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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3/2 - 4q-1/2 + 9q1/2 - 16q3/2 + 21q5/2 - 26q7/2 + 25q9/2 - 22q11/2 + 17q13/2 - 10q15/2 + 4q17/2 - q19/2

A2 (sl(3)) Invariant:

- q-4 + 2q-2 - 2 + 4q4 - 4q6 + 6q8 + 2q12 + 4q14 - 4q16 + 4q18 - 4q20 + 3q24 - 2q26 + q28

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-11z3 - a-11z5 - a-10z2 + 4a-10z4 - 4a-10z6 + 2a-9z - 8a-9z3 + 13a-9z5 - 9a-9z7 + 5a-8z2 - 13a-8z4 + 19a-8z6 - 12a-8z8 - 6a-7z3 + 7a-7z5 + 6a-7z7 - 9a-7z9 + 8a-6z2 - 27a-6z4 + 38a-6z6 - 15a-6z8 - 3a-6z10 - a-5z-1 - a-5z + 5a-5z3 - 14a-5z5 + 28a-5z7 - 16a-5z9 + a-4 + 3a-4z2 - 19a-4z4 + 30a-4z6 - 10a-4z8 - 3a-4z10 - a-3z-1 + 2a-3z - 4a-3z3 + 2a-3z5 + 9a-3z7 - 7a-3z9 - 7a-2z4 + 14a-2z6 - 7a-2z8 + a-1z - 6a-1z3 + 9a-1z5 - 4a-1z7 - z2 + 2z4 - z6

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 20                       1 j = 18                     3   j = 16                   7 1   j = 14                 10 3     j = 12               12 7       j = 10             14 11         j = 8           12 11           j = 6         9 14             j = 4       7 12               j = 2     3 10                 j = 0   1 6                   j = -2   3                     j = -4 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, 314]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 314]]
Out[4]=	PD[X[10, 1, 11, 2], X[20, 11, 21, 12], X[8, 21, 1, 22], X[16, 10, 17, 9], > X[14, 8, 15, 7], X[12, 4, 13, 3], X[18, 6, 19, 5], X[6, 18, 7, 17], > X[4, 14, 5, 13], X[22, 16, 9, 15], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 6, -9, 7, -8, 5, -3}, > {4, -1, 2, -6, 9, -5, 10, -4, 8, -7, 11, -2, 3, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 4 3/2 5/2 7/2 9/2 q - ------- + 9 Sqrt[q] - 16 q + 21 q - 26 q + 25 q - Sqrt[q] 11/2 13/2 15/2 17/2 19/2 > 22 q + 17 q - 10 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 4 6 8 12 14 16 18 20 -2 - q + -- + 4 q - 4 q + 6 q + 2 q + 4 q - 4 q + 4 q - 4 q + 2 q 24 26 28 > 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 314]][a, z]
Out[8]=	3 3 3 3 5 5 1 1 2 z 2 z 3 z z 2 z 4 z 4 z 2 z z 3 z -(----) + ---- - --- + --- + --- - - - ---- + ---- + ---- - ---- - -- + ---- + 5 3 7 5 3 a 7 5 3 a 7 5 a z a z a a a a a a a a 5 5 7 7 3 z z z z > ---- - -- + -- + -- 3 a 5 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 314]][a, z]
Out[9]=	2 2 2 2 3 -4 1 1 2 z z 2 z z 2 z 5 z 8 z 3 z z a - ---- - ---- + --- - -- + --- + - - z - --- + ---- + ---- + ---- + --- - 5 3 9 5 3 a 10 8 6 4 11 a z a z a a a a a a a a 3 3 3 3 3 4 4 4 4 8 z 6 z 5 z 4 z 6 z 4 4 z 13 z 27 z 19 z > ---- - ---- + ---- - ---- - ---- + 2 z + ---- - ----- - ----- - ----- - 9 7 5 3 a 10 8 6 4 a a a a a a a a 4 5 5 5 5 5 5 6 6 7 z z 13 z 7 z 14 z 2 z 9 z 6 4 z 19 z > ---- - --- + ----- + ---- - ----- + ---- + ---- - z - ---- + ----- + 2 11 9 7 5 3 a 10 8 a a a a a a a a 6 6 6 7 7 7 7 7 8 8 38 z 30 z 14 z 9 z 6 z 28 z 9 z 4 z 12 z 15 z > ----- + ----- + ----- - ---- + ---- + ----- + ---- - ---- - ----- - ----- - 6 4 2 9 7 5 3 a 8 6 a a a a a a a a a 8 8 9 9 9 10 10 10 z 7 z 9 z 16 z 7 z 3 z 3 z > ----- - ---- - ---- - ----- - ---- - ----- - ----- 4 2 7 5 3 6 4 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 3 6 3 q 4 6 6 2 10 q + 7 q + ----- + t + ----- + - + ---- + 12 q t + 9 q t + 14 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 > 12 q t + 11 q t + 14 q t + 11 q t + 12 q t + 7 q t + 14 5 14 6 16 6 16 7 18 7 20 8 > 10 q t + 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a314

L11a313 L11a315