L11n271

L11n270

L11n272

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Link Presentations

[edit Notes on L11n271's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X11,19,12,18 X17,9,18,22 X21,13,22,12 X13,21,14,20 X19,15,20,14 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 7, -8, 9, -4, 3, -6, 5, -9, 8, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-4 u v w^2+5 u v w-2 u v+2 u w^2-2 u w+2 v w^2-2 v w+2 w^3-5 w^2+4 w}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$-q^{10}+2 q^9-5 q^8+7 q^7-9 q^6+11 q^5-8 q^4+9 q^3-5 q^2+3 q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- a^{-10} z^{-2} - a^{-10} +3 z^2 a^{-8} +3 a^{-8} z^{-2} +5 a^{-8} -2 z^4 a^{-6} -4 z^2 a^{-6} -2 a^{-6} z^{-2} -5 a^{-6} -2 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} z^{-2} -2 a^{-4} +3 z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2}$ (db)
Kauffman polynomial	$z^7 a^{-11} -5 z^5 a^{-11} +9 z^3 a^{-11} -7 z a^{-11} +2 a^{-11} z^{-1} +2 z^8 a^{-10} -8 z^6 a^{-10} +9 z^4 a^{-10} -2 z^2 a^{-10} - a^{-10} z^{-2} + a^{-10} +z^9 a^{-9} +3 z^7 a^{-9} -26 z^5 a^{-9} +43 z^3 a^{-9} -27 z a^{-9} +8 a^{-9} z^{-1} +7 z^8 a^{-8} -23 z^6 a^{-8} +20 z^4 a^{-8} -8 z^2 a^{-8} -3 a^{-8} z^{-2} +5 a^{-8} +z^9 a^{-7} +10 z^7 a^{-7} -43 z^5 a^{-7} +52 z^3 a^{-7} -34 z a^{-7} +10 a^{-7} z^{-1} +5 z^8 a^{-6} -8 z^6 a^{-6} -z^4 a^{-6} -3 z^2 a^{-6} -2 a^{-6} z^{-2} +4 a^{-6} +8 z^7 a^{-5} -19 z^5 a^{-5} +18 z^3 a^{-5} -10 z a^{-5} +2 a^{-5} z^{-1} +7 z^6 a^{-4} -12 z^4 a^{-4} +9 z^2 a^{-4} + a^{-4} z^{-2} -3 a^{-4} +3 z^5 a^{-3} +4 z a^{-3} -2 a^{-3} z^{-1} +6 z^2 a^{-2} + a^{-2} z^{-2} -4 a^{-2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 21                   1 -1 19                 1   1 17               4 1   -3 15             3 1     2 13           6 4       -2 11         5 3         2 9       4 7           3 7     5 4             1 5     4               4 3 3 5                 -2 1 3                   3

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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