L11a497

L11a496

L11a498

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Link Presentations

[edit Notes on L11a497's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)\left(vw^{4}-2vw^{3}+2vw^{2}-2vw+2w^{3}-2w^{2}+2w-1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-3q^{-5}+3q^{4}+8q^{-4}-7q^{3}-11q^{-3}+11q^{2}+16q^{-2}-15q-17q^{-1}+19$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+3a^{4}z^{2}+2a^{4}z^{-2}+4a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-9a^{2}z^{4}-4z^{4}a^{-2}-16a^{2}z^{2}-6z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-14a^{2}-4a^{-2}+z^{8}+6z^{6}+15z^{4}+19z^{2}+4z^{-2}+14$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+8az^{9}+4z^{9}a^{-1}+5a^{4}z^{8}+13a^{2}z^{8}+6z^{8}a^{-2}+14z^{8}+3a^{5}z^{7}-3a^{3}z^{7}-10az^{7}+z^{7}a^{-1}+5z^{7}a^{-3}+a^{6}z^{6}-12a^{4}z^{6}-49a^{2}z^{6}-11z^{6}a^{-2}+3z^{6}a^{-4}-50z^{6}-7a^{5}z^{5}-14a^{3}z^{5}-19az^{5}-21z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+10a^{4}z^{4}+68a^{2}z^{4}+12z^{4}a^{-2}-5z^{4}a^{-4}+72z^{4}+3a^{5}z^{3}+22a^{3}z^{3}+50az^{3}+37z^{3}a^{-1}+4z^{3}a^{-3}-2z^{3}a^{-5}+3a^{6}z^{2}-13a^{4}z^{2}-50a^{2}z^{2}-11z^{2}a^{-2}+z^{2}a^{-4}-46z^{2}-17a^{3}z-35az-23za^{-1}-4za^{-3}+za^{-5}-a^{6}+8a^{4}+22a^{2}+5a^{-2}+19+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     2   2 7                   5 1   -4 5                 6 2     4 3               9 5       -4 1             10 6         4 -1           9 11           2 -3         7 8             -1 -5       4 9               5 -7     4 7                 -3 -9   1 6                   5 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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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