L11a491

L11a490

L11a492

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Link Presentations

[edit Notes on L11a491's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)-1)(t(3)-1)\left(t(3)^{4}+2t(1)t(3)^{3}-2t(3)^{3}-2t(1)t(3)^{2}+2t(3)^{2}+2t(1)t(3)-2t(3)-t(1)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{-11}+3q^{-10}-8q^{-9}+13q^{-8}-16q^{-7}+19q^{-6}-17q^{-5}+16q^{-4}-10q^{-3}+6q^{-2}-2q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-a^{10}z^{-2}-2a^{10}+3a^{8}z^{4}+8a^{8}z^{2}+4a^{8}z^{-2}+8a^{8}-2a^{6}z^{6}-7a^{6}z^{4}-10a^{6}z^{2}-5a^{6}z^{-2}-10a^{6}-a^{4}z^{6}-2a^{4}z^{4}+2a^{4}z^{-2}+2a^{4}+a^{2}z^{4}+3a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-4z^{4}a^{12}+z^{2}a^{12}+6z^{7}a^{11}-10z^{5}a^{11}+8z^{3}a^{11}-5za^{11}+a^{11}z^{-1}+7z^{8}a^{10}-11z^{6}a^{10}+10z^{4}a^{10}-11z^{2}a^{10}-a^{10}z^{-2}+6a^{10}+4z^{9}a^{9}+5z^{7}a^{9}-29z^{5}a^{9}+41z^{3}a^{9}-25za^{9}+5a^{9}z^{-1}+z^{10}a^{8}+13z^{8}a^{8}-39z^{6}a^{8}+55z^{4}a^{8}-45z^{2}a^{8}-4a^{8}z^{-2}+21a^{8}+7z^{9}a^{7}-6z^{7}a^{7}-18z^{5}a^{7}+43z^{3}a^{7}-35za^{7}+9a^{7}z^{-1}+z^{10}a^{6}+9z^{8}a^{6}-32z^{6}a^{6}+49z^{4}a^{6}-44z^{2}a^{6}-5a^{6}z^{-2}+22a^{6}+3z^{9}a^{5}-3z^{7}a^{5}-5z^{5}a^{5}+14z^{3}a^{5}-15za^{5}+5a^{5}z^{-1}+3z^{8}a^{4}-6z^{6}a^{4}+4z^{4}a^{4}-6z^{2}a^{4}-2a^{4}z^{-2}+6a^{4}+2z^{7}a^{3}-5z^{5}a^{3}+2z^{3}a^{3}+za^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     1   -1 -3                   5 1   4 -5                 7 3     -4 -7               9 3       6 -9             8 7         -1 -11           11 9           2 -13         7 10             3 -15       6 9               -3 -17     2 7                 5 -19   1 6                   -5 -21   2                     2 -23 1                       -1

Integral Khovanov Homology (db, data source)

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

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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