L11n315

L11n314

L11n316

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Link Presentations

[edit Notes on L11n315's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,17,8,16 X9,20,10,21 X11,18,12,19 X19,22,20,11 X15,9,16,8 X21,10,22,5 X17,14,18,15 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, -4, 8}, {-5, -2, 11, 9, -7, 3, -9, 5, -6, 4, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{2}-t(1)t(3)t(2)^{2}-t(3)t(2)^{2}+t(2)^{2}+t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)-t(1)t(3)t(2)+t(3)t(2)-t(2)-t(1)t(3)^{2}-t(3)^{2}+t(1)t(3)+t(3)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{-7}+2q^{-6}-2q^{-5}+2q^{-4}+q^{3}-q^{-3}-q^{2}+q^{-2}+2q+q^{-1}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}+a^{4}z^{4}+3a^{4}z^{2}+2a^{4}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+a^{2}+2a^{-2}-z^{4}-4z^{2}-2z^{-2}-4$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+3a^{4}z^{8}+a^{2}z^{8}+a^{7}z^{7}-4a^{5}z^{7}-6a^{3}z^{7}+z^{7}a^{-1}-11a^{6}z^{6}-19a^{4}z^{6}-7a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-5a^{7}z^{5}-a^{5}z^{5}+8a^{3}z^{5}-4z^{5}a^{-1}+16a^{6}z^{4}+33a^{4}z^{4}+11a^{2}z^{4}-5z^{4}a^{-2}-11z^{4}+6a^{7}z^{3}+9a^{5}z^{3}-3a^{3}z^{3}-5az^{3}+z^{3}a^{-1}-8a^{6}z^{2}-20a^{4}z^{2}-5a^{2}z^{2}+7z^{2}a^{-2}+14z^{2}-2a^{7}z-4a^{5}z+6az+4za^{-1}+2a^{6}+4a^{4}-2a^{2}-4a^{-2}-7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                         0 3                   2 1   1 1               2 1 1     2 -1             1 3 1       1 -3           2 2 2         2 -5         2 3 1           0 -7       1 1 1             1 -9     1 2 1               0 -11   1 1                   0 -13   1                     1 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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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