L11a503

L11a502

L11a504

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Link Presentations

[edit Notes on L11a503's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,12,19,11 X12,18,7,17 X20,16,21,15 X22,20,13,19 X16,22,17,21 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -6}, {11, -4, 7, -9, 6, -5, 8, -7, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(1)^2 t(2)^2 t(3)^3-t(1) t(3)^3-t(1)^2 t(2) t(3)^3+t(1) t(2) t(3)^3+t(3)^3-3 t(1)^2 t(2)^2 t(3)^2+2 t(1) t(2)^2 t(3)^2+2 t(1) t(3)^2+2 t(1)^2 t(2) t(3)^2-4 t(1) t(2) t(3)^2+2 t(2) t(3)^2-3 t(3)^2+3 t(1)^2 t(2)^2 t(3)-2 t(1) t(2)^2 t(3)-2 t(1) t(3)-2 t(1)^2 t(2) t(3)+4 t(1) t(2) t(3)-2 t(2) t(3)+3 t(3)-t(1)^2 t(2)^2+t(1) t(2)^2-t(1) t(2)+t(2)-1}{t(1) t(2) t(3)^{3/2}}$ (db)
Jones polynomial	$-q^4+2 q^3-5 q^2+9 q-11+15 q^{-1} -14 q^{-2} +14 q^{-3} -10 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^4 z^6+4 a^4 z^4+6 a^4 z^2+2 a^4 z^{-2} +5 a^4-a^2 z^8-6 a^2 z^6-15 a^2 z^4-z^4 a^{-2} -21 a^2 z^2-4 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -16 a^2-4 a^{-2} +2 z^6+10 z^4+18 z^2+4 z^{-2} +15$ (db)
Kauffman polynomial	$a^2 z^{10}+z^{10}+4 a^3 z^9+6 a z^9+2 z^9 a^{-1} +7 a^4 z^8+9 a^2 z^8+2 z^8 a^{-2} +4 z^8+7 a^5 z^7-a^3 z^7-12 a z^7-3 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-12 a^4 z^6-35 a^2 z^6-8 z^6 a^{-2} -25 z^6+3 a^7 z^5-8 a^5 z^5-15 a^3 z^5-11 a z^5-12 z^5 a^{-1} -5 z^5 a^{-3} +a^8 z^4-8 a^6 z^4+11 a^4 z^4+47 a^2 z^4+10 z^4 a^{-2} +37 z^4-2 a^7 z^3+2 a^5 z^3+23 a^3 z^3+39 a z^3+28 z^3 a^{-1} +8 z^3 a^{-3} -a^8 z^2+6 a^6 z^2-9 a^4 z^2-39 a^2 z^2-7 z^2 a^{-2} -30 z^2+a^5 z-16 a^3 z-33 a z-21 z a^{-1} -5 z a^{-3} -2 a^6+6 a^4+20 a^2+4 a^{-2} +17+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                       1 -1 7                     1   1 5                   4 1   -3 3                 5 1     4 1               6 4       -2 -1             9 5         4 -3           7 8           1 -5         7 7             0 -7       4 8               4 -9     3 6                 -3 -11     4                   4 -13 1 3                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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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