L11n339

L11n338

L11n340

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Link Presentations

[edit Notes on L11n339's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X7,14,8,15 X13,8,14,9 X19,22,20,13 X15,20,16,21 X21,16,22,17 X12,18,5,17 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 $u$, $v$, $w$, ...)	$-{\frac {(t(2)-1)(t(3)-1)\left(t(3)t(2)^{2}+t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-t(2)-t(1)t(3)\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$1-2q^{-1}+4q^{-2}-3q^{-3}+4q^{-4}-2q^{-5}+3q^{-6}-q^{-8}+q^{-9}-q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+2z^{2}a^{8}+4a^{8}z^{-2}+5a^{8}-2z^{2}a^{6}-5a^{6}z^{-2}-7a^{6}-z^{6}a^{4}-4z^{4}a^{4}-3z^{2}a^{4}+2a^{4}z^{-2}+a^{4}+z^{4}a^{2}+3z^{2}a^{2}+2a^{2}$ (db)
Kauffman polynomial	$z^{7}a^{11}-6z^{5}a^{11}+10z^{3}a^{11}-6za^{11}+a^{11}z^{-1}+z^{8}a^{10}-6z^{6}a^{10}+9z^{4}a^{10}-6z^{2}a^{10}-a^{10}z^{-2}+4a^{10}+2z^{7}a^{9}-16z^{5}a^{9}+33z^{3}a^{9}-22za^{9}+5a^{9}z^{-1}+z^{8}a^{8}-9z^{6}a^{8}+22z^{4}a^{8}-22z^{2}a^{8}-4a^{8}z^{-2}+15a^{8}+2z^{7}a^{7}-15z^{5}a^{7}+34z^{3}a^{7}-30za^{7}+9a^{7}z^{-1}+z^{8}a^{6}-5z^{6}a^{6}+11z^{4}a^{6}-19z^{2}a^{6}-5a^{6}z^{-2}+16a^{6}+3z^{7}a^{5}-12z^{5}a^{5}+15z^{3}a^{5}-13za^{5}+5a^{5}z^{-1}+z^{8}a^{4}-z^{6}a^{4}-6z^{4}a^{4}+2z^{2}a^{4}-2a^{4}z^{-2}+4a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+4z^{3}a^{3}+za^{3}+z^{6}a^{2}-4z^{4}a^{2}+5z^{2}a^{2}-2a^{2}$ (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                     1   -1 -3                   3 1   2 -5                 2 3     1 -7             1 3 1       1 -9           1 1 2         2 -11         1 5 3           1 -13       1   2             3 -15       1 2               -1 -17   1 1                   0 -19                         0 -21 1                       -1

Integral Khovanov Homology (db, data source)

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

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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