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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q1/2 + q3/2 - 4q5/2 + 5q7/2 - 8q9/2 + 9q11/2 - 10q13/2 + 10q15/2 - 7q17/2 + 5q19/2 - 3q21/2 + q23/2

A2 (sl(3)) Invariant:

q2 + q4 + 2q6 + 4q8 + 2q10 + 4q12 + q16 - 3q20 - 3q24 + q32 - q34

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-14z2 - a-14z4 + 4a-13z3 - 3a-13z5 - a-12z2 + 5a-12z4 - 4a-12z6 - 2a-11z3 + 5a-11z5 - 4a-11z7 - a-10 + 3a-10z2 - 10a-10z4 + 9a-10z6 - 4a-10z8 - 2a-9z + 9a-9z3 - 14a-9z5 + 9a-9z7 - 3a-9z9 + a-8z2 - 5a-8z4 + 5a-8z6 - a-8z10 - 2a-7z-1 + 3a-7z + 7a-7z3 - 18a-7z5 + 15a-7z7 - 4a-7z9 + 5a-6 - 13a-6z2 + 13a-6z4 - 5a-6z6 + 3a-6z8 - a-6z10 - 5a-5z-1 + 15a-5z - 20a-5z3 + 10a-5z5 + a-5z7 - a-5z9 + 5a-4 - 9a-4z2 + 2a-4z4 + 3a-4z6 - a-4z8 - 3a-3z-1 + 10a-3z - 12a-3z3 + 6a-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                     2   j = 20                   3 1   j = 18                 4 2     j = 16               6 3       j = 14             4 4         j = 12           5 6           j = 10         3 4             j = 8       2 5               j = 6     2 3                 j = 4   1 4                   j = 2                         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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 125]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 125]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[18, 8, 19, 7], X[20, 10, 21, 9], > X[22, 12, 5, 11], X[8, 20, 9, 19], X[10, 22, 11, 21], X[16, 14, 17, 13], > X[12, 18, 13, 17], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 4, -7, 5, -9, 8, -2, 11, -8, 9, -3, > 6, -4, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -Sqrt[q] + q - 4 q + 5 q - 8 q + 9 q - 10 q + 10 q - 17/2 19/2 21/2 23/2 > 7 q + 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 16 20 24 32 34 q + q + 2 q + 4 q + 2 q + 4 q + q - 3 q - 3 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 125]][a, z]
Out[8]=	3 3 3 3 5 2 5 3 z 2 z 10 z 7 z 3 z 3 z 9 z 5 z z ---- - ---- + ---- + -- + --- - ---- + --- + ---- - ---- - ---- + ---- + -- - 7 5 3 9 7 5 3 9 7 5 3 9 a z a z a z a a a a a a a a a 5 5 5 7 7 4 z 5 z z z z > ---- - ---- + -- - -- - -- 7 5 3 7 5 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 125]][a, z]
Out[9]=	2 2 -10 5 5 2 5 3 2 z 3 z 15 z 10 z z z -a + -- + -- - ---- - ---- - ---- - --- + --- + ---- + ---- + --- - --- + 6 4 7 5 3 9 7 5 3 14 12 a a a z a z a z a a a a a a 2 2 2 2 3 3 3 3 3 3 3 z z 13 z 9 z 4 z 2 z 9 z 7 z 20 z 12 z > ---- + -- - ----- - ---- + ---- - ---- + ---- + ---- - ----- - ----- - 10 8 6 4 13 11 9 7 5 3 a a a a a a a a a a 4 4 4 4 4 4 5 5 5 5 z 5 z 10 z 5 z 13 z 2 z 3 z 5 z 14 z 18 z > --- + ---- - ----- - ---- + ----- + ---- - ---- + ---- - ----- - ----- + 14 12 10 8 6 4 13 11 9 7 a a a a a a a a a a 5 5 6 6 6 6 6 7 7 7 10 z 6 z 4 z 9 z 5 z 5 z 3 z 4 z 9 z 15 z > ----- + ---- - ---- + ---- + ---- - ---- + ---- - ---- + ---- + ----- + 5 3 12 10 8 6 4 11 9 7 a a a a a a a a a a 7 7 8 8 8 9 9 9 10 10 z z 4 z 3 z z 3 z 4 z z z z > -- - -- - ---- + ---- - -- - ---- - ---- - -- - --- - --- 5 3 10 6 4 9 7 5 8 6 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 6 -2 q 6 8 8 2 10 2 10 3 4 q + 2 q + t + -- + 3 q t + 2 q t + 5 q t + 3 q t + 4 q t + t 12 3 12 4 14 4 14 5 16 5 16 6 > 5 q t + 6 q t + 4 q t + 4 q t + 6 q t + 3 q t + 18 6 18 7 20 7 20 8 22 8 24 9 > 4 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a125

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