L10a63

L10a62

L10a64

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Link Presentations

[edit Notes on L10a63's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X4,20,5,19 X14,5,15,6 X16,11,17,12 X18,13,19,14 X12,17,13,18 X6,15,1,16
Gauss code	{1, -4, 2, -5, 6, -10}, {4, -1, 3, -2, 7, -9, 8, -6, 10, -7, 9, -8, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{4}+5uv^{3}-5uv^{2}+5uv-2u+v^{4}-2v^{3}+2v^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	${\frac {11}{q^{9/2}}}-{\frac {12}{q^{7/2}}}+{\frac {10}{q^{5/2}}}+q^{3/2}-{\frac {9}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {9}{q^{11/2}}}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}-a^{7}z^{-1}+2z^{5}a^{5}+7z^{3}a^{5}+8za^{5}+3a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-10z^{3}a^{3}-9za^{3}-2a^{3}z^{-1}+z^{5}a+3z^{3}a+2za$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+3z^{3}a^{9}-5z^{6}a^{8}+6z^{4}a^{8}-3z^{2}a^{8}+a^{8}-6z^{7}a^{7}+10z^{5}a^{7}-10z^{3}a^{7}+5za^{7}-a^{7}z^{-1}-4z^{8}a^{6}+2z^{6}a^{6}+6z^{4}a^{6}-9z^{2}a^{6}+3a^{6}-z^{9}a^{5}-10z^{7}a^{5}+32z^{5}a^{5}-35z^{3}a^{5}+16za^{5}-3a^{5}z^{-1}-7z^{8}a^{4}+14z^{6}a^{4}-3z^{4}a^{4}-6z^{2}a^{4}+3a^{4}-z^{9}a^{3}-7z^{7}a^{3}+29z^{5}a^{3}-32z^{3}a^{3}+15za^{3}-2a^{3}z^{-1}-3z^{8}a^{2}+6z^{6}a^{2}+z^{4}a^{2}-3z^{2}a^{2}-3z^{7}a+10z^{5}a-10z^{3}a+4za-z^{6}+3z^{4}-2z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                     1 -1 2                   2   2 0                 3 1   -2 -2               6 2     4 -4             5 4       -1 -6           7 5         2 -8         5 6           1 -10       4 6             -2 -12     2 5               3 -14   1 4                 -3 -16   2                   2 -18 1                     -1

Integral Khovanov Homology (db, data source)

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

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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