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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X8,18,9,17 X14,8,15,7 X18,10,19,9 X10,12,5,11 X22,20,17,19 X16,22,11,21 X20,16,21,15 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q1/2 + 2q3/2 - 7q5/2 + 9q7/2 - 16q9/2 + 16q11/2 - 19q13/2 + 16q15/2 - 13q17/2 + 8q19/2 - 4q21/2 + q23/2

A2 (sl(3)) Invariant:

q2 + 3q6 + 5q8 + 4q10 + 12q12 + 8q14 + 13q16 + 12q18 + 7q20 + 10q22 + q24 + 4q26 + q28 - q30 + 2q32 - q34

HOMFLY-PT Polynomial:

- a-9z-3 - a-9z-1 + 2a-9z3 + a-9z5 + 3a-7z-3 + 6a-7z-1 + 4a-7z - a-7z3 - 3a-7z5 - a-7z7 - 3a-5z-3 - 9a-5z-1 - 10a-5z - 7a-5z3 - 4a-5z5 - a-5z7 + a-3z-3 + 4a-3z-1 + 6a-3z + 4a-3z3 + a-3z5

Kauffman Polynomial:

- a-14z4 + 2a-13z3 - 4a-13z5 - a-12z2 + 7a-12z4 - 8a-12z6 + 2a-11z - 8a-11z3 + 15a-11z5 - 11a-11z7 - 2a-10z2 - 2a-10z4 + 13a-10z6 - 10a-10z8 + a-9z-3 - 5a-9z-1 + 16a-9z - 32a-9z3 + 32a-9z5 - 5a-9z7 - 5a-9z9 - 3a-8z-2 + 10a-8 - 10a-8z2 - 17a-8z4 + 37a-8z6 - 14a-8z8 - a-8z10 + 3a-7z-3 - 12a-7z-1 + 31a-7z - 48a-7z3 + 28a-7z5 + 7a-7z7 - 7a-7z9 - 6a-6z-2 + 19a-6 - 18a-6z2 - 9a-6z4 + 22a-6z6 - 6a-6z8 - a-6z10 + 3a-5z-3 - 12a-5z-1 + 27a-5z - 36a-5z3 + 20a-5z5 - 2a-5z9 - 3a-4z-2 + 10a-4 - 9a-4z2 - 2a-4z4 + 6a-4z6 - 2a-4z8 + a-3z-3 - 5a-3z-1 + 10a-3z - 10a-3z3 + 5a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 24                       1 j = 22                     3   j = 20                   5 1   j = 18                 8 3     j = 16               10 7       j = 14             9 6         j = 12           9 12           j = 10         7 7             j = 8       4 11               j = 6     3 5                 j = 4   1 6                   j = 2   1                     j = 0 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 544]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 544]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[8, 18, 9, 17], X[14, 8, 15, 7], > X[18, 10, 19, 9], X[10, 12, 5, 11], X[22, 20, 17, 19], X[16, 22, 11, 21], > X[20, 16, 21, 15], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -3, 5, -6}, {6, -2, 11, -4, 9, -8}, > {3, -5, 7, -9, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 -Sqrt[q] + 2 q - 7 q + 9 q - 16 q + 16 q - 19 q + 15/2 17/2 19/2 21/2 23/2 > 16 q - 13 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 6 8 10 12 14 16 18 20 22 q + 3 q + 5 q + 4 q + 12 q + 8 q + 13 q + 12 q + 7 q + 10 q + 24 26 28 30 32 34 > q + 4 q + q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 544]][a, z]
Out[8]=	1 3 3 1 1 6 9 4 4 z 10 z -(-----) + ----- - ----- + ----- - ---- + ---- - ---- + ---- + --- - ---- + 9 3 7 3 5 3 3 3 9 7 5 3 7 5 a z a z a z a z a z a z a z a z a a 3 3 3 3 5 5 5 5 7 7 6 z 2 z z 7 z 4 z z 3 z 4 z z z z > --- + ---- - -- - ---- + ---- + -- - ---- - ---- + -- - -- - -- 3 9 7 5 3 9 7 5 3 7 5 a a a a a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 544]][a, z]
Out[9]=	10 19 10 1 3 3 1 3 6 3 5 -- + -- + -- + ----- + ----- + ----- + ----- - ----- - ----- - ----- - ---- - 8 6 4 9 3 7 3 5 3 3 3 8 2 6 2 4 2 9 a a a a z a z a z a z a z a z a z a z 2 2 2 12 12 5 2 z 16 z 31 z 27 z 10 z z 2 z 10 z > ---- - ---- - ---- + --- + ---- + ---- + ---- + ---- - --- - ---- - ----- - 7 5 3 11 9 7 5 3 12 10 8 a z a z a z a a a a a a a a 2 2 3 3 3 3 3 3 4 4 18 z 9 z 2 z 8 z 32 z 48 z 36 z 10 z z 7 z > ----- - ---- + ---- - ---- - ----- - ----- - ----- - ----- - --- + ---- - 6 4 13 11 9 7 5 3 14 12 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 2 z 17 z 9 z 2 z 4 z 15 z 32 z 28 z 20 z 5 z > ---- - ----- - ---- - ---- - ---- + ----- + ----- + ----- + ----- + ---- - 10 8 6 4 13 11 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 8 8 z 13 z 37 z 22 z 6 z 11 z 5 z 7 z z 10 z > ---- + ----- + ----- + ----- + ---- - ----- - ---- + ---- - -- - ----- - 12 10 8 6 4 11 9 7 3 10 a a a a a a a a a a 8 8 8 9 9 9 10 10 14 z 6 z 2 z 5 z 7 z 2 z z z > ----- - ---- - ---- - ---- - ---- - ---- - --- - --- 8 6 4 9 7 5 8 6 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 -2 q q 6 8 8 2 10 2 6 q + 3 q + t + -- + -- + 5 q t + 4 q t + 11 q t + 7 q t + t t 10 3 12 3 12 4 14 4 14 5 16 5 > 7 q t + 9 q t + 12 q t + 9 q t + 6 q t + 10 q t + 16 6 18 6 18 7 20 7 20 8 22 8 24 9 > 7 q t + 8 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a544

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