L11a97

L11a96

L11a98

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Link Presentations

[edit Notes on L11a97's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X20,17,21,18 X18,13,19,14 X14,19,15,20 X4,21,1,22 X10,5,11,6 X12,3,13,4 X22,11,5,12 X2,9,3,10 X8,15,9,16
Gauss code	{1, -10, 8, -6}, {7, -1, 2, -11, 10, -7, 9, -8, 4, -5, 11, -2, 3, -4, 5, -3, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(2)^5+6 t(1) t(2)^4-6 t(2)^4-13 t(1) t(2)^3+13 t(2)^3+13 t(1) t(2)^2-13 t(2)^2-6 t(1) t(2)+6 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{16}{q^{11/2}}-\frac{23}{q^{13/2}}+\frac{25}{q^{15/2}}-\frac{25}{q^{17/2}}+\frac{22}{q^{19/2}}-\frac{15}{q^{21/2}}+\frac{9}{q^{23/2}}-\frac{4}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z a^{13}+3 z^3 a^{11}+2 z a^{11}-2 a^{11} z^{-1} -2 z^5 a^9+7 z a^9+5 a^9 z^{-1} -4 z^5 a^7-11 z^3 a^7-10 z a^7-3 a^7 z^{-1} -z^5 a^5-z^3 a^5$ (db)
Kauffman polynomial	$-z^6 a^{16}+2 z^4 a^{16}-z^2 a^{16}-4 z^7 a^{15}+9 z^5 a^{15}-6 z^3 a^{15}+z a^{15}-7 z^8 a^{14}+15 z^6 a^{14}-11 z^4 a^{14}+4 z^2 a^{14}-a^{14}-6 z^9 a^{13}+3 z^7 a^{13}+14 z^5 a^{13}-13 z^3 a^{13}+2 z a^{13}-2 z^{10} a^{12}-17 z^8 a^{12}+48 z^6 a^{12}-41 z^4 a^{12}+14 z^2 a^{12}-14 z^9 a^{11}+15 z^7 a^{11}+12 z^5 a^{11}-14 z^3 a^{11}+4 z a^{11}-2 a^{11} z^{-1} -2 z^{10} a^{10}-22 z^8 a^{10}+52 z^6 a^{10}-34 z^4 a^{10}+2 z^2 a^{10}+5 a^{10}-8 z^9 a^9-2 z^7 a^9+25 z^5 a^9-24 z^3 a^9+13 z a^9-5 a^9 z^{-1} -12 z^8 a^8+16 z^6 a^8-3 z^4 a^8-7 z^2 a^8+5 a^8-10 z^7 a^7+17 z^5 a^7-16 z^3 a^7+10 z a^7-3 a^7 z^{-1} -4 z^6 a^6+3 z^4 a^6-z^5 a^5+z^3 a^5$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     4 1 -3 -8                   7     7 -10                 9 4     -5 -12               14 7       7 -14             12 10         -2 -16           13 13           0 -18         9 12             3 -20       6 13               -7 -22     3 9                 6 -24   1 6                   -5 -26   3                     3 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-8$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-7$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-6$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=-5$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{14}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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