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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,14,19,13 X14,8,15,7 X20,5,21,6 X4,19,5,20 X6,21,7,22 X16,9,17,10 X22,15,9,16 X8,18,1,17

Gauss Code:

{{1, -2, 3, -7, 6, -8, 5, -11}, {9, -1, 2, -3, 4, -5, 10, -9, 11, -4, 7, -6, 8, -10}}

Jones Polynomial:

- q-21/2 + 3q-19/2 - 6q-17/2 + 9q-15/2 - 11q-13/2 + 13q-11/2 - 13q-9/2 + 10q-7/2 - 9q-5/2 + 5q-3/2 - 3q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-30 - q-28 + 2q-26 - q-24 - q-20 - 3q-18 + 3q-16 - q-14 + 5q-12 + 2q-10 + 2q-8 + 2q-6 - q-4 + q-2 - 1

HOMFLY-PT Polynomial:

- a3z-1 + 2a3z + 7a3z3 + 5a3z5 + a3z7 + a5z-1 - 9a5z - 21a5z3 - 18a5z5 - 7a5z7 - a5z9 + 5a7z + 8a7z3 + 5a7z5 + a7z7

Kauffman Polynomial:

4a2z2 - 8a2z4 + 5a2z6 - a2z8 - a3z-1 - 3a3z + 18a3z3 - 28a3z5 + 16a3z7 - 3a3z9 + a4 + 11a4z2 - 25a4z4 + 8a4z6 + 5a4z8 - 2a4z10 - a5z-1 - 7a5z + 29a5z3 - 55a5z5 + 40a5z7 - 9a5z9 + 16a6z2 - 43a6z4 + 33a6z6 - 3a6z8 - 2a6z10 - 2a7z - a7z3 - 2a7z5 + 14a7z7 - 6a7z9 + 4a8z2 - 10a8z4 + 21a8z6 - 9a8z8 + a9z - 7a9z3 + 19a9z5 - 10a9z7 - 5a10z2 + 13a10z4 - 9a10z6 - a11z + 4a11z3 - 6a11z5 - 3a12z4 - a13z3

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 2                       1 j = 0                     2   j = -2                   3 1   j = -4                 6 2     j = -6               5 4       j = -8             8 5         j = -10           5 5           j = -12         6 8             j = -14       4 6               j = -16     2 5                 j = -18   1 4                   j = -20   2                     j = -22 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 325]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 325]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[18, 14, 19, 13], > X[14, 8, 15, 7], X[20, 5, 21, 6], X[4, 19, 5, 20], X[6, 21, 7, 22], > X[16, 9, 17, 10], X[22, 15, 9, 16], X[8, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -8, 5, -11}, > {9, -1, 2, -3, 4, -5, 10, -9, 11, -4, 7, -6, 8, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 6 9 11 13 13 10 9 5 -q + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 3 > ------- + Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 2 -24 -20 3 3 -14 5 2 2 2 -1 + q - q + --- - q - q - --- + --- - q + --- + --- + -- + -- - 26 18 16 12 10 8 6 q q q q q q q -4 -2 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 325]][a, z]
Out[8]=	3 5 a a 3 5 7 3 3 5 3 7 3 -(--) + -- + 2 a z - 9 a z + 5 a z + 7 a z - 21 a z + 8 a z + z z 3 5 5 5 7 5 3 7 5 7 7 7 5 9 > 5 a z - 18 a z + 5 a z + a z - 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 325]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 11 2 2 4 2 a - -- - -- - 3 a z - 7 a z - 2 a z + a z - a z + 4 a z + 11 a z + z z 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 16 a z + 4 a z - 5 a z + 18 a z + 29 a z - a z - 7 a z + 11 3 13 3 2 4 4 4 6 4 8 4 10 4 > 4 a z - a z - 8 a z - 25 a z - 43 a z - 10 a z + 13 a z - 12 4 3 5 5 5 7 5 9 5 11 5 2 6 > 3 a z - 28 a z - 55 a z - 2 a z + 19 a z - 6 a z + 5 a z + 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 8 a z + 33 a z + 21 a z - 9 a z + 16 a z + 40 a z + 14 a z - 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > 10 a z - a z + 5 a z - 3 a z - 9 a z - 3 a z - 9 a z - 7 9 4 10 6 10 > 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 6 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 22 8 20 7 18 7 18 6 16 6 16 5 14 5 q q q t q t q t q t q t q t q t 6 6 8 5 5 8 5 5 2 t > ------ + ------ + ------ + ------ + ------ + ----- + ---- + ---- + --- + 14 4 12 4 12 3 10 3 10 2 8 2 8 6 4 q t q t q t q t q t q t q t q t q 2 3 t 2 t 2 3 > --- + 2 t + -- + q t 2 2 q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a325

L11a324 L11a326