L11n354

L11n353

L11n355

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Link Presentations

[edit Notes on L11n354's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)\left(t(2)t(3)^{3}-t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-t(1)t(2)t(3)-t(2)t(3)+t(3)-t(1)t(2)^{2}+t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{-10}+2q^{-9}-2q^{-8}+3q^{-7}-q^{-6}+2q^{-5}+q^{-2}-q^{-1}+1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+2a^{8}z^{2}+a^{8}z^{-2}+2a^{8}+a^{6}z^{2}-2a^{6}z^{-2}-2a^{6}-a^{4}z^{6}-5a^{4}z^{4}-5a^{4}z^{2}+a^{4}z^{-2}-a^{4}+a^{2}z^{4}+4a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-5a^{11}z^{5}+6a^{11}z^{3}-2a^{11}z+2a^{10}z^{8}-11a^{10}z^{6}+17a^{10}z^{4}-11a^{10}z^{2}+3a^{10}+a^{9}z^{9}-3a^{9}z^{7}-6a^{9}z^{5}+16a^{9}z^{3}-7a^{9}z+4a^{8}z^{8}-25a^{8}z^{6}+46a^{8}z^{4}-35a^{8}z^{2}-a^{8}z^{-2}+12a^{8}+a^{7}z^{9}-4a^{7}z^{7}-4a^{7}z^{5}+19a^{7}z^{3}-14a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}-14a^{6}z^{6}+28a^{6}z^{4}-26a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+a^{5}z^{7}-8a^{5}z^{5}+14a^{5}z^{3}-10a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}-6a^{4}z^{4}+4a^{4}z^{2}-a^{4}z^{-2}+2a^{4}+a^{3}z^{7}-5a^{3}z^{5}+5a^{3}z^{3}-a^{3}z+a^{2}z^{6}-5a^{2}z^{4}+6a^{2}z^{2}-2a^{2}$ (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                         0 -3                 1 2 1   0 -5               1 1 1     1 -7             2 3 1       0 -9           2 2 2         2 -11         2 4 1           1 -13       2 1 1             2 -15     1 3 1               1 -17   1 1                   0 -19   1                     1 -21 1                       -1

Integral Khovanov Homology (db, data source)

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

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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