {13}{q^{15/2}}}+{\frac {12}{q^{13/2}}}-{\frac {13}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+3za^{11}+a^{11}z^{-1}-2z^{3}a^{9}+2za^{9}+2a^{9}z^{-1}-5z^{3}a^{7}-6za^{7}-2a^{7}z^{-1}-3z^{3}a^{5}-za^{5}-z^{3}a^{3}$ (db)
Kauffman polynomial	$-z^{8}a^{14}+6z^{6}a^{14}-13z^{4}a^{14}+12z^{2}a^{14}-4a^{14}-2z^{9}a^{13}+10z^{7}a^{13}-16z^{5}a^{13}+8z^{3}a^{13}+a^{13}z^{-1}-z^{10}a^{12}-2z^{8}a^{12}+27z^{6}a^{12}-54z^{4}a^{12}+38z^{2}a^{12}-9a^{12}-7z^{9}a^{11}+26z^{7}a^{11}-22z^{5}a^{11}-2z^{3}a^{11}+2za^{11}+a^{11}z^{-1}-z^{10}a^{10}-11z^{8}a^{10}+54z^{6}a^{10}-65z^{4}a^{10}+26z^{2}a^{10}-4a^{10}-5z^{9}a^{9}+4z^{7}a^{9}+28z^{5}a^{9}-35z^{3}a^{9}+11za^{9}-2a^{9}z^{-1}-10z^{8}a^{8}+24z^{6}a^{8}-10z^{4}a^{8}-z^{2}a^{8}+2a^{8}-12z^{7}a^{7}+28z^{5}a^{7}-20z^{3}a^{7}+8za^{7}-2a^{7}z^{-1}-9z^{6}a^{6}+11z^{4}a^{6}-z^{2}a^{6}-6z^{5}a^{5}+4z^{3}a^{5}-za^{5}-3z^{4}a^{4}-z^{3}a^{3}$ (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 -2                       1 1 -4                     3 1 -2 -6                   3     3 -8                 6 3     -3 -10               7 3       4 -12             6 7         1 -14           7 6           1 -16         4 6             2 -18       4 7               -3 -20     1 4                 3 -22   1 4                   -3 -24   1                     1 -26 1                       -1

Integral Khovanov Homology (db, data source)

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

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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