L11a125

L11a124

L11a126

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Link Presentations

[edit Notes on L11a125's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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