L11n337

L11n336

L11n338

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Link Presentations

[edit Notes on L11n337's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,18,12,19 X7,14,8,15 X13,8,14,9 X19,22,20,13 X15,20,16,21 X21,16,22,17 X17,12,18,5 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, -7, 8, -9, 3, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(2)-1)\left(t(1)t(3)^{3}+t(1)t(2)t(3)^{3}+2t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-2t(2)^{2}t(3)+2t(1)t(3)+t(2)^{2}+t(2)\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-13}+2q^{-12}-4q^{-11}+6q^{-10}-5q^{-9}+6q^{-8}-4q^{-7}+4q^{-6}-q^{-5}+q^{-3}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+4a^{12}z^{-2}+4a^{12}-5a^{10}z^{2}-5a^{10}z^{-2}-9a^{10}+a^{8}z^{4}+a^{8}z^{2}+2a^{8}z^{-2}+2a^{8}+a^{6}z^{6}+6a^{6}z^{4}+8a^{6}z^{2}+3a^{6}$ (db)
Kauffman polynomial	$a^{15}z^{7}-5a^{15}z^{5}+8a^{15}z^{3}-5a^{15}z+a^{15}z^{-1}+2a^{14}z^{8}-9a^{14}z^{6}+12a^{14}z^{4}-8a^{14}z^{2}-a^{14}z^{-2}+4a^{14}+a^{13}z^{9}+a^{13}z^{7}-21a^{13}z^{5}+36a^{13}z^{3}-23a^{13}z+5a^{13}z^{-1}+6a^{12}z^{8}-28a^{12}z^{6}+41a^{12}z^{4}-34a^{12}z^{2}-4a^{12}z^{-2}+18a^{12}+a^{11}z^{9}+3a^{11}z^{7}-32a^{11}z^{5}+57a^{11}z^{3}-39a^{11}z+9a^{11}z^{-1}+4a^{10}z^{8}-21a^{10}z^{6}+38a^{10}z^{4}-37a^{10}z^{2}-5a^{10}z^{-2}+21a^{10}+3a^{9}z^{7}-17a^{9}z^{5}+30a^{9}z^{3}-20a^{9}z+5a^{9}z^{-1}-a^{8}z^{6}+3a^{8}z^{4}-3a^{8}z^{2}-2a^{8}z^{-2}+5a^{8}-a^{7}z^{5}+a^{7}z^{3}+a^{7}z+a^{6}z^{6}-6a^{6}z^{4}+8a^{6}z^{2}-3a^{6}$ (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 -5                       1 1 -7                       1 1 -9                 2 1     -1 -11               3         3 -13             4 5 1       0 -15           3 1           2 -17         2 4 1           1 -19       4 3               1 -21     1 3                 2 -23   1 3                   -2 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

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

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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